For Ramin. From Jonathan December 9, 2014

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1 For Ramin From Jonathan December 9, Foundations. 1.0 Overview. Traditionally, knot diagrams are employed as a device which converts a topological object into a combinatorial one. One begins with a knot (a closed curve smoothly embedded in 3-space) and represents it as a 4-regular plane graph with a bit of information attached at each graph node. Diagrams of higherdimensional knots are also combinatorial objects. 1 We will take the opposite approach, regarding diagrams as fundamental objects and deriving topological objects to represent them. For example, Kauffman used this approach to introduce virtual knots, first establishing the rules for virtual knot diagrams and then defining a topological object (namely, curves in thickened stablized surfaces) to represent this class. Our approach is broken down into nine steps. 1. Generic maps. The domain of any diagrammatic theory is objects configured in generic position. A general definition for this notion is required. 2. Codimension-1 generic maps. Features common to generic maps in dimension-n, codimension-1 must be classified for use in a diagrammatic theory. In particular, we need to understand what types of singularities can occur. 3. Crossings and branches. The self-intersection set in a generic map consists of a collection of crossings which may intersect themselves and each other. A mechanism for separating and distinguishing the crossings is needed. The branches of 1 They are combinatorial in the sense that, like plane graphs, their essential structure can be naturally encoded as a finite string of data. Carter & Saito give method for encoding 2-knot diagrams; in higher dimensions the codes would presumably be more complex. The topic will not be discussed here. 1

2 a generic map also may intersect in the image, so the branch set needs to be separated into individual branches. 4. Moves. A move is a homotopy for which, at a single instant of time at a single point in the image, the map just barely fails to be generic. Moves can be classified into types. This notion must be made precise. 5. Moves for dimension n, codimension 1. There are only finitely many move types in dimension n. To define a diagrammatic theory, one first needs to catalog them. 6. Crossing data. A diagram consists of a generic map together with crossing data, which is a label on each crossing that designates it virtual or classical, and distinguishes an overpass and an underpass in case of the latter. As the diagram moves the crossing data is carried with it continously. 7. Forbidden crossings We now have the basic machinery for determining a diagrammatic theory. This is done in two steps. The first step is to set rules on crossing data. Each type of singularity (double-points, triple-points, etc.) is refined into subtypes depending on how crossing data is assigned to the crossings present at the singularity. When defining a particular theory, some of these subtypes can be declared forbidden. 8. Forbidden moves. The second step for determining a diagrammatic theory is to set rules on moves. Each move type is refined into several subtypes depending on the crossing data present at the move site. When defining a particular theory, some of these subtypes can be declared forbidden. (Some subtypes may already have been killed off by the crossing rules set in the previous step.) 9. Topological invariants. Now that your diagrammatic theory is determined (you ve chosen a dimension n and you ve set your crossing rules and your move rules) it s time to find a topological model. This is a functor from the category of diagrams (where morphisms are diagram equivalence under moves) to some topologically-flavored category (such as embeddings into R n+2, or embeddings into stabilized n + 2-folds). Finding topological invariants is the central problem and motivation for the present writing. Steps 1, 3, 4, and 6 establish a theoretical background for diagrammatic theories in general. In steps 2 and 5 this background is used to diagram singularities and diagram moves particular to dimension n. Steps 7 and 8 are where we set 2

3 the parameters of a particular diagrammatic theory. Step 9 is an open-ended quest for topological meaning in the theory. Notation and conventions. The following symbols will be used throughout this writing. X is a smooth manifold of dimension n, and Y is a smooth manifold of dimension m, both without boundary. We will usually take Y = R n+1 (with standard differentiable structure) but this will not be assumed in sections 1.1 ( Generic maps ) or 1.4 ( Moves ). f : X Y is a smooth map. B X is the branch set of X, that is, the set of points where the derivative of f vanishes. D X is the double-point set, which is a slight misnomer because D is actually the closure of the set of all points with multiplicity 2 or more: D = closure ({ x X f 1 (f(x)) {x} }). P D B is the set of pure double-points, that is, non-branch points with multiplicity exactly 2. Except when otherwise stated, the index of a homotopy (usually denoted s or t) always ranges over [0, 1]. 1.1 Generic maps. A map is considered generic if its essential topological properties are invariant under small perturbations. Definition: Generic map. A smooth map f : X Y is generic if for every homotopy {f t } (with f 0 = f, and f t smooth for all t) there exists a number T > 0 and isotopies {α t : X X} and {β t : Y Y } satisfying for all t < T. β t f t α t = f Note that we use homotopies consisting of smooth maps to test the genericity of f. This effectively bars potential false-negative results arising from testhomotopies with pathological behaviors such as tying a knot in X at a single point. A brief digression: If one wanted to develop a theory of generic maps in the PL category instead of Smooth, the above definition would need to be revised, 3

4 as follows. Definition: PL generic map. A PL map f : X Y is generic if for every nice homotopy {f t } (with f 0 = f) there exists a number T > 0 and PL isotopies {α t : X X} and {β t : Y Y } satisfying β t f t α t = f for all t < T. A homotopy {f t : X Y } of PL maps is considered nice if there exist an isotopy of triangulations {a t : A X}, where A is a fixed simplicial complex, a triangulation b : B (Y I) whose cross-sections are triangulations b t : B t (Y {t}) = Y (where B = t B t ), such that b t f t a t is simplicial for each t. The niceness condition imposed on the test homotopies emulates the smooth version s exclusion of false-negatives. We will not use the PL category again in this writing, and we presently return to the category Smooth. The next definition will be used in section 1.4 to describe local moves. Definition: Generic at y. A smooth map f : X Y is generic at the point y Y if the restricted map f f 1 (V ) is generic for some closed neighborhood V Y of y. We will find it convenient to slightly weaken the definition of generic in the case that Y has a boundary. This will be important, for example, when considering level-preserving maps X I Y I, and also when considering maps of closed disks D n B n+1 that map boundary to boundary. Definition: Generic map (continued). When Y has nonempty boundary, a map f : X Y is considered generic if it is generic throughout the interior of Y. As preparation for section 1.4 (Moves), let s take a quick look at continuous families of generic maps. Theorem 50. If {f s : X Y } is a homotopy of smooth maps, with f s generic for each s, then the associated level-preserving map F : X I Y I is generic. (By the preceding definition, the genericity of F does not depend on its behavior at the ends of I, that is, at s = 0, 1.) 4

5 Proof. Consider a homotopy {F t : X I Y I} (where each F t is level-preserving, F 0 = F, and F t is smooth for all t). Restrict the domain to X {s 0 } (for some s 0 I). The restricted homotopy {F t : X {s 0 } Y {s 0 }} is equal to a homotopy {(f s0 ) t } with (f s0 ) 0 = f s0. This homotopy consists of smooth maps, so we can choose isotopies {(α s0 ) t } and {(β s0 ) t } of X and Y, respectively, satisfying (β s0 ) t (f s0 ) t (α s0 ) t = (f s0 ) for all t < T (for some T > 0). Furthermore, these isotopies can be chosen for each s I so that they vary continuously with s. This gives two continuous families of isotopies {α s,t } and {β s,t } of X and Y respectively. The associated isotopies {A t : X I X I} and {B t : Y I Y I} satisfy B t F t A t = F for all t < T (for some T > 0). Thus F is generic. A weak converse of this theorem, called Theorem 100, will be stated in the discussion of moves (section 1.4). 1.2 Generic maps for dimension-n, codimension-1. The idea here is to understand what types of singularities occur in dimension-n knot diagrams. For example: In dimension 0, a knot diagram consists of a finite sequence of distinct points in the line R 1. In dimension 1, a knot diagram is some curves immersed in R 2, with finitely many transverse crossings of multiplicity 2. In dimension 2, a knot diagram is some surfaces mapped into R 3 with nonvanishing derivative except at finitely many branch points, where the image locally looks like the cone over a figure-8. The surface may cross itself transversely along finitely many immersed circles and open arcs; these crossings are of multiplicity 2 except at finitely many triple points where the image locally looks like the intersection of the three coordinate planes in R 3. These descriptions get longer and more complex as n gets bigger, so it would be nice to have a single, universal result about what features occur in these knot diagrams. Roseman has provided this in the form of a definition, now presented as a theorem. Theorem. A smooth map f : X R n+1 is generic if and only if: The restricted map f X B is generic in the usual sense of smooth topology. (This means all the crossings are transverse, and a crossing of multiplicity k locally looks like k of the n-dimensional coordinate hyperplanes at the origin of R n+1.) 5

6 Roseman s Rules for crossings and branches: a) B is a closed (n 2)-dimensional submanifold of X. b) B is a submanifold of D and for any b B there is a small (n 1)- dimensional open subdisk N with b N D such that N B has two components N 0 and N 1, each of which is an (n 1)-disk which is embedded by the restriction of f but with f(n 0 ) = f(n 1 ). c) D is the union of immersed closed (n 1)-dimensional manifolds in X with normal crossings. (A crossing is normal if the differential puts the tangent planes in general position.) d) f B is an immersion of B with normal crossings. e) B meets the crossing set of D transversely. f) The crossing set of f(b) is transverse to the crossing set of f(d B). Condition (b) from Roseman s Rules can be restated more evocatively. The image of a neighborhood of a point in B looks like the cone over a figure-8 curve, times an (n 2)-ball. Other types of branch point are never generic. Example. A 2-sphere can be smoothly mapped (via f) into R 3 so that its image is the suspension over some generic closed curve C in the plane. If C is a simple closed curve or a figure-8 curve, then f is generic. But if C is any other generic plane curve, then f fails to be generic at the two branch points. In this case, f can be perturbed into a different form such that every branch has the form of a cone over a figure-8 curve. 2 The example generalizes to higher dimensions. 1.3 Crossings and branches. The double-point set D is the union of immersed manifolds which are matched in pairs (or, in come cases, self-matched) by f to form crossings of f(x). These immersed manifolds may cross themselves and each other in X, so we need a way to separate D into individual crossings. For example, the depicted map f has a double-point set which is the union of six immersed circles. These circles can be paired up into three individual crossings, as shown. picture f rom Roseman Claim (Roseman). There exist: 2 Warning: This perturbation may not be possible respecting classical crossing data. Such diagrams can be regarded as projections of non-locally-flat knots, which may not be in the range of the theory s topological invariant. 6

7 a compact (n 1)-dimensional manifold Γ, not necessarily connected, possibly with boundary), an immersion γ : Γ R n+1 such that γ(γ) = f(d) and γ γ 1 (f(p )) is one-to-one (recall that P is the set of pure double-points), an (n 1)-dimensional manifold, not necessarily connected, but without boundary, an immersion δ : X such that δ( ) = D and δ(δ 1 (P )) is one-to-one, a unique map α : Γ such that the following diagram commutes: α Γ δ γ D f f(d) Furthermore, α δ 1 (B) is a two-to-one map onto Γ γ 1 (f(b)). Definition: Crossing. A crossing of f is the restriction of the above commuting diagram to a component of Γ, i α Γi δ γ where Γ i is a connected component of Γ f(d) i = γ(γ i ) D i = f 1 (f(d) i ) i = δ 1 (D i ). D i f f(d)i These restricted commuting diagrams separate the set D into pairs (or singles) of immersed connected manifolds, D i, which are the individual crossings of f. Definition: Branch. The components of the branch-set B are denoted B i. They are the individual branches of f. Note that each branch is a closed connected (n 2)-dimensional manifold embedded in X, and that two branches never intersect in X (although their images under f may intersect in R n+1, if n 5). 7

8 1.4 Moves In this section Y may be any smooth manifold. If f 0 and f 1 are generic maps, then there is a homotopy {f t : X Y } which transitions from one map to the other via a sequence of elementary moves. Furthermore, any homotopy from f 0 to f 1 can be slightly deformed into a sequence of elementary moves. These two facts will now be stated precisely. Definition: Local move. Let V be the union of finitely many disjoint n-dimensional disks V i. Let B be a (n + 1)-dimensional ball. (Everything has the standard smooth structure.) A homotopy of smooth maps {h t : V B} is called a local move if: h t ( V ) B for all t the associated level-preserving map H : V I B I is generic the maps h t are generic for all t except t =.5 the map h.5 fails to be generic at just one point in B the restricted maps h t V Vi, which omit one disk from the domain, are generic for all t including t =.5, for each i. Definition: Move type. Two local moves {h t }, {h t} are said to be of the same type if there are isotopies {ψ t }, {ω t } that make the following diagram commute for all t: V ψt V h t h t When {h t } is of the same type as either {h t} or {h 1 t}, then we say {h t } and {h t} are of the same type modulo time-reversal. Move types are equivalence classes; so are move types modulo time-reversal. Definition: To undergo a move. Let {f t : X Y } be a homotopy of smooth maps. Let {h s : V B} be a local move representing move type [h]. The homotopy {f t } is said to undergo the move [h] at time t = t 0, at the point y Y if there exist: a strictly-increasing continuous function t : I I with t(.5) = t 0 homotopies of embeddings {v s }, {b s } B ωt such that the following diagram commutes for all s B V vs X h s f t(s) B bs Y 8

9 and f t(s) v s (V ) = f t(s) (X) b s (B). Note that this is well-defined: The existence of t, {v s }, and {b s } does not depend on the choice of representative {h s } [h]. We now deliver a weak converse of theorem 50, as promised. Definition: Generic homotopy. A homotopy of smooth maps {f t : X Y } is called generic if f 0 and f 1 are both generic maps, and the associated level-preserving map F : X I Y I is generic. (Recall that F is considered generic if it is generic throughout the interior of Y I; so, we can ignore the levels 0, 1 I when making this judgment.) Theorem 100 Suppose {f t : X Y } is a generic homotopy. Then: f t is generic for all but finitely many t at these times, the set of points in Y where f t fails to be generic is discrete at each of these points in time and space, {f t } undergoes a move. Definition: Sequence of moves. If a generic homotopy undergoes only one move at a time (i.e., it is a Morse function), then it is called a sequence of moves. Theorem. Let {f t : X Y } be a generic homotopy, and also assume X is compact. Then the word discrete can be replaced with finite. Indeed, {f t } can be slightly deformed to become a sequence of moves. That is, there is a selfhomeomorphism α of X I which preserves x-coordinates, such that the homotopy {f t} is a sequence of moves, where f t = (π F α) X {t} for each t I. (Here, π : Y I Y is projection.) The self-homeomorphism α can be chosen to coincide with the identity map outside of a small neighborhood of each move site. 9

10 Evolution of crossings. (This part is surprisingly hard to explain!) As a knot diagram moves, its crossings move along with it continously. Let {f t : X Y } be a generic homotopy. Let t 0 I be a moment in time at which f t0 has at least one crossing. Suppose the following commuting diagram represents one crossing of f t0. Call this crossing c 0. i α Γi δ γ D i f t0 ft0 (D) i The homotopy {f t } is associated with a generic level-preserving map F : X I Y I, and this map has a crossing, call it C, whose cross-section at t = t 0 includes c 0 as a component. This is now explained. Suppose C is represented by the commuting diagram α i Γ i δ γ F F (D ) i D i The spaces i and Γ i each have a time function defined by composing δ or γ with projection to I, and these time functions commute with α, so that, in a sense, α is level-preserving. If we take the cross-section of Γ i at the time t = t 0, the result will include Γ i as a component. Broadening this view to regard a continuum of times, the inclusion of Γ i as a component is actually just one of a continuous time-indexed family of inclusion maps. If the crossing is ever involved in a move, then things get complicated. 1.5 Move types for dimension-n, codimension-1 We again assume, as usual, that Y = R n+1. Theorem (Roseman). There are only finitely many move types for each dimension n. The number of types increases with n. Roseman s method for cataloging these moves is apparently general enough to work in any dimension, but I find no evidence that anyone has actually done this for dimensions higher than n = 3. In this paper we will only study examples with n 2, so those moves are listed below. Theorem: Point-pass The only move for n = 0 is the point-pass : Two points that switch positions on a line. This move is time-symmetric, so the count of one move is the same whether or not we mod out by time-reversal. picture 10

11 Theorem: Reidemeister moves There are three moves for n = 1, the Reidemeister moves. Note that moves Ω 1 and Ω 2 are not time-symmetric, but Ω 3 is; thus we count five moves, total, without modding out time-reversal. pictures Theorem: Roseman moves There are seven moves for n = 2, the Roseman moves. Only the so-called tetrahedral move is time-symmetric, so we count thirteen moves without modding out by time-reversal. pictures Roseman has also studied the moves for n = 3; if I recall correctly there are twelve. True or false? For each n 1 there exist moves involving k disks, V = k i=1 V i, for k = 1,..., n + 2. There is exactly one move involving n + 2 disks, and this move is the only one with time-symmetry. Specifically, this is the simplex-dualizer move. 11

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