The Geometry and Topology of Wide Ribbons

Transcription

1 University of Iowa Iowa Research Online Theses and Dissertations Summer 2013 The Geometry and Topology of Wide Ribbons Susan Cecile Brooks University of Iowa Copyright 2013 Susan Cecile Brooks This dissertation is available at Iowa Research Online: Recommended Citation Brooks, Susan Cecile. "The Geometry and Topology of Wide Ribbons." PhD (Doctor of Philosophy) thesis, University of Iowa, Follow this and additional works at: Part of the Mathematics Commons

2 THE GEOMETRY AND TOPOLOGY OF WIDE RIBBONS by Susan Cecile Brooks A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree in Mathematics in the Graduate College of The University of Iowa August 2013 Thesis Supervisors: Associate Professor Oguz Durumeric Professor Emeritus Jonathan Simon

3 Graduate College The University of Iowa Iowa City, Iowa CERTIFICATE OF APPROVAL PH.D. THESIS This is to certify that the Ph.D. thesis of Susan Cecile Brooks has been approved by the Examining Committee for the thesis requirement for the Doctor of Philosophy degree in Mathematics at the August 2013 graduation. Thesis Committee: Oguz Durumeric, Thesis Supervisor Jonathan Simon, Thesis Supervisor Isabel Darcy Walter Seaman Maggy Tomova

4 ACKNOWLEDGEMENTS Thank you to my husband, CJ, for believing in me when I doubted myself. I could not have gotten through the past six years without your unending support. Thank you to my son, Charlie, for giving me the final push to finish. Thank you to my mother, Missy, for always reminding me of the end goal and uplifting me when I was down. Thank you to my father, Mike, for encouraging me throughout my first five years, and watching over me from above in my final year. Thank you to my siblings, friends, and in-laws for your support and comic relief when I needed a laugh. Thank you to my advisors, Oguz Durumeric and Jonathan Simon, who have provided me with both mathematical and everyday guidance throughout this journey. ii

5 ABSTRACT Intuitively, a ribbon is a topological and geometric surface that has a fixed width. In the 1960s and 1970s, Călugăreanu, White, and Fuller each independently proved a relationship between the geometry and topology of thin ribbons. This result has been applied in mathematical biology when analyzing properties of DNA strands. Although ribbons of small width have been studied extensively, it appears as though little to no research has be completed regarding ribbons of large width. In general, suppose K is a smoothly embedded knot in R 3. Given an arclength parametrization of K, denoted by γ(s), and given a smooth, smoothly-closed, unit vector field u(s) with the property that u (s) 0 for any s in the domain, we may define a ribbon of generalized width r 0 associated to γ and u as the set of all points γ(s) + ru(s) for all s in the domain and for all r [0, r 0 ]. These wide ribbons are likely to have self-intersections. In this thesis, we analyze how the knot type of the outer ribbon edge relates to that of the original knot K and the embedded resolutions of the unit vector field u as the width increases indefinitely. If the outer ribbon edge is embedded for large widths, we prove that the knot type of the outer ribbon edge is one of only finitely many possibilities. Furthermore, the possible set of finitely many knot types is completely determinable from u, independent of γ. However, the particular knot type in general depends on γ. The occurrence of stabilized knot types for large widths is generic; we show that the set of pairs (γ, u) for which the outer ribbon edge stabilizes for large widths (as a subset of all such pairs (γ, u)) is open and dense in the C 1 topology. Finally, we provide an algorithm for constructing a iii

6 ribbon of constant generalized width between any two given knot types K 1 and K 2. We conclude by providing concrete examples. iv

7 TABLE OF CONTENTS LIST OF TABLES vii LIST OF FIGURES viii CHAPTER 1 INTRODUCTION Motivation Basic Definitions Overview SELF-INTERSECTIONS OF THE OUTER RIBBON EDGE Requirements for Immersed Outer Ribbon Edges Self-Intersections for Large Widths and the Goal Post Property THE KNOT TYPE OF THE OUTER RIBBON EDGE Rescaling the Ribbon Understanding the Density of the Class of Stabilizing Ribbons Bounding the Knot Types of the Outer Ribbon Edge Relationship between the Unit Vector Field and the Normalized Outer Ribbon Edge Relationship between the Unit Vector Field and the Knot Types of the Outer Ribbon Edges CONSTRUCTING RIBBONS BETWEEN ANY TWO KNOTS Preliminaries Construction of the Ribbon THE SPECIAL CASE OF RIBBONS WITH N AS THE UNIT VEC- TOR FIELD: FRENET TYPE RIBBONS Locally Embedded Frenet Type Ribbons An Example of a Frenet Type Ribbon Having the Goal Post Property v

8 5.3 Perturbing Frenet Ribbons to Avoid the Goal Post Property EXAMPLES The Trefoil and Figure Eight Knot Example Example The Trefoil and Granny Knot Forms of Frenet Ribbons (1,2)-Torus Knot (1,3)-Torus Knot REFERENCES vi

9 LIST OF TABLES 6.1 Knot type of outer ribbon edge for particular radius ranges Knot type of outer ribbon edge for particular radius ranges Knot type of outer ribbon edge for particular radius ranges Knot type of outer ribbon edge for particular radius ranges Knot type of outer ribbon edge for particular radius ranges vii

10 LIST OF FIGURES 1.1 A thin ribbon A thin, closed ribbon Example of an isosceles triangle formed near a goal post Example of a suitable ε 3 -neighborhood Example of a possible ε 3 -neighborhood and suitable t value Example of a suitable bump function ϕ σ t0 may be isotoped to f J0 (σ t0 ) along the gray lines via H(s, τ) f J0 (σ t0 ) may be isotoped to f J0 (u) along the gray lines Example of an acceptable f function Outer ribbon edge vectors at the values λ k and λ k for ω(p k ) = Appropriate γ curve near S 2 representing the knot type of a trefoil A schematic of γ 0 and γ representing the knot type of a trefoil before and after isotopy Curve defined by γ for s [0, 4π] Limaçon ribbon for r = An ε(λ)-neighborhood about N 0 (λ) = N 0 ( λ) Example of a suitable bump function ϕ γ(s) for s [0, 2π] Figure eight defined by (x, y, z) for s [0, 2π] u(s) for s [0, 2π] Portions of the surface determined by d = u(s) (γ(s) γ(t)) near the approximate self-intersections values of u viii

11 6.5 γ(s) for s [0, 2π] Portions of the surface determined by d = u(s) (γ(s) γ(t)) near the approximate self-intersections values of u Granny defined by (x, y, z) for s [0, 2π] u(s) for s [0, 2π] Portions of the surface determined by d = u 0 (s) (γ(s) γ(t)) near the approximate self-intersections values of u γ(s) for s [0, 2π] N(s) for s [0, 2π] Portions of the surface determined by d = N(s) (γ(s) γ(t)) near the approximate self-intersections values of N γ(s) for s [0, 2π] N(s) for s [0, 2π] ix

12 1 CHAPTER 1 INTRODUCTION 1.1 Motivation Take a piece of paper. Cut a thin strip. This forms a ribbon. It has geometric attributes such as length L and width r. See Figure 1.1. L r Figure 1.1: A thin ribbon. Now glue the two ends together to make a closed ribbon as in Figure 1.2. Figure 1.2: A thin, closed ribbon.

13 2 At this point, there is topology as well as geometry to discuss. If the surface is topologically an annulus, then the two boundary curves have a topological linking number. Each individual edge has a topological knot type. (If the surface is topologically a möbius band, then the boundary is connected an hence a knot.) In 1961, Călugăreanu proved a relationship between the topology and geometry of thin ribbons [4]. Approximately ten years later, Fuller and White each independently discovered the same relationship [6, 16]. This topic has been further studied (see e.g. [1, 5]) and applied in mathematical biology with regards to DNA supercoiling (see e.g. [13, 15]). Although thin ribbons are well understood, it does not appear that research has been done regarding ribbons of arbitrarily large widths. For thin ribbons, the two boundary curves are isotopic; hence, they are ambient isotopic, and we say that they are of the same knot type. In this work, we consider the question of how the knot types are related if the ribbons are allowed to become arbitrarily wide. That is, we think in terms of a given starting curve as a boundary curve of a ribbon and analyze what happens to the other boundary curve as the ribbon gets wider. Since these ribbons have self-intersections in general, the two boundary curves may not be isotopic. In this thesis, under suitable conditions, we show the following. 1. For large widths r, we can specify an upper bound on the number of possible knot types of the outer boundary curve. In fact, under conditions that are generic, we show that the limiting knot type as r is unique.

14 3 2. Given two knot types, we can construct a ribbon of constant generalized width r (see definition below) whose boundary curves are exactly those knot types. 1.2 Basic Definitions We assume all maps are C smooth. (In fact, most of our results only require C 2 or C 3.) When we talk about a smooth, closed curve, we mean, in particular, that the curve is smoothly closed. Let K be a smoothly embedded knot in R 3, and let γ : [0, l 0 ] R 3 denote its arclength parametrization. Let u : [0, l 0 ] S 2 be any smooth unit vector field associated to γ that has the property that u (s) 0 for any s [0, l 0 ]. In addition, we will assume that u has only transversal self-intersections, each corresponding to exactly two parameter values. We will consider smoothly closed curves γ (and u) as C functions γ : D R 3 where D R/l 0 Z. In addition, for any pair of parameter values s and s, we will use the notation s s throughout the paper to denote the distance function of D. That is, s s will represent the smaller arclength between s and s on D. As in [5], we now define a ribbon whose edges are two closed curves, one of which is K. Definition. The ribbon R of generalized width r 0 associated to γ and u is the set of all points γ(s) + ru(s) for s D and r [0, r 0 ]. The outer ribbon edge O of R is the set of points θ(s) = γ(s) + r 0 u(s). We note that this notion of width coincides with the usual notion of width when u is perpendicular to γ, that is, when u γ = 0.

15 4 We will investigate how increasing the magnitude of r affects the knot type of the outer ribbon edge. Definition. Given r > 0, consider the disks of radius r which are normal to K and whose centers lie on K. For r sufficiently small, the disks are pairwise disjoint and together form a tubular neighborhood of K. The thickness radius of K, denoted R(K), is the supremum of such r values. We note that the above definition is as in [3]. The same notion is referred to as the injectivity radius in [8]. Such R(K) exist for smoothly embedded knots as a consequence of the Tubular Neighborhood Theorem as discussed in [7]. Recall the Frenet-Serret frame (T, N, B). Since γ is assumed to be an arclength parametrization, we have that γ (s) = 1. Then the unit tangent vector is T(s) = γ (s). The curvature of the curve γ at s is T (s). We will denote the curvature of γ at s by κ(s), and we require that κ(s) > 0 for all s to ensure the existence of the Frenet-Serret frame. The unit normal vector is defined to be N(s) = and the unit binormal vector is B(s) = T(s) N(s). This frame gives rise to the Frenet-Serret formulas: T (s) T (s), T (s) = κ(s)n(s) N (s) = κ(s)t(s) + τ(s)b(s) B (s) = τ(s)n(s) The coefficient τ(s) is called the torsion of γ at s.

16 5 1.3 Overview In Chapter 2, we introduce a notion called the goal post property. We show that this property holds when the set of width values for which the outer ribbon edge has self-intersections is unbounded. Hence, the goal post property is a potential obstruction to uniqueness of limiting knot type. However, we will show in the subsequent chapter that the absence of this bad property is generic. In Chapter 3, we introduce a different approach to study the knot type of the outer ribbon edge. Rather than analyzing the outer ribbon edge defined by γ + ru, we rescale the ribbon and study 1 γ + u. Using this rescaled outer ribbon edge, we r prove that for pairings (γ, u) not having the goal post property, the outer ribbon edge stabilizes to a unique knot type for large enough widths. Further, we show that this is a generic property of ribbons by proving that the set of pairings (γ, u) for which the goal post property is not satisfied is an open and dense subset of the set of all such pairings in the C 1 topology. We conclude the chapter by studying situations where goal posts might exist: for a given u as described in Section 1.2, we identify a finite set of knot types such that for large widths, each embedded outer ribbon edge must be of one of these particular knot types. In particular, the limiting knot type of the outer ribbon edge largely depends on the vector field u and is fine-tuned by how γ and u are coupled together. In Chapter 4, we provide a constructive proof of the fact that, given any two knot types K 1 and K 2, there exists a pairing (γ, u) such that γ represents the knot type K 1 and for large enough widths, the outer ribbon edge stabilizes to the knot

17 6 type K 2. Since we can use the Frenet-Serret formulas when calculating derivatives, Frenet ribbons are easier to analyze than general ribbons. Some sources define the Frenet ribbon as {γ + rn 1 r 1}. We shall consider the type of Frenet ribbon of width r 0 defined by Γ = {γ rn 0 r r 0 }. If r 0 is small enough, the ribbon is an embedded annulus. In Chapter 5, we focus on the pairing (γ, N), which represents a Frenet type ribbon of arbitrarily large width r 0. We show that the outer ribbon edge of such ribbons is always locally embedded. We also provide an explicit example of such a ribbon having the goal post property. We conclude the chapter by providing a constructive proof that given a pairing (γ 0, N 0 ) having the goal post property, γ 0 may be modified by an arbitrarily small amount so that the new pairing (γ, N) no longer has the goal post property. Thus, the condition of not having the undesirable goal post property is generic among Frenet type ribbons as well as more general ribbons. We conclude by providing several examples in Chapter 6. The examples include pairings (γ, u) where: 1. γ is a parametrization of the left trefoil, u is a spherical projection of a figure eight knot, and the outer ribbon edge stabilizes to an unknot 2. γ is a parametrization of the granny knot, u is a spherical projection of the left trefoil, and the outer ribbon edge stabilizes to an unknot 3. (two examples) γ is a parametrization of an unknot as a (1, q) torus curve, the

18 7 ribbon is Frenet type, and the outer ribbon edge stabilizes to a right trefoil for q = 2 or 8 19 for q = 3.

19 8 CHAPTER 2 SELF-INTERSECTIONS OF THE OUTER RIBBON EDGE Our goal is to understand the behavior of the outer ribbon edge for arbitrarily large widths. We first analyze how the knot type of the outer ribbon edge relates to the knot type of K for small widths. Proposition 1. For small enough r 0, the knot type of the outer ribbon edge O is the same as the knot type of the initial knot K. Proof. In order for the knot type of O to differ from that of K, a self-intersection must have occurred along a ribbon edge γ +ru for some r (0, r 0 ). By definition, for r R(K), the ribbon is not self-intersecting. As the outer ribbon edge may then be isotoped along the ribbon surface to the original knot, then by the isotopy extension theorem, the knot types of the two ribbon edges must be the same. We would like to show that for almost all ribbons, the knot type of O stabilizes for sufficiently large r. We will show that the set of parametrizations γ and unit vector fields u that result in a stabilized knot type is dense in the C 1 topology as described in Section 3.2. First, we focus on the local behavior of the outer ribbon edge. 2.1 Requirements for Immersed Outer Ribbon Edges A question that one might ask is the following: For what widths are the outer ribbon edges immersed? In this section, we show that for u normal to γ, the outer ribbon edges are eventually immersed for large enough widths.

20 9 If u is normal to γ, we may express u in the following way: u = (u N)N + (u B)B. Notice that (u N) 2 + (u B) 2 = 1 since u = 1, u T = 0, and N and B are orthonormal. Thus, (u N, u B) : D S 1 is a differentiable function, so there exists a unique differentiable function α : D = R/l 0 Z R/2πZ such that (cos α, sin α) = (u N, u B). Proposition 2. Suppose K is a smooth space curve with nonvanishing curvature parametrized by arclength s, and let u be any C 1 vector field normal to the tangent vector field γ. Since the curvature of γ is nonzero, the Frenet frame (T, N, B) exists. Further, since u is normal to T, we can write u(s) = cos α(s)n(s) + sin α(s)b(s) for a smooth function α : D R/2πZ as described above. Then for r > 0, the only way that the outer ribbon edge function θ can fail to be an immersion is if the following two conditions hold: rκ(s) cos α(s) = 1 α (s) + τ(s) = 0 Proof. Recall that the outer ribbon edge of R is given by θ(s) = γ(s)+ru(s). Making use of the Frenet-Serret formulas, we compute θ (s):

21 10 θ (s) = γ (s) + ru (s) = T(s) + r[ α (s) sin α(s)n(s) + cos α(s)n (s) + α (s) cos α(s)b(s) + sin α(s)b (s)] = T(s) + r[ α (s) sin α(s)n(s) + cos α(s)( κ(s)t(s) + τ(s)b(s)) +α (s) cos α(s)b(s) + sin α(s)( τ(s)n(s))] = T(s)[1 rκ(s) cos α(s)] + N(s)[ rα (s) sin α(s) rτ(s) sin α(s)] +B(s)[rτ(s) cos α(s) + rα (s) cos α(s)] Since T(s), N(s), and B(s) are linearly independent, in order for θ (s) = 0, it must be the case that the coefficients of each vector equal zero. That is, we must have the following: 1 rκ(s) cos α(s) = 0 r sin α(s)(α (s) + τ(s)) = 0 r cos α(s)(τ(s) + α (s)) = 0 Recall that r > 0, and the first equation implies that cos α(s) 0, so we may divide both sides of the third equation by r cos α(s). Thus, the above three equations may be reduced to the following two equations: rκ(s) cos α(s) = 1 α (s) + τ(s) = 0 Hence, the only way that θ(s) can fail to be an immersion is when the above conditions are satisfied.

22 11 Notice that the first equation holds for at most one r. Hence, for large r, ribbon edges are immersed. Corollary 1. For γ and u as above, if u(s) is never exactly ±B(s), then θ(s) is eventually immersed. Proof. Since u(s) B(s) for all s D, cos α(s) is bounded away from zero. Further, in Section 1.2 we assumed that κ(s) > 0 for all s D. Thus, the bad radii from the first equation in Proposition 2, namely are bounded away from infinity. r = 1 κ(s) cos α(s), In Chapter 5, we will show that in the special case where u = N, θ is always an immersion. 2.2 Self-Intersections for Large Widths and the Goal Post Property We now investigate when global (rather than local) self-intersections of the outer ribbon edges occur. To do so, we introduce some new terms. Definition. Points γ(s) and γ( s) along the knot are said to have the goal post property with respect to the vector field u if s s in D (and hence γ(s) γ( s)) u(s) = u( s)

23 12 u(s) (γ(s) γ( s)) = 0. At times, we will refer to a pairing (γ, u) as having the goal post property if there exists s s so that γ(s) and γ( s) have the goal post property with respect to u. Before we proceed to our next result, we include the following result regarding the definition of the derivative of a continuous function and convergent sequences. Lemma 1. If f : I R R n is C 1 and we have sequences {s n } and {t n } with the property that s n s 0 and t n s 0, then f(s n ) f(t n ) lim = f (s 0 ). n s n t n Proof. We begin by proving the result for n = 1. The Mean Value Theorem implies that there exists a p n between s n and t n such that f(s n ) f(t n ) s n t n = f (p n ). Now, since p n is between s n and t n, and since {s n } and {t n } both converge to s 0, the Squeeze Theorem implies that {p n } also converges to s 0. Finally, after applying limits to both sides, we have that f(s n ) f(t n ) lim n s n t n = lim n f (p n ) = f (s 0 ). Finally, for n > 1, we apply the above argument to each component of f. As noted in Proposition 1, the only time the knot type of the outer ribbon edge O can change from that of the initial knot K is when γ + ru for r (0, r 0 )

24 13 passes through itself. That is, when γ(s) + ru(s) = γ( s) + ru( s) for distinct s and s and for some fixed r. This implies the following equality: γ(s) γ( s) = r(u( s) u(s)) (2.1) Definition. Let r(s, s) represent the real number r > 0 which corresponds to a selfintersection of the outer ribbon edge at s and s (if it exists). That is, if r(s, s) exists, then for s s, Equation 2.1 holds. Notice that if r(s, s) exists, then it is unique. Indeed, by definition, if r(s, s) exists, then the following is true: γ(s) γ( s) = r(s, s)(u( s) u(s)). That is, we have r(s, s) = γ(s) γ( s) u( s) u(s). Since γ is an embedded parametrization and s s, we have that γ(s) γ( s) 0 Thus, Equation 2.1 implies that u(s) u( s). It follows that r(s, s) is a unique real number. We are now ready to state our next theorem, which says if the self-intersections of outer ribbon edges occur for arbitrarily large r, this results in points having the goal post property. Proposition 3. If there exists a sequence {(s k, s k )} k N with s k s k such that lim r(s k, s k ) =, then there exists a subsequence {(s kn, s kn )} kn N converging to some (λ, λ) with λ λ such that γ(λ) and γ( λ) have the goal post property.

25 14 Proof. Suppose there exists a sequence {(s k, s k )} k N with lim{r(s k, s k )} =. By compactness, there exist (λ, λ) D and convergent subsequences {s kn } λ and { s kn } λ over D such that r(s kn, s kn ) n. For brevity, we will denote s kn by s n with the understanding that we will be working with the convergent subsequences throughout the remainder of the proof. By Equation 2.1, we obtain the following: u(s n ) u( s n ) γ(s n ) γ( s n ) 1 n. (2.2) Recall we are assuming γ (s) = 1 and u (s) > 0 for all s D. We first show that λ λ. Indeed, suppose that λ = λ. We note the following: u(s n ) u( s n ) γ(s n ) γ( s n ) = u(s n) u( s n ) s n s n s n s n γ(s n ) γ( s n ). Since K is parametrized with respect to arclength, lim n s n s n γ(s n ) γ( s n ) = 1. Further, Lemma 1 implies that u(s n ) u( s n ) lim n s n s n = u (λ). Thus, we have that u(s n ) u( s n ) lim n γ(s n ) γ( s n ) = u (λ). Finally, we have that u u(s n ) u( s n ) (λ) = lim n γ(s n ) γ( s n ) lim 1 n n = 0,

26 15 so it must be the case that u (λ) = 0, which contradicts our assumptions on u. Now, since λ λ, we have that γ(λ) γ( λ). Hence, γ(λ) γ( λ) represents a (non-trivial) chord on the knot, and we know that its length is bounded by the diameter of the knot. Returning to Equation 2.2 and taking limits on both sides, we have 0 lim n u(s n ) u( s n ) γ(s n ) γ( s n ) u(λ) u( λ) = γ(λ) γ( λ) lim 1 n n = 0. Thus, we have that u(λ) u( λ) γ(λ) γ( λ) = 0. Since γ(λ) γ( λ), we conclude that u(λ) = u( λ). In order to conclude that γ(λ) and γ( λ) have the goal post property, we will investigate the isosceles triangles whose vertices are γ(s n ), γ( s n ), and γ(s n ) + r(s n, s n )u(s n ) = γ( s n ) + r(s n, s n )u( s n ), as shown in Figure 2.1. γ(λ) + ru(λ) γ( λ) + ru( λ) γ(λ) β n β n γ( λ) γ(s n) γ( s n) K K Figure 2.1: Example of an isosceles triangle formed near a goal post.

27 16 We know that the length of the base edge of the isosceles triangle, given by γ(s n ) γ( s n ), is bounded above while the length of the two equal sides, given by r(s n, s n ), diverges to as n increases. Thus, the apex angle becomes arbitrarily small as n increases. Denote the length of the base edge by M n, and denote the measure of the base angles by β n. Consider the right triangle formed by half of the base edge, one of the equal edges of length r(s n, s n ), and the altitude of the isosceles triangle. We then have that 0 cos(β n ) = M n 2r(s n, s n ). Since M n is bounded and r(s k, s k ) diverges to infinity while n increases, we have lim cos(β n) = 0. n This implies that lim β n = π 2. Observe that the directions u(s n) of the equal edges converge to lim n u(s n ) = u(λ) since u C 1. A similar result holds for u( s n ). Similarly, the directions of the base edge converge to lim n γ(s n ) γ( s n ) = γ(λ) γ( λ). These facts imply the following equality: u(λ) u( λ) = lim n u(s n ) u( s n ) = lim n γ(s n ) γ( s n ) r(s n, s n ) Thus, we conclude that at γ(λ) and γ( λ), both u(λ) and u( λ) are equal and perpendicular to the chord γ(λ) γ( λ); hence, γ(λ) and γ( λ) have the goal post property with respect to u. = 0. The following result is a direct consequence of Propositions 2 and 3.

28 17 Corollary 2. If the equations of Proposition 2 are not satisfied and there are no pairs of points having the goal post property for a given pairing (γ, u), then the outer ribbon edge is locally embedded for all r > 0 and is eventually globally embedded for large enough r. Remark. The existence of points having the goal post property need not imply that the outer ribbon edge has self-intersections. For example, consider the graph of z = x 2, which is a downward-facing parabola in the xz-plane in R 3 whose vertex is located at the origin. We will denote this parabola by P 1. Also consider the parabola z = (y 10) 2, which is a copy of P 1, shifted ten units and rotated ninety degrees so that it lies in the yz-plane. Denote this parabola by P 2. Notice that P 1 and P 2 have identical unit outward normal vectors at their vertices, as they are both parallel to the z-axis. Further, the outward normal vectors at both vertices are perpendicular to the chord from P 1 at (0, 0, 0) to P 2 at (0, 10, 0), as the chord is parallel to the y-axis. Thus, the vertices of P 1 and P 2 have the goal post property. However, the outward normal rays along the parabolas will never intersect at any width. Remark. We note that on a standard helix, the outward normal rays {γ(t) rn(t) r 0} never intersect for different values of t. Indeed, using a standard parametrization of the helix, namely (cos t, sin t, t), we see that the (unit) normal vectors are given by ( cos t, sin t, 0). Thus, the normal vectors lie in the xy-plane and trace out the equator on S 2 by the normal indicatrix. Furthermore, we know that the z-coordinate of the point on the helix from which the normal vectors emanate is given by t, as seen in the parametrization. Thus, for distinct t, each normal line lies

29 18 on a horizontal plane with z-coordinate t. Hence, no two distinct normal lines will ever intersect. We also note that the normal vectors of the helix are 2π-periodic. That is, u(t) = u(t + 2π). Furthermore, we have that each pair of points γ(t) and γ(t + 2π) have the goal post property, even though nearby, the normal lines never intersect.

30 19 CHAPTER 3 THE KNOT TYPE OF THE OUTER RIBBON EDGE As stated in Section 1.2, we will assume the following throughout the rest of the thesis. Assume that u is a smooth unit vector field in R 3 along γ (not necessarily normal to γ ). Further assume that u represents a generic curve on S 2 with u (s) 0 for all s. That is, the curve corresponding to u has only finitely many self-intersections, all of which are transversal, and each self-intersection corresponds to exactly two parameter values. 3.1 Rescaling the Ribbon Recall that the outer ribbon edge was defined to be θ(s) = γ(s) + ru(s) for r > 0 and s D. Our goal is to determine the behavior of the outer ribbon edge as r increases indefinitely. To do this, we will change perspectives completely. Definition. Let the rescaled outer ribbon edge be defined by σ t (s) = tθ(s) = tγ(s)+ u(s) where t = 1 r. Further, define its normalization to be ˆσ t(s) = σt(s) σ t(s). As needed for Theorem 1 and Lemma 5, the next result states that for small enough t > 0, σ t and ˆσ t exist and are regular curves. Proposition 4. There exists t 0 > 0 such that for t < t 0, ˆσ t exists, σ t 0, and ˆσ t 0; that is, σ t and ˆσ t are regular curves. Proof. Since γ(d) is compact, we know there exists an M R such that M = max γ. To ensure that ˆσ t is defined, we require that t (0, 1 ). In other words, we M

31 20 are considering ribbons of width r > M. We also want σ t to be regular. By definition, σ t fails to be regular when tγ (s) = u (s). Since γ(s) is an arclength parametrization, this condition implies that u (s) = t and u (s) γ (s). By compactness, we know there exists an m u R such that 0 < m u < u for all s D. So when t (0, m u ), σ t is regular. Finally, we want ˆσ t to be regular. By definition we have that ˆσ t = ( ˆσ t = σ t σ t + σ t (σ t σ t ) 1 2 ( ) = σ t σ t + σ t σ t σ t σ t 3 = 1 σ t ( σ t σ t(σ t σ t ) σ t 2 ) ) σt. Thus, σ t Notice that ˆσ t = 0 implies that σ t = σt(σ t σt) σ t 2 and hence σ t σ t. Since σt(σ t σt) σ t 2 is exactly proj σt σ t, we conclude that σ t σ t implies ˆσ t = 0. Thus, we need σ t σ t 0. By definition of σ t, this implies that t 2 [γ (s) γ(s)] + t[γ (s) u(s) + u (s) γ(s)] + u (s) u(s) 0. (3.1) Recall that since u is required to be unit and regular, u (s) u(s) = u(s) u (s) = u (s) > m u > 0. By compactness, there exists M γ, M γ,u R such that γ (s) γ(s) < M γ < and γ (s) u(s) + u (s) γ(s) < M γ,u <. Finally, we have the following: ˆσ t(s) σ t (s) u (s) u(s) t 2 (γ (s) γ(s)) + t(γ (s) u(s) + u (s) γ(s)) m u t 2 γ (s) γ(s) t γ (s) u(s) + u (s) γ(s)) > m u t 2 M γ tm γ,u

32 21 Thus, we may solve for t to find the existence of a t 0 R satisfying 0 < t 0 min( 1 M, m u) such that for t (0, t 0 ), ˆσ t is regular. Recall that for distinct parameter values s and s, the points γ(s) and γ( s) are said to have the goal post property if u(s) is identical to u( s) and if u(s) is normal to the chord γ(s) γ( s). We are now able to prove that if no two points have the goal post property, then the knot type of the rescaled outer ribbon edge stabilizes for small enough t values. That is, if the pairing (γ, u) does not have the goal post property, then there exists an r 0 such that for all r > r 0, γ + ru remains in the same knot class. Theorem 1. Given a parametrization γ and a vector field u such that no pair of distinct points γ(s) and γ( s) has the goal post property, the rescaled outer ribbon edge, σ t, stabilizes to a unique knot type for t ( 0, 1 R ), where R = sup({0} {r(s, s) (s, s) satisfies γ(s) γ( s) = r(s, s) (u( s) u(s))}) <. Proof. Since no pair of distinct points γ(s) and γ( s) has the goal post property, Proposition 3 implies the existence of a finite number R R such that R = sup({0} {r(s, s) (s, s) satisfies γ(s) γ( s) = r(s, s)(u( s) u(s))}) <. In other words, there does not exist a distinct pair of parameter values s and s such that r(s, s) > R. Hence, σ t has no self-intersections for t < 1 R. Finally, we have that for all 0 < t 1 t t 2 < 1 R, σ t is an ambient isotopy of σ t1 to σ t2, implying that the knot type of σ t is unique.

33 22 Remark. If {r(s, s) (s, s) satisfies γ(s) γ( s) = r(s, s)(u( s) u(s))} =, then for all t (0, m u ), the knot type of σ t is unique, even if the ribbon is not embedded. 3.2 Understanding the Density of the Class of Stabilizing Ribbons We now focus on investigating how often the previous theorem holds true. Let X = {γ : D C1 R 3 γ 0, γ is C 1 -closed}. We define the metric on X to be d X (γ, θ) = max s D ( γ(s) θ(s) + γ (s) θ (s) ). Let X 0 = {γ : D C1 R 3 γ = 1, γ is embedded and C 1 -closed} and X 1 = {u : D C1 R 3 u = 1, u 0, u is C 1 -closed}. Notice that X 0, X 1 X. Finally, let Y = {(γ, u) γ X 0, u X 1 }. We define a metric on Y by setting d Y ((γ, u), (θ, v)) = d X (γ, θ) + d X (u, v).

34 23 In particular, we are interested in the following subset of Y : u has only transversal self-intersections corresponding Y good = (γ, u) Y to double points, and distinct s, s D with γ(s) and γ( s) having the goal post property. Notice that Theorem 1 simply states that for a knot K with (γ, u) Y good, the rescaled outer ribbon edge associated to K stabilizes to a unique knot type. Proposition 5. Y good is nonempty. Proof. It is clear that X is nonempty. Thus, let γ X be arbitrary. Recall that u : D S 2 where D R/l 0 Z. For this argument, we will use [0, l 0 ] in place of D for ease of notation. Now, let u : [0, l ] vector field with no self-intersections. Then let u : [0, l 0 ] C 1 S 2 be any arbitrary, C 1 -closed, regular, unit C 1 S 2 be a rescaled version of u defined by u(s) = u ( l 0 l s). Since u has no self-intersections, it follows that u has no self-intersections. Thus, it is vacuously true that there are no two distinct points γ(s) and γ( s) having the goal post property. Whence, (γ, u) Y good. Next, we would like to show that Y good is open. We will do so by using the fact that given two curves that are relatively close with respect to C 1 distance, their self-intersections are also relatively close. First, we introduce some notation. Notation. Let {{λ α i, λ α i } : i = 1,..., l} denote the finite set of distinct parameter values such that u α (λ i ) = u α ( λ i ). For brevity, we will denote this set by {{λ α i, λ α i }} l i=1. When no confusion shall arise, we will omit the α superscript and simply write {{λ i, λ i }} l i=1.

35 24 Lemma 2. Let γ be a C 1 -closed curve in R 3 such that γ 0, and let {{ρ i, ρ i }} i Λ denote the set of distinct parameter values such that γ(ρ i ) = γ( ρ i ). If {{ρ i, ρ i }} i Λ, then ε > 0, δ > 0 such that θ where θ is a C 1 -closed curve in R 3, (d X (γ, θ) δ and θ(s θ ) = θ( s θ ) with s θ s θ = i 0 Λ such that θ(s θ ) γ(ρ i0 ) < ε). If {{ρ i, ρ i }} i Λ =, then δ > 0 such that d X (γ, θ) δ = s θ, s θ such that s θ s θ and θ(s θ ) = θ( s θ ). Proof. First, suppose that {{ρ i, ρ i }} i Λ. By way of contradiction, suppose ε > 0 such that δ n = 1 with n N, θ n n where θ n is a C 1 -closed curve in R 3 so that d X (γ, θ n ) 1 and θ n n(s n ) = θ n ( s n ) with s n s n and i Λ, θ n (s n ) γ(ρ i ) ε. We note the following: 1 0 lim n d X (γ, θ n ) lim δ n = lim n n n = 0. Thus, by definition of d X (γ, θ n ), we see that as n, we have that θ n C 1 γ. We consider the sequence {(θ n, s n, s n )} n=1. By compactness, we pass to a convergent subsequence {(θ nk, s nk, s nk )} {(θ 0, s 0, s 0 )} for n k N. For brevity, we will denote {(θ nk, s nk, s nk )} nk N by {(θ k, s k, s k )} k N. Case 1. s 0 = s 0 We first note that since D = S 1, there exists a D D such that D is homeomorphic to some interval [a, b] R, and all of s k, s k, s 0, s 0 int(d ). Now, since θ k is a curve in R 3, we may represent it as θ k (s) = (x k (s), y k (s), z k (s)). Thus, by assumption, we have that x k (s k ) = x k ( s k ), y k (s k ) = y k ( s k ), and z k (s k ) = z k ( s k ) for s k s k. Since θ k is C 1 continuous on D = [a, b], Rolle s Theorem implies that

36 25 there exists s kx between s k and s k in D so that x k (s k x ) = 0. Similarly, there exists s ky and s kz between s k and s k in D so that y k (s k y ) = 0 and z k (s k z ) = 0. Since s k and s k both converge to s 0 int(d ) D, then by the Squeeze Theorem, s kx, s ky, and s kz all converge to s 0. We conclude that γ (s 0 ) = 0 since θ k γ, which contradicts our assumption that γ (s) 0 for all s. Thus, this case cannot happen. Case 2. s 0 s 0. Since θ k γ and since θ k is uniformly convergent, we have that θ k (s k ) γ(s 0 ) and θ k ( s k ) γ( s 0 ). Further, since θ k (s k ) = θ k ( s k ), we have that γ(s 0 ) = γ( s 0 ). Then by definition of the set {{ρ i, ρ i }} i Λ, s 0 s 0 implies there exists an i 0 γ such that s 0 = ρ i0 and s 0 = ρ i0. Further, since θ k (s k ) γ(ρ i0 ), then for k large enough, we have that θ k (s k ) γ(ρ i0 ) < ε, which contradicts our assumption. Finally, suppose that {{ρ i, ρ i }} i Λ =. By way of contradiction, suppose that for all n N, there exists (θ n, s n, s n ) such that d X (θ n, γ) 1 and θ n n(s n ) = θ( s n ). As above, we consider the subsequence {(θ k, s k, s k )} k=1 where s k s 0, s k s 0, and θ k γ. Case 1. s 0 = s 0. The proof of this case follows analogously to Case 1. Case 2. s 0 s 0. As in Case 2, since θ k γ, we have θ k (s k ) γ(s 0 ) and θ k ( s k ) γ( s 0 ). Further, since θ k (s k ) = θ k ( s k ), we have that γ(s 0 ) = γ( s 0 ). However, the fact that s 0 s 0 and γ(s 0 ) = γ( s 0 ) implies that {s 0, s 0 } {{ρ i, ρ i }} i Λ, which contradicts the fact that {{ρ i, ρ i }} i Λ =.

37 26 Notation. Let B(r, x) R 3 denote the ball of radius r centered at point x. Proposition 6. Y good is open in Y. Proof. We wish to show that Y \ Y good is closed. As such, it suffices to show that if {(γ n, u n )} n N is a sequence of points in Y \ Y good and (γ n, u n ) d Y (γ 0, u 0 ) Y, then (γ 0, u 0 ) Y \ Y good. Type I: Assume each (γ n, u n ) Y \ Y good and there exists a sequence of distinct parameter values {{λ n, λ n }} n N such that the pairs γ n (λ n ), γ n ( λ n ) have the goal post property with respect to their corresponding u n vector fields. By compactness, we may pass to subsequences {λ nk } nk N and { λ nk } nk N converging to λ 0 and λ 0, respectively. These subsequences induce the subsequences γ nk γ 0 and u nk u 0, both of which converge uniformly in the C 1 sense. Thus, we have that γ nk (λ nk ) γ 0 (λ 0 ), γ nk ( λ nk ) γ 0 ( λ 0 ), and u nk (λ nk ) u 0 (λ 0 ). Case 1. λ 0 = λ 0 Since u nk (λ nk ) = u nk ( λ nk ) for λ nk λ nk (by the goal post property), we know that u nk has self-intersections. Thus, since u nk C 1 u 0, this case follows analogously to Case 1 of Lemma 2 for (u nk, λ nk, λ nk ) to obtain u (λ 0 ) = 0, which contradicts our assumption. Case 2. λ 0 λ 0 It suffices to show that γ 0 (λ 0 ) and γ 0 ( λ 0 ) have the goal post property with respect to u 0. However, since each pair γ nk (λ nk ), γ nk ( λ nk ) has the goal post property with respect to their corresponding u nk vector fields, we know that u nk (λ nk ) = u nk ( λ nk ) and u nk (λ nk ) (γ nk (λ nk ) γ nk ( λ nk )) = 0. Since the dot product is continuous,

38 27 we have that u 0 (λ 0 ) = u 0 ( λ 0 ) and u 0 (λ 0 ) (γ 0 (λ 0 ) γ 0 ( λ 0 )) = 0. Type II: Assume each (γ n, u n ) Y \ Y good and there exists a sequence of three distinct parameter values (λ n, λ n, λn ) such that u n (λ n ) = u n ( λ n ) = u n ( λn ). Compactness implies the existence of convergent subsequences {λ nk } nk N λ 0, { λ nk } nk N λ 0, and { λnk } nk N λ0. If any two of λ 0, λ 0, λ0 are equal, we obtain a contradiction as in Type I, Case 1. Thus, λ 0, λ 0, λ0 are all distinct, implying that u 0 contains a triple point. Hence, (γ 0, u 0 ) Y \ Y good. Type III: Assume each (γ n, u n ) Y \ Y good and there exists a sequence of distinct parameter values {{λ n, λ n }} n N such that u n has a non-transversal self-intersection. That is, u n (λ n ) = u n ( λ n ) and u n(λ n ) = c n u n( λ n ) for all n with c n 0. By compactness, there exist convergent subsequences {λ nk } nk N λ 0 and { λ nk } nk N λ 0. Case 1. λ 0 = λ 0 This case is similar to Type I, Case 1, which results in u (λ 0 ) = 0 and contradicts our assumption. Case 2. λ 0 λ 0 We wish to analyze the limit of the equation u n k (λ nk ) = c nk u n k ( λ nk ). Our convergent subsequences imply u n k (λ nk ) u 0(λ 0 ) and u n k ( λ nk ) u 0( λ 0 ). However, we do not know the value of limit of c nk or if it even exists. First, suppose that lim c nk = 0. Then there exists a subsequence c n k 0 so that u n (λ n k k ) = c n k u n ( λ n k k ) 0. However, since u n k(λ n k ) u 0(λ 0 ), this implies that u 0(λ 0 ) = 0, which contradicts our assumptions on u.

39 28 Similarly, suppose that lim c nk =. Then there exists a subsequence c n k + so that we have that u n ( λ n ) = 1 k k c n k u n ( λ n ) 0. k k However, since u n ( λ n ) u k k 0( λ 0 ), this implies that u 0( λ 0 ) = 0, which contradicts our assumptions on u. We conclude that 0 < lim c nk lim c nk <. Hence, there exists a c 0 R such that c 0 0 and c n k c 0 for some subsequence {c n k } k=1. Taking the limit of the equation u n (λ n ) = c k k n c 0 u 0( λ 0 ). Further, since u n k (λ n k ) = u n ( λ k n k u k n k ( λ n k ) results in u 0(λ 0 ) = ) for all n k, we have that u 0(λ 0 ) = u 0 ( λ 0 ). It follows that u 0 has a non-transversal self-intersection at λ 0 and λ 0. Thus, (γ 0, u 0 ) Y \ Y good. Finally, if any sequence (γ n, u n ) Y \ Y good fails to be in Y good for any combination of Types I, II, or III, then there exists a subsequence which fails only from one type (since 3 <, and by the pigeonhole principle). We now provide an alternative and more constructive proof which introduces functions that will be used in later proofs. Let Z denote the space D D \, where represents the diagonal of D D. We define a metric on X Z to be d X Z ((γ, s, s), (γ, s, s )) = d X (γ, γ ) + s s + s s. Now, we define the continuous function G : X 0 Z S 2 by G(γ, s, s) = γ(s) γ( s) γ(s) γ( s).

40 29 We note that near the diagonal, given a fixed γ, Lemma 1 and the fact that γ is parametrized by arclength implies the following: lim G(γ, s, s) = γ ( s) = lim G(γ, s, s). s s + s s Proposition 7. Y good is open in Y. Proof. Let (γ 0, u 0 ) Y good. First, suppose that {{λ 0 i, λ 0 i }} =. Then by Lemma 2, we know that there exists a δ > 0 such that for all u X 1 with d X (u 0, u) δ, then u also has no self-intersections. It follows that given any (γ, u) B dy (δ, (γ 0, u 0 )), (γ, u) Y good. Hence, Y good is open in Y. Now, suppose that {{λ 0 i, λ 0 i }} l i=1. Then by definition of Y good, we have that min G(γ 0, λ 0 i, λ 0 i ) u 0 (λ 0 i ) 2 = ε > 0, 1 i l where denotes the standard distance of R 3. That is, G(γ 0, λ 0 i, λ 0 i ) is not normal to u 0 (λ 0 i ) since the pairing (γ 0, u 0 ) has no goal posts. Let {λ 0 i, λ 0 i } {{λ 0 i, λ 0 i }} l i=1 be arbitrary. By continuity of G, for ε > 0 as ( above, there exists a δ 1 > 0 such that for all (γ, s, s) X 0 Z, d X Z (γ, s, s), (γ0, λ 0 i, λ 0 i ) ) < δ 1 implies G(γ, s, s) G(γ 0, λ 0 i, λ 0 i ) < ε. 2 Let B(r, y) = {x R 3 x y < r}. We can choose r 0 ( 0, 2) ε sufficiently small such that u ( 1 0 l i=1b(r 0, u 0 (λ 0 i ) ) is the union of two open intervals, each of diameter less than δ 1 3 about λ 0 i, λ 0 i. (We are using the convention that u 1 0 (s) denotes the preimage for u 0 at the parameter value s.) Such a small choice of r 0 > 0 is possible because the radii of such intervals tend to 0 as r 0 0. This can be seen by applying

41 30 the Rank Theorem [2] to u 0 locally about each of the λ 0 i and λ 0 i (since u 0 0) to obtain (ψ u 0 ϕ 1 )(s) = (s, 0, 0) for appropriate choices of local diffeomorphisms ψ and ϕ. (We caution the reader that this fails for a given u 0 if u 0 = 0 is allowed.) Notice that there are only two such intervals since D is compact and u 0 has no triple intersection points. We note that by Lemma 2, for r 0 > 0 as above, there exists a δ 2 (0, r 0 2 ] such that for all u X, if d X (u 0, u) δ 2 and if u(λ i ) = u( λ i ) for λ i λ i, then there exist λ 0 i 0, λ 0 i 0 such that u(λ i ) u 0 (λ 0 i 0 ) < r 0 2. However, we would like for the intersection of u at λ i and λ i to be transversal. As such, by the Stability Theorem [7] and the fact that u 0 self-intersects transversally at u 0 (λ 0 i ), we may choose δ 3 > 0 so that this property holds for u with d X (u 0, u) δ 3. Since u 0 has no triple (or higher order) point intersections, we can choose δ 3 small enough to secure that u has no triple point intersections. Thus, let δ 4 = min(δ 2, δ 3 ). Assuming d X (u 0, u) δ 4 ( r 0 2 ), we have the following: u 0 (λ i ) u 0 (λ 0 i 0 ) u 0 (λ i ) u(λ i ) + u(λ i ) u 0 (λ 0 i 0 ) < r r 0 2 = r 0 Thus, by choice of r 0, we have that λ 0 i 0 λ i < δ 1 3 and λ 0 i 0 λ i < δ 1 3. We repeat the above process for each {λ 0 j, λ 0 j} {{λ 0 i, λ 0 i }} l i=1 to first find δ 1 = min 1 j l δ 1 (j) > 0. Using δ 1 > 0 for all {λ 0 j, λ 0 j}, we find r 0 = min 1 j l r 0 (j); then using r 0 for all {λ 0 j, λ 0 j}, we find δ 4 = min 1 j l r 0 (j)

42 31 Now, let δ = min( δ 13, δ 4 ) > 0, and consider B dy (δ, (γ 0, u 0 )). Let (γ, u) B dy (δ, (γ 0, u 0 )) be arbitrary. If {{λ i, λ i }} i Λu = for u, then it is vacuously true that (γ, u) Y good. Thus, suppose {{λ i, λ i }} i Λu, and choose {λ i, λ i } arbitrarily. Since δ δ 4, there exists an i 0 {1, 2,..., l} such that u 0 (λ 0 i 0 ) u(λ i ) < r 0 2 by Lemma 2. By choice of r 0, this implies that λ 0 i 0 λ i < δ 13 and λ 0 i 0 λ i < δ 13. Further, since δ δ 13, we have that (γ, u) B dy (δ, (γ, u)) implies that d X (γ 0, γ) < δ 13. Thus, we have that d X Z ((γ 0, λ 0 i 0, λ 0 i 0 ), (γ, λ i λi )) = d X (γ 0, γ) + λ 0 i λ i + λ 0 i λ i < δ δ δ 1 3 = δ 1. This implies that G(γ 0, λ 0 i, λ 0 i ) G(γ, λ i, λ i ) < ε 2. Finally, since u 0 (λ 0 i 0 ) u(λ i ) < r 0 2 < ε 4 and G(γ 0, λ 0 i, λ 0 i ) G(γ, λ i, λ i ) < ε 2, we conclude that G(γ, λi, λ i ) u(λ i ) 2 > 0, and hence G(γ, λi, λ i ) u(λ i ) 0. Thus, for any arbitrary {λ i, λ i }, we have that γ(λ i ) and γ( λ i ) do not have the goal post property. Moreover, by choice of δ 4, all self-intersections of u are transversal and correspond to double points, implying that u has finitely many self-intersections. Hence, (γ, u) Y good. Proposition 8. Y good is dense in Y. Proof. Let (θ, v) Y \ Y good be arbitrary. Then either v has a non-transversal self-intersection, there is a self-intersection corresponding to three or more parameter values, or there exist distinct parameter values such that (θ, v) has the goal post property. First, suppose that v has a non-transversal self-intersection. Since the Transversality Theorem [7] implies that mappings into Euclidean space are generic, then given

43 32 any δ > 0, there exists (θ, v 1 ) B Y ( δ 2, (θ, v)) such that v 1 has only transversal selfintersections, thus implying that v 1 has only finitely many self-intersections. By the same reasoning, if there exists a self-intersection corresponding to three or more parameter values, then there exists a nearby unit vector field whose self-intersections all correspond to double points. As such, we will assume that v 1 contains finitely many transversal self-intersections, all of which correspond to exactly two parameter values. We note that given any two points p 1, q 1 S 2 with the property that (p 1, q 1 ) = π and given any point p 2 S2, there exists an α 1 (p, p 1, q 1 ) > 0 such that for all α with 0 < α < α 1 (p, p 1, q 1 ), the rotation R p,α on S 2 about p with rotation angle α will result in (R p,α (p 1 ), q 1 ) π 2 unless p = ±p 1 or p = ±q 1. Now, if (θ, v 1 ) Y good, then we are done. Consider the set of self-intersections of v 1, namely P 1 = {v 1 (ρ i ) = v 1 ( ρ i )} l i=1 S 2. For brevity, we will denote the points v 1 (ρ i ) as p i. Further, consider the set of points P 2 = {G(θ, ρ i, ρ i )} l i=1 S 2. Similarly, we will denote G(θ, ρ i, ρ i ) by q i. Let J 0 = {i {1, 2,..., l} (θ(ρ i ) θ( ρ i )) v 1 (ρ i ) = 0}. Suppose there exists a j J 0. This implies that (p j, q j ) = π 2. Now, choose any arbitrary point p S 2 \ (P 1 P 1 P 2 P 2 ). By choice of p, we may rigidly rotate the vector field v 1 at an angle α such that 0 < α < min j J0 α 1 (p, p j, q j ) about p. We now consider R p,α (v 1 ). Due to the rigid rotation, the self-intersections of R p,α (v 1 ) occur at the same parameter values as the selfintersections of v 1, but because of the rotation, (R p,α (p i ), q i ) π 2 for all j J 0. For a given i J c 0 = {1, 2,..., l} \ J 0, we have that (p i, q i ) π 2. Since

44 33 (p i, q i ) π 2 is an open condition, there exists an α 2(p, p i, q i ) > 0 such that for all α with 0 < α < α 2 (p, p i, q i ), (R p,α (p i ), q i ) π 2. Let ᾱ 1 = min j J0 α 1 (p, p j, q j ) and ᾱ 2 = min i J c 0 α 2 (p, p i, q i ), and choose α 3 (0, min(ᾱ 1, ᾱ 2 )) such that d X (R p,α3 (v 1 ), v 1 ) = d Y ((θ, v 1 ), (θ, R p,α3 (v 1 )) < δ 2. Then for all α (0, α 3], (θ, R p,α (v 1 )) Y good and d Y ((θ, v 1 ), (θ, R p,α3 (v 1 )) < δ. Since δ > 0 was arbitrary, we have that Y good is dense in Y. 3.3 Bounding the Knot Types of the Outer Ribbon Edge In this section, we would like to determine what can be said about the rescaled outer ribbon edge when points having the goal post property exist. Recall that σ t = u + tγ is the rescaled outer ribbon edge and ˆσ t is its radial projection to S 2. The following lemma states facts regarding chords between u, σ t, ˆσ t, and their derivatives. Lemma 3. Let σ t = u + tγ and ˆσ t = σt σ t for t ( 0, min ( 1 2M, m u)) be given, where γ(s) M for all s, as in Section 3.1. Then the following hold: 1. σ t u tm for all s. 2. σ t u = t for all s. 3. σ t ˆσ t σ t 1 tm for all s. 4. There exists a C 0 > 0 such that σ t ˆσ t tc 0 for all s.

45 34 5. (a) d X (u, σ t ) t(1 + M) so that σ t u in C 1 topology as t 0, that is, with respect to d X. (b) d X (u, ˆσ t ) t(1 + 2M + C 0 ) so that ˆσ t u in C 1 topology as t 0, that is, with respect to d X. Proof. 1. Recall that u = γ = 1, and σ t = u + tγ. It follows that σ t u = tγ tm for all s. 2. Since σ t = u + tγ and γ = 1, we have that σ t u = tγ = t for all s. 3. First, by definition of σ t, we note the following: u t γ σ t u + t γ. Using the fact that u = 1, we obtain t γ σ t 1 t γ. If follows that σ t 1 tm. We also have the following: σ t ˆσ t = σ t 1 1 σ t = σ t 1 tm. 4. By definition, σ t ˆσ t = ( σ t σ ) t σ t σ t = σ t σ t + σ t σ t 3 (σ t σ t ) ) = σ t (1 1σt + σ t σ t 3 (σ t σ t )

46 35 Observe the following: (a) If t < 1 2M, then σ t 1 2 by part 3. (b) σ t σ t = (u + tγ ) (u + tγ) = t(u γ + u γ + tγ γ ), since u u = 0 (by u = 1). Making use of these observations and compactness, we obtain the following for appropriate choices of C 0, C 1, C 2 R: σ t ˆσ t σ t 1 1 σ t + 1 σ t 2 σ t σ t σ t σ t σ t σ t 2 t(u γ + u γ + tγ γ ) tc 0 σ t σ t σ t 1 + t(c 1 + C 2 t) Note that the last inequality follows from the fact that σ t is bounded above and σ t is bounded above and away from zero. 5. (a) This is an immediate consequence of parts 1 and 2. (b) This follows from parts 1 through 4. Now, we introduce some notation that will be used in the upcoming lemmas. Note that even when σ t is embedded, ˆσ t may have self-intersections. Notation. Recall that {{λ i, λ i }} l i=1 denotes the finite set of distinct parameter values such that u(λ i ) = u( λ i ). Further, let {{µ t,j, µ t,j }} j Γt denote the set of distinct parameter values such that ˆσ t (µ t,j ) = ˆσ t ( µ t,j ).

47 Relationship between the Unit Vector Field and the Normalized Outer Ribbon Edge Lemma 4. For every ε > 0, there exists a δ > 0 such that for every t (0, δ), the existence of a pair {µ t,j0, µ t,j0 } {{µ t,j, µ t,j }} j Γt implies the existence of an i 0 {1, 2,..., l} such that ˆσ t (µ t,j0 ) u(λ i0 ) < ε. Proof. This is an immediate consequence of Lemmas 2 and 3, since d X (u, ˆσ t ) t(1 + 2M + C 0 ). We now have that we may choose a particular ribbon width such that every neighborhood of a self-intersection of a ˆσ t curve contains a self-intersection of the u curve, provided that t is small enough. However, we improve upon this. Indeed, our next lemma states that for every self-intersection of the u curve, there is a small enough neighborhood so that there is a unique crossing of the ˆσ t curve within that neighborhood, provided that t is small enough. Lemma 5. There exists an ε 0 > 0 such that for all ε (0, ε 0 ), there exists a corresponding δ 0 > 0 so that for every t (0, δ 0 ), given any pair {λ i0, λ i0 } {{λ i, λ i }} l i=1, there exists a unique pair {µ t,j0, µ t,j0 } {{µ t,j, µ t,j }} j Γt such that ˆσ t (µ t,j0 ) u(λ i0 ) < ε. Proof. Let λ i0 and λ i0 be one of the l distinct pairs of parameter values such that u(λ i0 ) = u( λ i0 ). Since i 0 is fixed, we will omit the subscript for the remainder of the proof and simply write λ and λ in place of λ i0 and λ i0. Choose ε 1 (0, 1) so that the ball of radius ε 1 centered at u(λ) in R 3 contains only one of the finitely

48 37 many intersections of u(s) (namely the intersection at u(λ) and u( λ)). We make the requirement that ε 1 < 1 so that the neighborhood covers less than a hemisphere of S 2. This ensures that when we project the neighborhood centered at u(λ) of radius ε 1 to the corresponding tangent space, T u(λ) S 2, the radial projection is a diffeomorphism of bounded derivative. We will choose a particular isometry to identify the tangent plane with the standard xy-plane, or R 2, and F will be the composition of the radial projection followed by the isometry onto R 2. By assumption, the crossing at u(λ) and u( λ) is transversal so that u (λ) is not parallel to ±u ( λ). Thus, ( u (λ), u ( λ) ) (0, π). As such, we choose the isometry so that (F u)(λ) = 0, (F u) (λ) is contained within the open first quadrant while (F u) ( λ) is contained within the open fourth quadrant, and the angle between them is bisected by the positive x-axis. In doing so, we ensure that both the x and y components of (F u) (λ) and (F u) ( λ) are nonzero. Notice that ( (F u) (λ), (F u) ( λ) ) = ( u (λ), u ( λ) ) (0, π) since u(λ) is the point of tangency to S 2. We will now focus on the two connected components of (F u)(s) in the xy-plane, one whose domain is within a small, open, connected neighborhood of λ while the other s domain is similarly about λ. If necessary, choose a smaller ε 2 - ball centered at u(λ) in R 3 such that both of the connected components are graphs of functions of one variable within the xy-plane. Notice that such an ε 2 exists by the Implicit Function Theorem since both the x and y components of (F u) (λ) are nonzero. Let f 1 (x) be the function whose graph is the restricted portion of the

49 38 connected component of (F u)(s) about s = λ, and let f 2 (x) be the other function corresponding to (F u)(s) about s = λ. Then there exists an a 0 > 0 such that f k : [ a 0, a 0 ] R for k = 1, 2 are defined by the Inverse Function Theorem. Let f 1(0) = m = f 2(0). By choice of isometry, we have that m > 0. Since both F and u, and hence f 1 and f 2, are C 1, we may further reduce the radius of the original ball about u(λ) in R 3 by choosing some ε 3 (0, ε 2 ] and a 0 (0, a 0 ] such that the following properties hold for all x [ a 0, a 0]: (x, f k (x)) F (B(ε 3, u(λ)) S 2 ) for k = 1, 2, f 1(x) m < m 2, and f 2(x) ( m) = f 2(x) + m < m 2. Notice that for x > 0, these conditions imply that mx 2 < f 1(x) < 3mx 2 and 3mx 2 < f 2 (x) < mx 2, and for x < 0, we have 3mx 2 < f 1 (x) < mx 2 and mx 2 < f 2(x) < 3mx 2. Now, choose a 1 (0, a 0] so that Rect a1 = [ a 1, a 1 ] [ 2ma 1, 2ma 1 ] F (B(ε 3, u(λ)) S 2 ) = B( ε 3, 0; R 2 ). Notice that by the choice of F, equality holds and ε 3 = ε 3 (ε 3 ). criteria. See Figure 3.1 below for an example of an ε 3 -neighborhood meeting the above

50 39 2ma ma 1 ma ma 1 ε 3 a1 a ma 1 f 2 ma 1 f ma 1 2ma 1 Figure 3.1: Example of a suitable ε 3 -neighborhood. We wish to show that within this ε 3 -neighborhood, there exists a self-intersection of the two arcs of (F ˆσ t )(s) for s contained in small, open, connected neighborhoods about λ and λ. However, we will first show that the arcs of (F ˆσ t )(s) are functions within this neighborhood. Let I 1 and I 2 be subintervals of D containing λ and λ, respectively, such that the following hold: {(F u)(s) s I 1 } = {(x, f 1 (x)) Rect a1 a 1 x a 1 } {(F u)(s) s I 2 } = {(x, f 2 (x)) Rect a1 a 1 x a 1 } Recall the assumption that t < min ( 1 2M, m u). By Lemma 3, we have that ˆσ t and F (ˆσ t ) converge in a C 1 sense to u and F (u), respectively. This implies that for a sufficiently small δ 1 R with 0 < δ 1 < min ( 1 2M, m u), t (0, δ1 ) ensures that (F ˆσ t )(s) F (B(ε 3, u(λ)) S 2 ) for s I 1. Now, choose a 2 > 0 slightly but strictly

51 40 less than a 1. We will now focus on Rect a2 Rect a1, which is defined similarly to Rect a1. Next, we consider the mapping x t,1 = π 1 F ˆσ t : I 1 R, which maps s to x, where π 1 is the standard projection mapping onto the first coordinate. We may similarly define x 1 = π 1 F u : I 1 [ a 1, a 1 ] R. Note that x t,1 is a smooth, oneto-one mapping. Since the image of an interval under a continuous map is an interval and x 1,t x 1 uniformly, there exists a positive δ 2 δ 1 so that t (0, δ 2 ) implies x t,1 (I 1 ) [ a 2, a 2 ]. (Observe that (F u)(s 0 ) and (F ˆσ t )(s 0 ) are not necessarily vertically aligned. For this reason, we must consider the smaller rectangle, or Rect a2.) By choice of ε 3, we have that m < f 2 k < 3m for k = 1, 2. Thus, we know 2 that (F u)(s) is not vertical on R 2 for any s I 1. It follows that for s I 1, x 1(s) = (π 1 F u) (s) 0, implying that its inverse, denoted s 1 (x), exists and is smooth. Since (F ˆσ t ) (s) converges in the C 0 sense to (F u) (s), we know that there exists a positive δ 3 δ 2 such that t (0, δ 3 ) implies that (F ˆσ t )(s) is not vertical for all s I 1. Thus, we conclude that x t,1(s) = (π 1 F ˆσ t ) (s) 0 for all s I 1. Since x t,1 (s) is a regular, one-to-one mapping, we know that its inverse, s t,1 (x), exists and is a smooth function for all t (0, δ 3 ). We restrict these inverses to the domain [ a 2, a 2 ] for all t (0, δ 3 ). We now define the function g t,1 : [ a 2, a 2 ] Rect a2 as g t,1 (x) = (F ˆσ t s t,1 )(x). In a sense, g t,1 is much like f 1 except that g t,1 corresponds to the graph of ˆσ t on the plane rather than the graph of u. In fact, notice that g t,1 converges to f 1 as t approaches zero in the C 1 sense. This follows since x t,1(s) 0 for all s I 1 implies

52 41 that s t,1 converges to s 1 in the C 1 sense. We note that the above argument may be repeated with the interval I 2, which would result in the function g t,2. Of course, δ 2 may have to be adjusted appropriately. Since g t,k converges to f k for k = 1, 2 in C 1 sense, we may choose a positive δ 4 δ 3 such that for t (0, δ 4 ), the following hold: ma 2 4 < g t,1 (a 2 ) < 7ma 2 4 while 7ma 2 4 < g t,1 ( a 2 ) < ma 2 4, and 7ma 2 4 < g t,2 (a 2 ) < ma 2 4 while ma 2 4 < g t,2 ( a 2 ) < 7ma 2 4. See Figure 3.2 below. 2ma 1 7 g t,1 4 ma 2 ma 1 f ma 2 ε a 3 2 a ma 2 f 2 ma 1 2ma 1 g t,2 7 4 ma 2 Figure 3.2: Example of a possible ε 3 -neighborhood and suitable t value. For t (0, δ 4 ), when looking at the continuous function h t = g t,1 g t,2 on [ a 2, a 2 ], since g t,1 (a 2 ) and g t,1 ( a 2 ) lie in the open first and third quadrant respectively and g t,2 (a 2 ) and g t,2 ( a 2 ) lie in the open second and fourth quadrant, we have

53 42 that h t (a 1 ) < 0 and h t (a 2 ) > 0. The Intermediate Value Theorem implies that there exists a point x 0 ( a 2, a 2 ) so that h t (x 0 ) = 0. Thus, we see that within the ε 3 -neighborhood and for t (0, δ 4 ), there exists a self-intersection of g t,1 and g t,2. Further, since g t,1(x) converges uniformly to f 1(x), we may choose δ 5 (0, δ 4 ] such that for t (0, δ 5 ) and for x [ a 2, a 2 ], f 1(x) g t,1(x) < m 2 and f 2(x) g t,2(x) < m 2. Now, making the requirement that t (0, δ 5 ) and focusing on Rect a2 F (B(ε 3, u(λ i )) S 2 ), it must be the case that no more than one self-intersection of the g t,1 and g t,2 graphs occurs. Indeed, if there were more than one self-intersection of g t,1 and g t,2, say at x 0 and x 0 within [ a 2, a 2 ], then Rolle s Theorem implies that there exists a point x 0 ( a 2, a 2 ) between x 0 and x 0 such that h t(x 0 ) = (g t,1 g t,2)(x 0 ) = 0. However, for x [ a 2, a 2 ] we have the following: 2m = m + f 2(x) f 2(x) + g t,2(x) g t,2(x) + g t,1(x) f 1(x) + m + f 1(x) g t,1(x) m + f 2(x) + f 2(x) g t,2(x) + g t,2(x) g t,1(x) + f 1(x) m + f 1(x) g t,1(x) < m 2 + m 2 + g t,2(x) g t,1(x) + m 2 + m 2 = 2m + g t,2(x) g t,1(x) Thus, we have that h t(x) = g t,2(x) g t,1(x) > 0 for all x [ a 2, a 2 ]. Whence, the crossing at x 0 is unique, as was to be shown. Now, choose the positive real number ε 3 < ε 3 so that the radius of F (B(ε 3, u(λ)) S 2 ) is equal to the minimum of a 2 and ma 2. Following the same proof as above, choose a 1 and a 2 so that 0 < a 2 < a 1 and Rect a 2 Rect a 1 F (B(ε 3, u(λ)) S 2 ) Rect a2.

54 43 Further, choose δ m > 0 for m = 1, 2, 3, 4, 5 such that δ m δ m+1 and such that the g t,k functions exist and are bounded as described above for k = 1, 2. The above proof implies that for t (0, δ 5), there exists a self-intersection of the g t,k graphs within Rect a 2 F (B(ε 3, u(λ)) S 2 ). Further, since we previously proved that the self-intersection in Rect a2 is unique, this implies that the self-intersection in F (B(ε 3, u(λ)) S 2 ) is unique, as well. Since F (B(ε 3, u(λ)) S 2 ) is diffeomorphic to B(ε 3, u(λ)) S 2, we conclude that there exists a unique self-intersection of ˆσ t inside B(ε 3, u(λ)) S 2 for all t (0, δ 5). Finally, we may repeat this process of finding ε 3(i) and δ 5(i) for each of the i = 1, 2,..., l parameter values within {λ i, λ i }. Let ε 0 = min{ε i 3(i)} and δ 0 = min{δ i 5(i)}. Since each B(ε 0, u(λ i )) is diffeomorphic to F (B(ε 0, u(λ i )) S 2 ), then for t (0, δ 0 ) each of the l B(ε 0, u(λ i )) contains exactly one of the self-intersections of the u curve as well as a unique self-intersection of the ˆσ t curve. As a consequence of Lemma 4 and Lemma 5, we are finally able to describe the relationship between the self-intersections of ˆσ t and the self-intersections of u. Corollary 3. There exists a δ > 0 such that for t (0, δ), the self-intersections of the u curve are in one-to-one correspondence with the self-intersections of the ˆσ t curve, each appearing in B(ε 0, u(λ i )) for ε 0 of Lemma 5. Proof. Choose ε 0 and δ 0 as in Lemma 5, and choose δ 1 (ε 0 ) as in Lemma 4. Let δ = min{δ 0, δ 1 (ε 0 )}. Lemma 5 implies that for t (0, δ), for each self-intersection of the u curve, say u(λ i ) = u( λ i ), there is a unique self-intersections of the ˆσ t curve within

55 44 B(ε 0, u(λ i )). Further, Lemma 4 implies that the are no additional self-intersections of ˆσ t outside of the B(ε 0, u(λ i )) neighborhoods for i = 1,..., l. Notation. As a consequence of Corollary 3, since the self-intersections of ˆσ t are in one-to-one correspondence with the self-intersections of u for sufficiently small t, we may denote the set of parameter values at which the self-intersections of ˆσ t occur by {{µ t,i, µ t,i }} l i= Relationship between the Unit Vector Field and the Knot Types of the Outer Ribbon Edges Definition. For values of t such that σ t is embedded and for each pair {µ t,i0, µ t,i0 } {{µ t,i, µ t,i }} l i=1, consider the values σ t (µ t,i0 ) σ t ( µ t,i0 ). The parameter value that results in the larger magnitude of σ t is defined to be the over-crossing parameter value, while the parameter value that results in the smaller magnitude of σ t is defined to be the under-crossing parameter value. Notation. Given a pair of parameter values {µ t,i0, µ t,i0 } {{µ t,i, µ t,i }} l i=1, suppose that µ t,i0 is the over-crossing parameter value and µ t,i0 is the under-crossing parameter value. We will denote this distinction by (µ t,i0, +1) and ( µ t,i0, 1), respectively. So far, we have been focusing on the relationship between the self-intersections of the u curve and the self-intersections of the ˆσ t curve. However, we would like to understand the end behavior of the rescaled outer ribbon edge σ t. Finally, we are able to give a bound on the number of possible knot types of the rescaled outer ribbon edge, assuming that the outer ribbon edge is embedded in R 3.

56 45 Theorem 2. There exists a ˇδ > 0 such that for all t (0, ˇδ), if σ t is an embedded curve, then it is isotopic to one of at most 2 l possible knot types. Further, for a given t (0, ˇδ), the knot type of σ t is uniquely determined by the over-crossing and under-crossing parameter values {µ t,i, µ t,i } l i=1. These knots have at most l crossings. Remark. We begin by providing the overall idea of the proof. First, note that given any C 1 u curve on S 2 with l transverse self-intersections, each self-intersection can be resolved in two ways isotopically to obtain an embedded curve. Indeed, each selfintersection involves two arcs, one arc with parameter values about λ i and the other arc with parameter values about λ i. We may deform one of the arcs to the inside of S 2 while deforming the other arc to the outside of S 2. In order to do so, a choice must be made as to which value, λ i or λ i, shall be the over-crossing parameter value and which shall be the under-crossing parameter value, respectively. As such, there are 2 l possible ways to resolve u isotopically in this fashion so that it is an embedded curve in R 3. Our goal is to show that for small enough t, the outer ribbon edge σ t is ambient isotopic to one of these 2 l resolved u curves in R 3. However, some of these 2 l curves can be isotopically the same in R 3 and hence in the same knot class. Proof. Consider ε 0 2 > 0 where ε 0 is as in Lemma 5. Using Corollary 3, choose ˇδ 0 > 0 similar to δ > 0 for ε 0 2. Note that Lemma 5 ensures that any pair of neighborhoods B( ε 0 2, u(λ i )) and B( ε 0 2, u(λ j )) are disjoint for i j. For a given i {1, 2,..., l}, there exists an η i > 0 such that the following conditions hold: s λ i < η i = u(s) B( ε 0 4, u(λ i ))

57 46 s λ i < 2η i = u(s) B( ε 0 2, u(λ i )) We will also assume that similar conditions hold for λ i by choosing η i > 0 appropriately. Let η = min 1 i l η i. For a given i {1, 2,..., l}, let I i = {s D s λ i < η or s λ i < η}. Let ε i = min u(λ i ) u(s) on D \ I i. Note that such an ε i > 0 exists since u is assumed to be generic and thus does not contain any triple points, and D \ I i is compact. Furthermore, it must be the case that ε i ε 0 4. We note that if u(s) B(ε i, u(λ i )), then s I i. Let ε = min 1 i l ε i. Now, for a given i {1, 2,..., l}, using Corollary 3, choose ˇδ i (0, ˇδ 0 ] corresponding to ε 3. If necessary, decrease ˇδ i so that 0 < ˇδ i ε 3M all s D. Fix t (0, ˇδ i ). First, we note the following: where γ(s) M for σ t (µ t,i ) u(µ t,i ) = tγ(µ t,i ) < ˇδ i M ε 3M M ε 3. Furthermore, the triangle inequality and the fact that σ t (s) and ˆσ t (s) are parallel for each s imply the following: ˆσ t (µ t,i ) σ t (µ t,i ) = σ t (µ t,i ) 1 = u(µ t,i )+tγ(µ t,i ) u(µ t,i ) tγ(µ t,i ) < ε 3. Observe that the choice of ε above with respect to Corollary 3 implies that u(λ i ) ˆσ(µ t,i ) < ε 3.

58 47 Finally, we observe the relationship between u(λ i ) and u(µ t,i ): u(λ i ) u(µ t,i ) = u(λ i ) ˆσ t (µ t,i ) + ˆσ t (µ t,i ) σ t (µ t,i ) + σ t (µ t,i ) u(µ t,i ) u(λ i ) ˆσ t (µ t,i ) + ˆσ t (µ t,i ) σ t (µ t,i ) + σ t (µ t,i ) u(µ t,i ) < ε 3 + ε 3 + ε 3 = ε Hence, our choice of ε implies that µ t,i I i. Similarly, we conclude that µ t,i I i. Furthermore, the proof of Lemma 5 implies that µ t,i and µ t,i are sufficiently far away from each other, implying that one of the parameter values lies in the set {s D s λ i < η} while the other lies in the set {s D s λ i < η}. Finally, ( ) let ˇδ = min ˇδ 1,..., ˇδ 1 l,. 5M Now, choose a C conditions hold: bump function ϕ : R [0, 1 ] such that the following 2 ϕ(s) = 0 for s satisfying s 2η 0 < ϕ(s) < 1 4 for s satisfying η < s < 2η 1 4 ϕ(s) 1 2 for s satisfying s η See Figure 3.3 for an example of a bump function meeting the above criteria. For a given C curve v(s) : D R 3 and for λ D, we define the following alteration of v(s) by identifying D = R/l 0 Z: f λ,±1 (v)(s) = (1 ± ϕ(s λ))v(s).

59 48 ϕ(s) η η 0 η 2η s Figure 3.3: Example of a suitable bump function ϕ. Observe the following facts: f λ,±1 (u)(s) f λ,±1 (u)(s) f λ,±1 (σ t )(s) f λ,±1 (σ t )(s) = u(s) (3.2) = ˆσ t (s) = σ t(s) σ t (s) (3.3) Now, fix t 0 (0, ˇδ) such that σ t0 defines an embedded curve in R 3. Our choice of ˇδ > 0 ensures that each pair λ i, λ i corresponds to a pair µ t0,i, µ t0,i, in respective orders. As in the previous definition, we know that each pair of parameter values µ i,t0, µ i,t0 can be split up into an over-crossing parameter value and an under-crossing parameter value. This gives rise to the following function: J 0 : l i=1{λ i, λ i } 1:1 l i=1{µ i,t0, µ i,t0 } {+1, 1}, where J 0 (λ i ) = +1 if µ i,t0 is an over-crossing parameter value for σ t, and J 0 (λ i ) = J 0 ( λ i ). For a given C curve v(s) : D R 3, let f J0 (v) = (f λl,j 0 (λ l ) f λl,j 0 ( λ l ) f λ1,j 0 (λ 1 ) f λ1,j 0 ( λ 1 ))(v)

60 49 Note that our choice of η implies that statements similar to equations (3.2) and (3.3) hold for f J0. Our goal is to show that σ t0 = fj0 (σ t0 ) = f J0 (u) where = denotes ambient isotopy. We begin by defining a homotopy between σ t0 and f J0 (σ t0 ). Indeed, let H : D [0, 1] R 3 be defined by H(s, τ) = τσ t0 (s) + (1 τ)f J0 (σ t0 (s)). Much like the curves f J0 (σ t ) and f J0 (u), the curve H(s, τ) has the following property: H(s, τ) H(s, τ) = ˆσ t 0 (s). We wish to show that H is an isotopy. Figure 3.4 depicts a local schematic diagram of the isotopy. f J0 (σ t0 ) H(s, τ) σ t0 Figure 3.4: σ t0 may be isotoped to f J0 (σ t0 ) along the gray lines via H(s, τ) Indeed, by definition of f, we see that f is the identity function for s D \ l i=1i i. Furthermore, since the radial projection of H(s, τ) onto S 2 is identically ˆσ t0, the existence of a self-intersection of H(s, τ) for a particular τ would imply the

61 50 existence of a self-intersection of ˆσ t0 at the same parameter values. As such, suppose there exists a τ 0 such that H(s, τ 0 ) is not an embedded curve. This implies that H(µ t0,i 0, τ 0 ) = H( µ t0,i 0, τ 0 ) for some pair {µ t0,i 0, µ t0,i 0 }. Our choice of ˇδ > 0 and η implies that parameter values corresponding to self-intersections of σ t0 must lie within I i0. Thus, the following hold: µ t0,i 0 λ i0 < η = ϕ(µ t0,i 0 λ i0 ) 1 4 µ t0,i 0 λ i0 < η = ϕ( µ t0,i 0 λ i0 ) 1 4 However, by definition of ϕ (and by extension, f), any possible self-intersections of f J0 (σ t0 ) are avoided as the arc of the curve σ t0 (s) for s nearby the over-crossing parameter value is bumped away from the origin while the arc of the curve σ t0 (s) for s nearby the under-crossing parameter value is bumped toward the origin, thus avoiding any possible self-intersections. In other words, the following is true: [ τ 0 + (1 τ 0 ) ( 1 + J 0 (λ i0 )ϕ(µ t0,i 0 λ i0 ) )] σ t0 (µ t0,i 0 ) [ τ 0 + (1 τ 0 ) ( 1 + J 0 ( λ i0 )ϕ( µ t0,i 0 λ i0 ) )] σ t0 ( µ t0,i 0 ), since J 0 (λ i0 ) = J 0 ( λ i0 ). Hence, H is an ambient isotopy and f J0 (σ t0 ) = σ t0. Next, we wish to show that f J0 (σ t0 ) and f J0 (u) are ambient isotopic. We claim that for t [0, t 0 ], f J0 (σ t ) serves as an isotopy from f J0 (u) to f J0 (σ t0 ). See Figure 3.5 for a local schematic diagram of the isotopy. Indeed, suppose there exists a t 1 [0, t 0 ] such that f J0 (σ t1 ) is not an embedded curve. As with f J0 (u) and f J0 (σ t0 ), we note that the projection of f J0 (σ t1 ) onto S 2

62 51 f J0 (u) f J0 (σ t0 ) Figure 3.5: f J0 (σ t0 ) may be isotoped to f J0 (u) along the gray lines. is identically ˆσ t1. Thus, since t 1 < t 0 < ˇδ, Corollary 3 implies that f J0 ( σt1 (µ t1,i 1 ) ) = f J0 ( σt1 ( µ t1,i 1 ) ) for some pair of parameter values {µ t1,i 1, µ t1,i 1 } {{µ t1,i, µ t1,i}} l i=1. However, Corollary 3, as well as our choice of η, ε, and ˇδ, implies that µ t,i1, µ t,i1 I i1. Thus, we have the following: µ t1,i 1 λ i1 < η = ϕ(µ t1,i 1 λ i1 ) 1 4 µ t1,i 1 λ i1 < η = ϕ( µ t1,i 1 λ i1 ) 1 4 Without loss of generality, suppose that J 0 (λ i1 ) = +1 and J 0 ( λ i1 ) = 1. Using the fact that t 1 < ˇδ 1, we have 5M f J0 (σ t1 (µ t1,i 1 )) = ( 1 + J 0 (λ i1 )ϕ(µ t1,i 1 λ i1 ) ) σ t1 (µ t1,i 1 ) = ( 1 + ϕ(µ t1,i 1 λ i1 ) ) σ t1 (µ t1,i 1 ) 5 4 σ t 1 (µ t1,i 1 ) 5 4 (1 t 1M) > 1

63 52 Similarly, we have f J0 (σ t1 ( µ t1,i 1 )) = ( 1 + J( λ i1 )ϕ( µ t1,i 1 λ i1 ) ) σ t1 ( µ t1,i 1 ) = ( 1 ϕ( µ t1,i 1 λ i1 ) ) σ t1 ( µ t1,i 1 ) 3 4 σ t 1 (µ t1,i 1 ) 3 4 (1 + t 1M) < 9 10 This contradicts the fact that f J0 ( σt1 (µ t1,i 1 ) ) = f J0 ( σt1 ( µ t1,i 1 ) ). Thus, f J0 (σ t ) is an isotopy of f J0 (σ t0 ) to f J0 (u). Hence, σ t0 is isotopic to f J0 (u), where J 0 is determined from the over-crossing and under-crossing parameter values {{µ t,i, µ t,i }} l i=1. For a given σ t0, J 0 is uniquely determined by the over and under-crossing values at {µ t0,i, µ t0,i}, and hence, f J0 (u) is uniquely determined among 2 l possibilities {f J (u) J : {{λ i, λ i }} l i=1 { 1, +1}, J(λ i ) = J( λ i )}. Thus, the fact that f J0 (u) is one of the 2 l embedded resolutions f J (u) of u, and σ t0 = fj0 (σ t0 ) = f J0 (u) implies that σ t0 is one of at most 2 l possible knot types. Thus, the previous theorem provides an understanding of how the knot type of the outer ribbon edge is governed by the parameter values at which the selfintersections of ˆσ t occur, which are denoted by {{µ t,i, µ t,i }} l i=1.

64 53 CHAPTER 4 CONSTRUCTING RIBBONS BETWEEN ANY TWO KNOTS In this chapter, we provide an algorithm for constructing a ribbon from one fixed knot type to another fixed knot type that has the property that the ribbon has a fixed width and contains no goal posts. 4.1 Preliminaries Lemma 6. Suppose that γ(s 0 ) u(s 0 ) > γ( s 0 ) u( s 0 ) for s 0 s 0. If t (0, M 0 (s 0, s 0 )), where M 0 (s 0, s 0 ) = 2 γ(s 0) u(s 0 ) γ( s 0 ) u( s 0 ), then σ γ( s 0 ) 2 γ(s 0 ) 2 t (s 0 ) > σ t ( s 0 ). Proof. Since γ(s 0 ) u(s 0 ) > γ( s 0 ) u( s 0 ), we have that 2t ( γ(s 0 ) u(s 0 ) γ( s 0 ) u( s 0 ) ) > 0 Further, since t < M 0 (s 0, s 0 ), we have 2t ( γ(s 0 ) u(s 0 ) γ( s 0 ) u( s 0 ) ) + t 2( γ(s 0 ) 2 γ( s 0 ) 2) > 0 Using the fact that u(s 0 ) = 1 = u( s 0 ), we obtain the following: u(s 0 ) 2 + 2t ( γ(s 0 ) u(s 0 ) ) + t 2 γ(s 0 ) 2 > u( s 0 ) 2 + 2t ( γ( s 0 ) u( s 0 ) ) + t 2 γ( s 0 ) 2 However, this is equivalent to u(s 0 ) + tγ(s 0 ) 2 > u( s 0 ) + tγ( s 0 ) 2 In other words, we have that σ t (s 0 ) 2 > σ t ( s 0 ) 2, so we conclude that σ t (s 0 ) > σ t ( s 0 ), as was to be shown.

65 54 Lemma 7. Let {p 1, p 2,..., p k } be a collection of points in R 2 where the coordinates of a point p j are given by (a j, b j ). Suppose that a 1 < a 2 < < a k. Further, let each point p j have an associated sign, denoted by ω(p j ), where ω(p j ) = ±1. Then there exists an isotopy of R 2 that separates the points such that all points with sign equal to +1 lie in the open upper half space and all points with sign equal to 1 lie in the open lower half space. Proof. We begin by dividing the plane into vertical strips with the following vertical lines: x = a 1 1, x = a k + 1, x = a j + a j+1 2 for j = 1,..., k 1. This collection of k + 1 lines divides the plane into k vertical strips, each containing one of the k points, as well as two unbounded portions of the plane. Let I j R denote the closed interval along the x-axis between the vertical lines containing a j. Now, we choose a smooth function f : R R in the following way: If b j > 0 and ω(p j ) = +1, then f(x) = 0 for x I j. If b j < 0 and ω(p j ) = 1, then f(x) = 0 for x I j. If b j 0 and ω(p j ) = 1, then f(x) is a smooth bump function passing through the point (a j, b j + 1) for x I j. If b j 0 and ω(p j ) = +1, then f(x) is a smooth bump function passing through the point (a j, b j 1) for x I j. If x / j I j, then f(x) = 0.

66 55 See Figure 4.1 for an example of an appropriate function f given the pairs (p 1, +1), (p 2, +1), (p 3, 1), (p 4, 1), and (p 5, 1) as indicated in the plane. (p 3, 1) y (p 2, +1) (p 5, 1) f x (p 4, 1) (p 1, +1) I 1 I 2 I 3 I 4 I 5 Figure 4.1: Example of an acceptable f function. Now, let F : I R 2 R 2 be defined by F (t, x, y) = (x, y tf(x)). Notice that F is an isotopy of R 2 and that F (1, x, f(x)) = (x, 0). Further, we have that F (1, p j ) H + if ω(p j ) = +1, and F (1, p j ) H if ω(p j ) = 1, as was desired. Lemma 8. Let A = {(x, y, z) R 3 z < 1 2 } and B = {(x, y, z) R3 1 4 < x2 + y 2 + (z 1) 2 < 9 and (x, y, z) / positive z-axis}. Then there exists a smooth isotopy 4

67 56 H : A [0, 1] R 3 such that H(x, 0) = x, H(x, 1) is a diffeomorphism of A onto B, H((x, y, 0), 1) is the inverse of the stereographic projection ϕ(u, v, w) = ( u, v, 0), 2 w 2 w and H((x, y, z), 1) = (z + 1)ϕ 1 (x, y, 0); that is, H(x, 1) maps vertical line segments in A to radial line segments in B. Proof. We will provide such a function. For t (0, 1], let r t = 1 t and let S t be the portion of the sphere of radius r t and center (0, 0, r t ) restricted to 0 z < 2. We define H(x, y, z, 0) = (x, y, t). For t (0, 1], define H((x, y, 0), t) to be the intersection point of S t with the line segment between (x, y, 0) and (0, 0, 2) (which contains ϕ 1 (x, y, 0)). Let N(x, y, t) denote the unit inward normal of S t at H((x, y, 0), t). Define H((x, y, z), t) = H((x, y, 0), t) + zn(x, y, t), i.e. taking vertical segments at (x, y, 0) to normal segments to S t. H((x, y, 0), t) is a diffeomorphism of the xy-plane onto S t, whose inverse is a projection from (0, 0, 2). The normals change smoothly on S t, and the normal segments do not intersect with each other since the focal point (or the center of S t ) is at a distance r t > Construction of the Ribbon Theorem 3. Given an initial knot type K 1 and an end result knot type K 2, there exists a pair of curves γ and u, where γ is a smoothly embedded curve in R 3 whose knot type is K 1 and u is a smooth curve on S 2 satisfying (γ, u) Y good and there exists a T 1 > 0 so that for t (0, T 1 ), the rescaled outer ribbon edge defined by σ t = u + tγ

68 57 stabilizes to the unique knot type K 2, as in Theorem 1. Proof. We begin by considering a special case which is illustrative of the process that we will be using throughout this proof. Case 1. K 1 is the unknot, and K 2 is a knot type with the property that there exists a regular projection of K 2 on the plane containing an arc that traverses each crossing once before traversing any crossing a second time. Step 1: Separating double points of ũ into groups Let ũ denote a smoothly embedded curve in R 3 whose knot type is K 2, and let P (ũ) be a regular projection of ũ on the plane containing an arc, denoted A, which traverses all of the crossings once before traversing any of them a second time. Without loss of generality, we will assume that ũ is chosen such that P (ũ) has a regular parametrization, with transversal intersections and no triple (or higher order) self-intersections. Let P denote the starting point of A, where P is not a double point of a crossing. Suppose that P (ũ) has l crossings. Fix an orientation along the diagram, and beginning at P, trace along A. While doing so, enumerate the double points of the crossings of P (ũ) along A in the order in which they occur. Let {p 1, p 2,..., p l } denote the double points of first l crossings of P (ũ), which lie along A. For each j {1, 2,..., l}, in addition to the p j label, we will assign a sign, denoted by ω(p j ), to indicate whether the arc is crossing over or under at the first time of the crossing along A. Let ω(p j ) = +1 denote that p j is an over-crossing double point, and let ω(p j ) = 1 denote that p j is an under-crossing double point. Thus, each double point of A has an associated pair

69 58 (p 1, ω(p 1 )), (p 2, ω(p 2 )),... (p l, ω(p l )). Once we traverse through the double point labeled p l, we begin to traverse through each crossing a second time, but possibly in a different order. We will denote these double points (which lie outside of A) by the pairs ( pτ(1), ω( p τ(1) ) ), ( p τ(2), ω( p τ(2) ) ),..., ( p τ(l), ω( p τ(l) ) ), where (τ(1), τ(2),..., τ(l)) is a permutation of (1, 2,..., l). We note that p k and p k represent the same point in the plane; however, p k denotes the point along arc A passing through the crossing the first time, and p k denotes the point along the arc that passes through the crossing a second time. Further, we note that it must be the case that ω( p k ) = ω(p k ). Now, in the plane of P (ũ), consider the l points labeled {p 1, p 2,..., p l }. Further, consider the set S consisting of all of the lines connecting each pair of those points. Let L be a line that is not perpendicular to any line in S. Rotate the plane so that L coincides with the x-axis. Let the coordinates of a particular point p j be denoted by (a j, b j ). Notice that our choice of L implies that each a j is distinct for j {1, 2,..., l}. By Lemma 7, we know that there exists an isotopy of R 2 such that after applying the isotopy, p j moves into H + if ω(p j ) = +1, and p j moves into H if ω(p j ) = 1. (We note that we may apply Lemma 7 regardless of the fact that the a j values are not in increasing order. The crucial point is that the a j values are distinct; if necessary, we may reorder the points to apply Lemma 7 and then go back to the original order.) Since we can isotope the points {p 1, p 2,..., p l } as described above, without loss of generality, we will assume that each p j already lies in the appropriate

70 59 open half space according to ω(p j ). Further, every isotopy f of the xy-plane can be extended to an isotopy f on R 3 by using the same isotopy on each plane z = c, sending vertical lines to vertical lines; that is, f(x, y, z, t) = (f(x, y, t), z). Hence, the isotopy of moving {p 1,..., p k } into the correct half plane extends to R 3, isotoping ũ to an equivalent knot. Step 2: Mapping groups of double points into small disks on S 2 Enclose the sets of points {p j ω(p j ) = +1} and {p j ω(p j ) = 1} within corresponding closed rectangles of finite area not intersecting L. Let R + denote the closed rectangle containing the points in H + such that R + H +, and let R denote the closed rectangle enclosing the points in H such that R H. We can then choose a (possibly very large) open disk, D + such that R + D + D + H + and an open disk D such that R D D H. Consider S 2 \ {(0, 1, 0)}, and let D + be the open disk centered at (0, 0, 1) with polar angle π from (0, 0, 1). Similarly, let 12 D be the open disk centered at (0, 0, 1) with polar angle π from (0, 0, 1). Finally, let 12 L be the equator passing through (±1, 0, 0) and (0, 1, 0) while omitting (0, 1, 0). Let A = {(x, y, z) z < 1} and 2 B 0 = {(x, y, z) 1 4 < x2 + y 2 + z 2 < 9 4 and (x, y, z) / negative y-axis}. We claim that there exists a smooth isotopy H 1 : A I R 3 such that H 1 (, 1) maps vertical lines in A to radial segments in B 0 and H 1 (D +, 1) = D +, H 1 (D, 1) = D, and H 1 (L, 1) = L. Indeed, by using Lemma 8 as well as rigid translation and rotation type iso-

71 60 topies, we can construct an isotopy H 0 : A I R 3 deforming A onto B 0, taking vertical line segments of A to radial segments of B 0 at t = 1, and H 0 (, 1) restricted to the xy-plane has the inverse ϕ : S 2 \ {(0, 1, 0)} R 3, a stereographic projection. Compare ϕ(d +), ϕ(d ), and ϕ(l ) to D +, D, and L, respectively. There exists a C isotopy f of R 2 isotoping D +, D, and L to ϕ(d +), ϕ(d ), and ϕ(l ), respectively. (This is possible by translating and rotating to first match the lines L and L. If the disks are on opposite sides, we can rotate 180 and then isotope the disks to the target disks.) We then extend f to an isotopy of B 0, namely f defined by f(x, y, z, t) = (f(x, y, t), z) as discussed above. Then define a C isotopy H 1 : A I R 3 by applying f followed by H 0. This concludes the proof of the claim. Without loss of generality, we can assume ũ A by using a vertical compression isotopy. H 1 provides an isotopy from ũ to H 1 (ũ, 1) B 0. Let ψ = ϕ 1 f : R 2 S 2 \ {(0, 1, 0)}. Since P (ũ) has a regular parametrization with transversal self-intersections and without triple points, the same is true for ψ(p (ũ)) which we define as u. Step 3: Defining γ Thus, we have taken a regular projection P (ũ) of our embedded ũ curve, picked out the double points corresponding to the first l over-crossings or under-crossings along the arc A, and separated them in a way so that after mapping P (ũ) diffeomorphically to u in S 2 \ {(0, 1, 0)}, all of the over-crossing points along A lie within a small disk at the north pole, and all of the under-crossing points along A lie within a small disk

72 61 at the south pole. After defining the curve γ, we will adjust the parametrization for u appropriately to create the association γ(s) u(s). Since we are assuming that K 1 is the unknot, let γ : D S 2 be the smooth arclength parametrization of the great circle where γ(0) = γ(2π) lies on (0, 0, 1), γ( π ) lies on (1, 0, 0), and γ(π) lies on (0, 0, 1). We specify 2l particular parameter 2 values: Let λ j = π(j 1) 12l so that the set of parameter values {λ 1, λ 2,..., λ l } are contained in [ 0, π 12). Let λ τ(j) = π + π(j 1) 12l so that the set of parameter values { λ τ(1), λ τ(2),..., λ τ(l) } are contained in [ π, π + π 12), where {τ(j)} l j=1 describes the order P (ũ) traversed the crossings the second time. Our choice of λ j and λ τ(j) implies that the set {γ(λ 1 ), γ(λ 2 ),..., γ(λ l )} is contained in a cap of polar angle π 12 at the north pole of S2, and the set {γ( λ τ(1) ), γ( λ τ(2) ),..., γ( λ τ(l) )} is contained in a cap of polar angle π at the south pole of 12 S2. Reparametrize u such that for all j {1, 2,..., l}, the point p j corresponds to u(λ j ) and the point p τ(j) corresponds to u( λ τ(j) ). We can do this since the order of λ 1, λ 2,..., λ l, λ τ(1), λ τ(2),..., λ τ(l) is preserved and the rest of the parametrization is to adjust the speed in between.

73 62 Step 4: Analyzing the magnitude of σ t (s) for s {λ i, λ i } l i=1 At this stage, we have that for any λ j, both γ(λ j ) and u(λ j ) are contained either in a small disk centered at the north pole or the south pole of S 2. The same holds true for any λ τ(j). This implies that u(λ j ) γ(λ j ) ±1 (and similarly for λ τ(j) ). Pick any pair of parameter values λ k, λ k corresponding to a self-intersection of u. Then we have that u(λ k ) = u( λ k ). Further, by choice of γ, we have that γ(λ k ) γ( λ k ). In particular, this implies that u(λ k ) γ(λ k ) u( λ k ) γ( λ k ). Note that this implies that (γ, u) does not have the goal post property. See Figure 4.2 for a local schematic diagram of the outer ribbon edge vectors at the parameter values λ k and λ k for ω(p k ) = +1. γ(λ k ) u(λ k ) + tγ(λ k ) σ t u u( λ k ) + tγ( λ k ) γ( λ k ) Figure 4.2: Outer ribbon edge vectors at the values λ k and λ k for ω(p k ) = +1. We know that as t tends to zero, σ t = u + tγ tends to u. However, since u is not an embedded curve, we claim that we have chosen u and γ is such a way that near the self-intersections of u, σ t has the correct over-crossing and under-crossing

74 63 configuration so that its knot type is that of K 2. Case 1.1. u(λ k ) is contained in a cap near the north pole of S 2, that is, ω(p k ) = +1. By construction of γ, we have that γ(λ k ) is contained in a cap near the north pole of S 2. Thus, we have that u(λ k ) γ(λ k ) > u( λ k ) γ( λ k ). By Lemma 6, we conclude that σ t (λ k ) > σ t ( λ k ). Step 5: Analyzing the magnitude of σ t (s) for s {µ t,i, µ t,i } l i=1 Next, we would like to understand how the length of σ t (µ t,k ) compares to that of σ t ( µ t,k ) so that we can apply Theorem 2. Indeed, by our choice of H 1 and because ω(p k ) = +1, we have that 0 (u(λ k ), (0, 0, 1)) < π. Further, recall that u(λ 12 k) = u( λ k ) and by construction of γ, we have that 0 (γ(λ k ), (0, 0, 1)) < π. 12 This implies that 0 (u(λ k ), γ(λ k )) < π 6. However, by construction of γ, we have that 0 (γ( λ k ), (0, 0, 1)) < π 12. Since u( λ k ) and γ( λ k ) are contained in opposite polar caps, we have that 5π 6 < (u( λ k ), γ( λ k )) π. Further, since both u and γ are of unit length at λ k and λ k, we conclude the following: 0.85 < 3 2 < u(λ k) γ(λ k ) u( λ k ) γ( λ k ) < 2

75 64 Now, choose η 1 > 0 so that for all k, if s λ k < η 1 and s λ k < η 1, then the following hold: 0.8 < u(s) γ(s) 1 1 u( s) γ( s) < 0.8 Finally, choose T 0 > 0 so that for all t (0, T 0 ), µ t,k λ k < η 1 by using the argument in the beginning of the proof of Theorem 2 for finding ˇδ. By choice of η 1, we know that the following hold: 0.8 < u(µ t,k ) γ(µ t,k ) 1 1 u( µ t,k ) γ( µ t,k ) < 0.8 In particular, we have that u(µ t,k ) γ(µ t,k ) > u( µ t,k ) γ( µ t,k ). Let I i = {s D s λ i < η 1 } and let I i = {s D s λ i < η 1 } as in Theorem 2. Then let M 0 = min{m 0 (s, s)}, where the minimum is taken over all (s, s) l i=1īi Īi and where M 0 (s, s) is as defined in Lemma 6. Note that M 0 > 0 exists since M 0 (s, s) is continuous taking values in (0, ) and l i=1īi Īi is compact. Then for t (0, T 1 ) where T 1 = min(t 0, M 0 ), Lemma 6 implies σ t (µ t,k ) > σ t ( µ t,k ). Case 1.2. u(λ k ) is contained in a cap near the south pole of S 2. This case is analogous to Case 1.1. Indeed, if u(λ k ) is near the south pole of S 2, we know that u(λ k ) γ(λ k ) < u( λ k ) γ( λ k ). Thus, for t (0, T 1 ) Lemma 6 and

76 65 an argument similar to Case 1.1 implies σ t (λ k ) < σ t ( λ k ) and σ t (µ t,k ) < σ t ( µ t,k ). In summary, γ(λ k ) is near the north pole of S 2 and γ( λ k ) is near the south pole of S 2 by construction of γ. For t (0, T 1 ), we have the following: If u(λ k ) is near the north pole, then σ t (µ t,k ) > σ t ( µ t,k ). If u(λ k ) is near the south pole, then σ t (µ t,k ) < σ t ( µ t,k ). Step 6: Analyzing crossing configuration of σ t Finally, for u(λ k ) near the north pole, by construction of u, we have that ω(p k ) = +1. By definition of ω, this implies that p k is an over-crossing double point of P (ũ), and p k is an under-crossing double point of P (ũ). In contrast, for u(λ k ) near the south pole, we have that ω(p k ) = 1. In this case, p k is an under-crossing double point of P (ũ), and p k is an over-crossing double point of P (ũ). σ t has the same overcrossing and under-crossing configuration as ũ, and σ t is isotopic to one of the 2 l embedded resolutions of u, namely f J0 (u), by Theorem 2. The isotopy H 1 used in reverse direction isotopes f J0 (u) to a curve ṽ in A. Since H 1 (, 1) takes vertical segments in A to radial segments in Y 0, f J0 (ψ(p (ũ)))(s) u(s) for all s implies that P (ṽ) = P (ũ). ṽ and ũ also have the same over-crossing and under-crossing configuration as σ t. Hence, ũ = ṽ = f J0 (u) = σ t for all t (0, T 1 ) where = denotes isotopic. In other words, the knot type of σ t for t (0, T 1 ) is K 2. This concludes Case 1.

77 66 Case 2. K 1 is any knot type, and K 2 is a knot type with the property that there exists a regular projection of K 2 on the plane containing an arc that traverses each crossing once before traversing any crossing a second time. (Obviously, this case contains Case 1, with less restrictions on K 1.) Let γ be a smoothly embedded curve in R 3 whose knot type is that of K 1. Recall that Theorem 2 and Lemma 5 imply that the knot type of σ t is determined by the behavior of γ near the self-intersections of u which lie near the polar caps. As such, we can isotope the curve so that γ follows the great circle through (0, 0, ±1) and ( 1, 0, 0) except for a sufficiently small ball centered at (1, 0, 0). For an example of a suitable γ curve whose knot type is the trefoil, see Figure 4.3. S 2 γ Figure 4.3: Appropriate γ curve near S 2 representing the knot type of a trefoil.

78 67 Once the curve defined by γ has been fixed, we may proceed by defining the parameter values {λ 1, λ 2,..., λ l } and { λ τ(1), λ τ(2),..., λ τ(l) } as explained in Case 1 which depend on u alone and not γ. The remainder of the arguments in Case 1 apply to such a curve. General Case: K 1 and K 2 represent arbitrary knot types. Let ũ denote a smoothly embedded curve in R 3 whose knot type is K 2, and let P (ũ) be a regular projection of ũ on the plane. Without loss of generality, we will assume P (ũ) is a regular curve with transversal self-intersections and no triple (or higher order) self-intersections. Let q denote a point along P (ũ) that is not a double point of a crossing, and fix an orientation along the diagram. As in Case 1, we may label each double point of P (ũ) as either (p i, ω(p i )) or ( p i, ω( p i )). Recall that the set {p 1, p 2,... p l } denotes the points in the plane corresponding to double points of the projection which are traversed first when tracing the diagram beginning with q, and let { p 1, p 2,..., p l } denote the same set of points in the plane but representing the double points along the projection which are traversed second when tracing along the diagram beginning with q. We label p 1,..., p l in the order that they appear on the arc beginning at q, with the understanding that some p j may occur before all of the p i are traversed. Further, ω(p i ) and ω( p i ) indicate whether the point represents an over-crossing point or an under-crossing point. Consider the set of points {p 1, p 2,..., p l }. As in Case 1, following steps 1 and 2, we have u = ψ(p (u)) in S 2 \ {(0, 1, 0)} so that of all of the (first time) overcrossing points contained within the set lie within a small disk centered at the north

79 68 pole while all of the (first time) under-crossing points lie within a small disk centered at the south pole. Choose any regular parametrization of u : D S 2 so that u(λ i ) = p i, u( λ i ) = p i, and 0 λ i, λ j l 0 2 for all i, j {1, 2,..., l}. By our choice of the order of p i, we have that 0 λ 1 < < λ l l 0 2. Unlike Cases 1 and 2, the following condition no longer holds: 0 λ 1 < λ 2 < < λ l < λ τ(1) < λ τ(2) < < λ τ(l) l 0 2. Even though λ i < λ i for all i, there is no guarantee that λ i < λ j for all i, j {1, 2,..., l}. Indeed the λ i parameter values and λ j parameter values can be in mixed order. For example, it might be case that λ i < λ i < λ j. However, we know that the knot type of σ t depends on the behavior near the self-intersections of u. Now, let γ 0 be a smoothly embedded curve in R 3 whose knot type is that of K 1. As in Case 2, we can isotope γ 0 so that it follows the great circle through (0, 0, ±1) and ( 1, 0, 0) on S 2 except for a sufficiently small ball in R 3 centered at (1, 0, 0). Denote this ball by X. We choose a parametrization of γ 0 : D R 3 so that γ 0 (s) for s [0, l 0 2 ] is along the arc of the great circle mentioned above, γ 0 (s) for s ( l 02, l 0 ) is contained in X, and, as usual, γ0 (0) = γ 0 (l 0 ). Thus, γ 0 looks like the example shown in Figure 4.3. Our goal is to have the set of points {γ(λ i )} l i=1 contained in a small cap, denoted C N, at the north pole of S 2 and the set of points {γ( λ i )} l i=1 contained in a small cap, denoted C S, at the south pole of S 2. However, since the λ i and λ j can be in mixed order as discussed above, we must modify our argument in Case 2 and

80 69 isotope γ 0 to γ as follows. Let ϕ 1 : S 2 \ X R 2 be a diffeomorphism, and let γ 0 (s) = ϕ 1 (γ 0 (s)) for s (0, l 0 2 ). By applying Lemma 7, we can isotope γ 0 to γ satisfying the following conditions: The isotopy is the identity outside a compact set. γ(λ i ) ϕ 1 (C N ) and γ( λ j ) ϕ 1 (C S ) for all i, j {1, 2,..., l}. Finally, let γ(s) = ϕ 1 ( γ(s)) for s (0, l 0 2 ) and γ(s) = γ 0 (s) for γ 0 (s) X. Note that continuity of γ follows from the first condition listed above. See Figure 4.4 for a schematic diagram of the above isotopy. γ(λ i ) γ(λ j ) γ 0 (λ j ) S 2 S 2 γ 0 γ 0 ( λ j ) γ γ 0 (λ i ) γ 0 ( λ i ) γ( λ i ) γ( λ j ) Figure 4.4: A schematic of γ 0 and γ representing the knot type of a trefoil before and after isotopy.

81 70 The remainder of the proof follows as in the previous cases. Remark. We believe that the argument made in Case 1.1 and Case 1.2 regarding the over and under-crossing patterns of σ t at the parameter values {{λ i, λ i }} l i=1 can be made in general using Lemma 6. However, such generality was not needed for the above proof. We made use of the fact that u(λ k ) γ(λ k ) u( λ k ) γ( λ k ), which is stronger than simply assuming u(λ k ) γ(λ k ) > u( λ k ) γ( λ k ) or u(λ k ) γ(λ k ) < u( λ k ) γ( λ k ).

82 71 CHAPTER 5 THE SPECIAL CASE OF RIBBONS WITH N AS THE UNIT VECTOR FIELD: FRENET TYPE RIBBONS Suppose we are given a knot K and a unit-speed parametrization γ for K with the properties that γ is C 3 and κ > 0. A natural unit vector field that one might consider in order to produce a ribbon is the outward unit normal vector field. Thus, suppose that u = N. Throughout this chapter, we will investigate the special properties of the pairing of γ and N and determine what the results of this paper imply about ribbons formed by such pairings. We begin by posing questions regarding γ and N. 1. For any arbitrary γ, does u = N imply that the outer ribbon edge is locally embedded? 2. Given γ, is it possible to have points γ(s) and γ( s) satisfying the goal post condition with respect to the vector field N? 3. If there exists points γ(s) and γ( s) satisfying the goal post condition with respect to N, is there a way to modify γ so that (γ, N) Y good? 5.1 Locally Embedded Frenet Type Ribbons We begin by recalling Proposition 2, whose statement was the following: Suppose K is a smooth space curve with nonvanishing curvature parametrized by arclength s, and let u be any C 1 vector field normal to the tangent vector field γ. Since the curvature of γ is nonzero, the Frenet

83 72 frame (T, N, B) exists. Further, since u is normal to T, we can write u(s) = cos α(s)n(s)+sin α(s)b(s) for a smooth function α : D R/2πZ as described above. Then for r > 0, the only way that the outer ribbon edge function θ can fail to be an immersion is if the following two conditions hold: rκ(s) cos α(s) = 1 α (s) + τ(s) = 0 We now provide a result that is a special case of the above proposition. Corollary 4. Given the pairing (γ, N), the outer ribbon edge and hence the rescaled outer ribbon edge, σ t, are always an immersion. Proof. First, we note that when the unit vector field is defined to be u(s) = cos α(s)n(s)+ sin α(s)b(s), it must be the case that α(s) = π. Thus, we have that α (s) = 0 and cos α(s) = 1. After substituting these values into the conditions listed in Proposition 2, the two conditions which must be true in order for the outer ribbon edge to fail to be a local embedding become rκ(s) = 1 τ(s) = 0 However, as specified in Chapter 1, we know that r 0 and κ(s) > 0. Thus, the first equation never holds, implying that the outer ribbon edge is always an immersion.

84 An Example of a Frenet Type Ribbon Having the Goal Post Property Next, we provide a specific example that indicates that it is possible to have goal posts when u = N. Example. A smooth curve with points having the goal post property. γ(s) = Consider the following parametrization: ( ( ( s ( s ( ( s cos(s) sin, cos + sin(s) sin, 2)) 2) 2)) 1 ) 2 cos(s), where s [0, 4π]. This parametrization gives rise to the curve shown in Figure 5.1, whose knot class is that of the unknot. Figure 5.1: Curve defined by γ for s [0, 4π] As seen in the figure, the curve is symmetric about the xy-plane. One may

85 74 readily check that the points γ( π 6 11π ) and γ( ) have the goal post property with respect 6 to N. In this particular example, the goal posts result in a non-embedded outer ribbon edge for widths larger than 0.8. As can be seen in Figure 5.2, self-intersections of the limaçon ribbon occur along the line defined by the apexes of the isosceles triangles formed near the points satisfying the goal post condition. The darkened inner curve on the ribbon corresponds to the initial curve γ. Figure 5.2: Limaçon ribbon for r = 2.

86 Perturbing Frenet Ribbons to Avoid the Goal Post Property The previous example proves that goal posts are possible when u = N. We note that method used to perturb a particular (γ, u) Y so that (γ, u) Y good as in Proposition 8 can no longer by applied to the pairing (γ, N), as a rigid rotation of our unit vector field comes with the same rigid rotation of γ. Such rotations maintain existing goal posts. The crucial point is that u is directly related to γ when u = N. Thus, alternative means must be used, which brings us to our next proposition. Proposition 9. Let (γ 0, N 0 ) Y be given, where N 0 (s) has finitely many selfintersections on S 2, each of which correspond to exactly two parameter values. Of these, let {{λ ij, λ ij }} k j=1 for k l denote the set of parameter values at which goal posts occur. For an appropriate choice of bump function ϕ and a unit vector u which is not perpendicular to {±N 0 (λ ij )} k j=1, there exists a δ 0 > 0 such that for all δ (0, δ 0 ), γ δ = γ 0 + δϕu satisfies (γ δ, N δ ) Y good. In particular, we can choose ϕ so that except for 2k small neighborhoods away from the self-intersections of N 0, we have that ϕ = 0, and hence, N δ = N 0 and γ δ is either a translation of γ or it is identical to γ. Proof. Suppose there exists a pair of parameter values λ λ such that γ 0 (λ) and γ 0 ( λ) have the goal post property with respect to N 0. By definition, this implies that N 0 (λ) = N 0 ( λ) and N 0 (λ) (γ 0 (λ) γ 0 ( λ)) = 0. Since N 0 has only finitely many self-intersections, there exists an ε(λ) > 0 such that B S 2(ε(λ), N 0 (λ)) contains only one self-intersection of N 0, namely N 0 (λ) = N 0 ( λ) and λ and λ

87 76 belong to two different components of N 1 0 (B S 2(ε(λ), N 0 (λ)). Choose a 1 (λ) and a 2 (λ) satisfying a 1 (λ) < λ < a 2 (λ) so that the following conditions hold: { N 0 (s) s [a 1 (λ), a 2 (λ)]} B S 2(ε(λ), N 0 (λ)) and { N 0 (a 1 (λ)), N 0 (a 2 (λ))} B S 2(ε(λ), N 0 (λ)) \ B S 2 ( ) ε(λ), N 3 0(λ) Further, choose η(λ) > 0 so that { N 0 (s) s [a 1 (λ) η(λ), a 1 (λ) + η(λ)] [a 2 (λ) η(λ), a 2 (λ) + η(λ)]} See Figure 5.3. B(ε(λ), N 0 (λ)) \ B ( ) ε(λ) 3, N 0(λ). B(ε(λ), N 0 (λ)) ( ) ε(λ) B 3, N 0(λ) N 0 (a 2 (λ)) N 0 (λ) = N 0 ( λ) N 0 (a 2 (λ) η(λ)) N 0 (a 2 (λ) + η(λ)) N 0 (a 1 (λ)) N 0 (a 1 (λ) + η(λ)) N 0 (a 1 (λ) η(λ)) Figure 5.3: An ε(λ)-neighborhood about N 0 (λ) = N 0 ( λ). Let I 1 (λ) D denote the interval [a 1 (λ) η(λ), a 2 (λ) + η(λ)] where D is the domain of both N 0 and γ 0. Our goal is to define a modification γ δ of γ 0 on I 1 (λ)

88 77 such that N δ (λ) = N δ ( λ) but N δ (λ) (γ δ (λ) γ δ ( λ)) 0. Choose a C bump function ϕ : D [0, 1] such that the following hold: ϕ(s) = 1 for all s [a 1 (λ) + η(λ), a 2 (λ) η(λ)] 0 < ϕ(s) < 1 for all s I 1 (λ) \ [a 1 (λ) + η 2 (λ), a 2 (λ) η(λ)] ϕ(s) = 0 for all s D = D \ I 1 (λ) See Figure 5.4 for an example of a bump function satisfying the above criteria. ϕ(s) 1 a 1 (λ) η(λ) a 1 (λ) + η(λ) λ a 2 (λ) η(λ) a 2 (λ) + η(λ) s Figure 5.4: Example of a suitable bump function ϕ. Note that our conditions on ϕ imply that ϕ (s) = 0 for s D [a 1 (λ) + η(λ), a 2 (λ) η(λ)], and ϕ (s) is bounded otherwise. The same holds true for ϕ (s). Fix a unit vector u such that u is not perpendicular to ±N 0 (λ). We now define the following modification of γ 0 for δ > 0: γ δ (s) = γ 0 (s) + δϕ(s)u.

89 78 We first note that Lemma 2 implies that there exists a δ 1 > 0 such that for δ (0, δ 1 ), γ δ is an embedded curve. Actually, γ δ converges to γ 0 in a C -sense. Consequently, N δ converges to N 0 uniformly in C 1 -sense. Notice that for s D, γ δ (s) = γ 0 (s), and for s [a 1 (λ) + η(λ), a 2 (λ) η(λ)], γ δ (s) is a translation of γ 0 (s). Thus, for s D [a 1 (λ) + η(λ), a 2 (λ) η(λ)], N δ (s) = N 0 (s). Let I 2 (λ) I 1 (λ) denote the set [a 1 (λ) η(λ), a 1 (λ) + η(λ)] [a 2 (λ) η(λ), a 2 (λ) + η(λ)]. We have that N δ (s) N 0 (s) implies s I 2 (λ). Let ( ( )) ε(λ) δ 2 (λ) = min dist N 0 (s), B s I 2 (λ) 3, N 0(λ). Now, since N δ δ 0 N 0 uniformly in C 1 -sense, we can choose δ 3 (λ) > 0 corresponding to ε(λ) 4 as in Lemma 2 applied to N 0 so that all self-intersections of N δ are within ε(λ) 4 neighborhoods of self-intersections of N 0. We further choose δ 3 (λ) > 0 small enough so that for all δ (0, δ 3 (λ)), N δ (s) ( N 0 (s)) < δ 2 (λ). ( ) Observe that our choices of δ 2 and δ 3 imply that N δ (s) / B ε(λ), N 3 0(λ) for all s I 2 (λ) and for all δ (0, δ 3 ). Finally, let δ 0 = min(δ 1, δ 3 ). For δ (0, δ 0 ), we note the following properties of γ δ and N δ at λ and λ: γ δ (λ) = γ 0 (λ) + δu γ δ ( λ) = γ 0 ( λ) N δ (λ) = N 0 (λ) = N δ ( λ) = N 0 ( λ) The above equalities imply the following: N δ (λ) (γ δ (λ) γ δ ( λ)) = N 0 (λ) [(γ 0 (λ) γ 0 ( λ)) + δu ] = N 0 (λ) δu 0

90 79 The second equality follows from the fact that γ 0 (λ) and γ 0 ( λ) were assumed to have the goal post property with respect to N 0. Lastly, we note that since u 0 was chosen so that it was not normal to N 0, we conclude that N δ (λ) (γ δ (λ) γ δ ( λ)) 0. Thus, γ δ (λ) and γ δ ( λ) do not have the goal post property with respect to N δ. We note that no new goal posts were introduced after modifying γ 0 to γ δ. Indeed, since δ < δ 3, which was chosen corresponding to ε(λ) 4 as in Lemma 2, we conclude that any self-intersections of N δ must lie within an ε(λ) -neighborhood of a 4 self-intersection of N 0. Recall that the only portions of N δ (s) that differed from N 0 (s) are when s I 2 (λ). Thus, the only way a new goal post could have been ( ) introduced is if a new self-intersection was formed within B ε(λ), N(λ) by the set 4 ( ) { N δ (s) s I 2 (λ)}, which is disjoint from B ε(λ), N 3 0(λ). Therefore, no such intersection may exist. Thus, by modifying the pairing (γ 0, N 0 ) to (γ δ, N δ ), we have reduced the overall number of goal posts by one for δ (0, δ 0 ). Since we can make each successive alteration of γ small enough so as to not reintroduce any previously eliminated goal posts and since we may choose our unit vector u so that it is not perpendicular to {±N 0 (λ ij )} k j=1, we can proceed inductively to eliminate all goal posts.

91 80 CHAPTER 6 EXAMPLES We now present concrete examples to illustrate the theory developed above. We note that we extensively used the programs Maple by Maplesoft [9] and KnotPlot by R. Scharein [12] when generating the examples contained in this chapter. In general, we will explore particular examples of ribbons by following these steps: 1. Approximate the parameter values at which u has self-intersections 2. Calculate and graph the values of the dot product u(s) (γ(s) γ(t)) for s and t values near the approximate double points of u to check that the pairing (γ, u) contains no goal posts 3. Gradually expand the ribbon to see the knot types that the outer ribbon edge takes on We note that the knot type of the outer ribbon edge was determined by visual inspection and computationally. We first exported a matrix from Maple containing three-dimensional vertices of a polygonal approximation of the outer ribbon edge. We then imported the data points into KnotPlot in order to calculate the HOMFLY polynomial as well as relax the imported knot to better visualize the knot type. In the tables below, we only listed the radius values for which the knot type is evident. We omitted some intermediate radius values for which the knot type could not be positively identified.

92 81 Unlike the theory above, the parametrizations used for γ are not given by arclength; that was a convenience for proofs but is not necessary to illustrate the phenomena. Also, for each example, we checked that the parametrizations γ and u are regular by graphing the magnitude of their derivatives. Throughout this chapter, we analyze the configuration of the outer ribbon edge defined by γ + ru rather than the normalized outer ribbon edge defined by tγ + u where t = 1. Note that large values of r correspond to small values of t. For such r values of r and t, it is difficult to determine the crossing configuration of the outer ribbon edge. An obvious, but still open question for pairings (γ, u) containing no goal posts remains: How big must r be to guarantee that the knot type has stabilized? 6.1 The Trefoil and Figure Eight Knot In this section, we choose a particular parametrization γ of a left-handed trefoil knot, and u is a parametrization of a curve on S 2 obtained by normalizing a particular parametrization of the figure eight knot Example 1 Let the parametrization of the trefoil be given by γ(s) = ( 10 cos s 2 cos 5s + 15 sin 2s, 15 cos 2s + 10 sin s 2 sin 5s, 10 cos 3s), for s [0, 2π] as given in [11]. Figure 6.1 displays the curve given by γ. For the parametrization of the figure-eight knot, we use the following para-

93 82 Figure 6.1: γ(s) for s [0, 2π] metric coordinate equations as given in [14]: x = 32 cos s 51 sin s 104 cos 2s 34 sin 2s cos 3s 91 sin 3s y = 94 cos s + 41 sin s cos 2s 68 cos 3s 124 sin 3s z = 16 cos s + 73 sin s 211 cos 2s 39 sin 2s 99 cos 3s 21 sin 3s This parametrization gives rise to the following configuration of the figure eight knot shown in Figure 6.2. Figure 6.2: Figure eight defined by (x, y, z) for s [0, 2π]

94 83 To make u a unit vector field, we use the normalized parametrization of the above equations. We first note that u is generic, as it has four transversal selfintersections. Figure 6.3 displays the non-embedded space curve winding about a wire frame of S 2. Figure 6.3: u(s) for s [0, 2π] Next, we note that there does not exist a set of parameter values satisfying the goal post property. Indeed, using the fsolve command in Maple, the self-intersections of u are found to occur approximately at the following parameter values: λ 1 = λ 1 = λ 2 = λ 2 = λ 3 = λ 3 = λ 4 = λ 4 = Recall that two distinct parameter values s and t have the goal post property if u(s) = u(t) and u(s) (γ(s) γ(t)) = 0. Using the plot3d and dotprod commands in Maple, we obtain graphs of

95 84 the value of the dot product d = u(s) (γ(s) γ(t)) about neighborhoods of the approximate self-intersection parameter values listed above. Figure 6.4 displays these portions of the generated surface. Notice that the value of the dot product d, which is displayed along the z-axis, is bounded away from zero. This confirms that the pairing (γ, u) does not have the goal post property. Thus, (γ, u) Y good, where Y good is defined in Section 3.1. Now that we have confirmed that no goal posts exist, Theorem 1 implies that the outer ribbon edge stabilizes to a unique knot type. Furthermore, Theorem 2 states that the knot type of the stabilized outer ribbon edge must be one of at most 2 4, or 16, possible knot types. However, since we know that there are only four knot types with four or fewer crossings, namely the unknot, the left and right trefoils, and the figure eight, we can reduce the number of possible knot types from sixteen to four. We now investigate the knot type of the outer ribbon edge defined by θ(s) = γ(s) + ru(s) as r increases. Table 6.1 shows the various knot types that the outer ribbon edge actually takes on, eventually stabilizing to the unknot Example 2 While maintaining the parametrization of u, we now investigate how changing the parametrization of γ affects the knot type of the outer ribbon edge. Let γ be parametrized in the following way: γ(s) = ((cos 3s + 2) cos 2s, (cos 3s + 2) sin 2s, sin 3s).

96 Figure 6.4: Portions of the surface determined by d = u(s) (γ(s) γ(t)) near the approximate self-intersections values of u 85

97 86 Radius Range Knot Type Left Trefoil Unknot Unknot ,575 Left Trefoil 2,293 Unknot Table 6.1: Knot type of outer ribbon edge for particular radius ranges Figure 6.5 shows the curve defined by γ. Figure 6.5: γ(s) for s [0, 2π] Since the parametrization of u did not change, the approximate parameter values at which self-intersections of u occur are the same as those listed in Section Similar to the previous example, we see that (γ, u) does not have the goal post

98 87 property, as displayed by Figure 6.6. Thus, (γ, u) Y good. Now we are ready to determine the intermediate and final knot types of the outer ribbon edge. See Table 6.2 for a complete list of the knot types of the outer ribbon edge for various ranges of radii. Radius Range Knot Type Left Trefoil Unknot Figure Eight 2,167 Unknot Table 6.2: Knot type of outer ribbon edge for particular radius ranges Although this ribbon also stabilized to the unknot, we see that it took on the knot type of the figure eight for a range of intermediate radius values. This differs from our example in Section The Trefoil and Granny Knot In this section, γ is the parametrization of the granny knot as given in [14], and u is the parametrization of the curve on S 2 obtained by normalizing the parametrization of the trefoil used in Section Let the parametrization γ of the granny knot

99 Figure 6.6: Portions of the surface determined by d = u(s) (γ(s) γ(t)) near the approximate self-intersections values of u 88

100 89 be given by the following parametric coordinate equations: x = 22 cos s 128 sin s 44 cos 3s 78 sin 3s y = 10 cos 2s 27 sin 2s + 38 cos 4s + 46 sin 4s z = 70 cos 3s 40 sin 3s The above parametrization gives rise to the following configuration of the granny knot shown in Figure 6.7. Figure 6.7: Granny defined by (x, y, z) for s [0, 2π] For an image of the embedded trefoil being used in this example, refer to Figure 6.1. As in Section 6.1.1, we note that the normalized parametrization of this trefoil is generic, as it has three transversal self-intersections. See Figure 6.8. For this particular parametrization of the trefoil, u has self-intersections near

101 90 Figure 6.8: u(s) for s [0, 2π] the following pairs of parameter values: λ 1 = λ 1 = λ 2 = λ 2 = λ 3 = λ 3 = Unlike any of our previous examples, we see that γ(λ 2 ) and γ( λ 2 ) have the goal post property with respect to u. Indeed, one may verify that Furthermore, we have the following: ( π ) ( ) 3π u(λ 2 ) = u = (0, 1, 0) = u = u( λ 2 ). 2 2 ( π ) γ(λ 2 ) γ( λ 2 ) = γ γ 2 ( ) 3π = ( 100, 0, 80). 2 Since (γ, u) has the goal post property, we will apply the method described in the proof of Proposition 8. Although we know that we may perturb u by an arbitrarily small amount in order to eliminate the goal post condition, for ease of notation, we will rotate the initial parametrization of the trefoil thirty degrees about the z-axis.

102 91 As such, our new parametrization of our unit vector field becomes 3 x = 2 ( 10 cos s 2 cos 5s + 15 sin 2s) 1 ( 15 cos 2s + 10 sin s 2 sin 5s) 2 y = ( 10 cos s 2 cos 5s + 15 sin 2s) + ( 15 cos 2s + 10 sin s 2 sin 5s) 2 z = 10 cos 3s Let u 0 denote the rotated unit vector field. Since we applied a rigid rotation to u, the parameter values corresponding to self-intersections of u 0 remain the same as those corresponding to the self-intersections of u. However, we now have the following: ( ( π ) u 0 (λ 2 ) = u 0 = 1 ) ( 3 3π 2 2, 2, 0 = u 0 2 ) = u 0 ( λ 2 ). Thus, goal posts no longer exist at γ(λ 2 ) and γ( λ 2 ). As in the previous examples, Figure 6.9 displays the portions of the surface defined by the dot product d = u 0 (s) (γ(s) γ(t)) for s and t near the approximated values {λ i, λ i } 3 i=1, which confirms that there are no goal posts. Whence, (γ, u 0 ) Y good. Finally, we are ready to determine the intermediate and final knot types of the outer ribbon edge. See Table 6.3 for a complete list of the knot types of the outer ribbon edge for various ranges of radii. 6.3 Forms of Frenet Ribbons In this section, we analyze a class of examples whose unit vector field u is the outward unit normal vector field. In particular, we analyze Frenet type ribbons formed from unknots that resemble a slinky about S 1. That is, they are (1, q)-torus knots.

103 Figure 6.9: Portions of the surface determined by d = u 0 (s) (γ(s) γ(t)) near the approximate self-intersections values of u 92

104 93 Radius Range Knot Type 0-22 Granny 23 Unknot Table 6.3: Knot type of outer ribbon edge for particular radius ranges (1,2)-Torus Knot We use the following parametrization of the (1, 2)-torus knot: γ(s) = ((cos 2s + 2) cos s, (cos 2s + 2) sin s, sin 2t), for s [0, 2π]. Figure 6.10 displays the curve given by γ. Figure 6.10: γ(s) for s [0, 2π] The outward unit normal vector field, denoted N, associated to γ as defined above gives rise to the curve on S 2 shown in Figure For the given parametrization of γ, the approximate parameter values at which

105 94 Figure 6.11: N(s) for s [0, 2π] N has self-intersections are the following: λ 1 = λ 1 = λ 2 = λ 2 = λ 3 = λ 3 = λ 4 = λ 4 = Figure 6.12 indicates that (γ, N) does not have the goal post property. Table 6.4 displays that knot types that the outer ribbon edge takes on for this particular (γ, N) pairing. Radius Range Knot Type Unknot > 1.1 Right Trefoil Table 6.4: Knot type of outer ribbon edge for particular radius ranges

106 Figure 6.12: Portions of the surface determined by d = N(s) (γ(s) γ(t)) near the approximate self-intersections values of N 95

107 (1,3)-Torus Knot We use the following parametrization of the (1, 3)-torus knot: γ(s) = ((cos 3s + 2) cos s, (cos 3s + 2) sin s, sin 3t), for s [0, 2π]. Figure 6.13 displays the curve given by γ. Figure 6.13: γ(s) for s [0, 2π] The outward unit normal vector field associated to γ as defined above gives rise to the curve on S 2 shown in Figure Notice that N has only transversal self-intersections. Furthermore, the selfintersections were found to occur approximately at the following sets of parameter values: λ 1 = 0 λ1 = λ 4 = λ4 = λ 7 = λ7 = λ 2 = λ2 = λ 5 = λ5 = λ 8 = λ8 = λ 3 = λ3 = λ 6 = λ6 = λ 9 = λ9 = Due to the large number of self-intersections of N, we will omit the graphs