KNOTS AND THEIR CURVATURES

Transcription

1 KNOTS AND THEIR CURVATURES LIVIU I. NICOLAESCU ABSTRACT. I dscuss an old result of John Mlnor statng roughly that f a closed curve n sace s not too curved then t cannot be knotted. CONTENTS. The total curvature of a olygonal curve 2. A robablstc nterretaton of the total curvature 2 3. The total curvature of a smooth closed curve 4 4. Total curvature and knottng 6 References 7. THE TOTAL CURVATURE OF A POLYGONAL CURVE An (orented) olygonal knot (or curve) s a closed curve C n R 3, wthout selfntersectons, obtaned by successvely jonng n dstnct onts,..., n, n+ = R 3 va straght lne segments [ 2 ],..., [ n n ], [ n, ]. The onts are called the vertces of the olygonal knot C. We denote by V C the set of vertces. To each orented edge [, + ], n, we assocate the unt vector γ := + +. Denote by S 2 the unt shere n R 3 centered at the orgn. We obtan n ths fashon a ma γ = γ C : V C S 2, γ( ) = γ. Ths s known as the Gauss ma of the olygonal knot C. Let [0, π) be the angle between γ and γ + ; see Fgure. We obtan n ths fashon a ma = C : V C [0, π), ( ) =. We defne the total curvature of C to be the ostve real number K(C) = C () = n. (.) V C = Date: Started August 8, Comleted Comleted on Aug. 9, Last modfed on Setember, 2009.

2 2 LIVIU I. NICOLAESCU FIGURE. A lanar olygonal knot. Observe that f C s a convex, lanar olygonal curve then K(C) =. We can gve a smle geometrc nterretaton to the total curvature. The onts γ and γ + on S 2 determne a great crcle (thnk Equator) on the shere obtaned by ntersectng the shere wth the lane Π through the orgn and contanng these two onts. Ths great crcle s dvded nto two arcs by the onts γ and γ +. We let σ denote the shorter of the two arcs. Note that = length(σ ). The collecton of curves σ trace a closed curve σ C on S 2 called the gaussan mage of C. We deduce K(C) = length(σ C). 2. A PROBABILISTIC INTERPRETATION OF THE TOTAL CURVATURE Every unt vector u S 2 determnes a lnear ma L u : R 3 R, L u (x) = u x, where denotes the dot roduct n R 3. Ths nduces by restrcton a contnuous ma A vertex of C s a local mnmum of l u f We now defne l u = L u C : C R. l u () l u (x), for all x C stuated n a neghborhood of. µ C : S 2 V C R, S 2 V C (u, ) µ C (u, ) = We set {, f s a local mnmum of l u, 0, otherwse. µ C : S 2 R, µ C (u) = the number of vertces of C that are local mnma of l u. Let us ont out that that µ C (u) = ) for some u s. Observe that µ C (u) = V C µ C (u, ). (2.) Let us have a look at the functon µ C. Frst let us call a unt vector u S 2 nondegenerate (wth resect to C) f the restrcton l u : V C R,.e., the functon l u takes dfferent values on dfferent

3 KNOTS AND THEIR CURVATURES 3 vertces of C. Otherwse, we say that u s degenerate (wth resect to C). We denote by C S 2 the collecton of degenerate vectors. Note that u s degenerate f and only f there exst, j V C such that u ( j ),.e., u s erendcular to the lne l j determned by and j. In other words, u belongs to the great crcle E j S 2 obtaned by ntersectng S 2 wth the lane through orgn erendcular to l j. Thus C = E j. <j n In artcular, the set C has zero area,.e., most vectors u S 2 are nondegenerate. Set Let us ont out that S 2 C := S 2 \ C. u S 2 C µ C (u) <. The set S 2 C s the comlement of fntely many great crcles, and thus conssts of the nterors of fntely many shercal olygons, S 2 C = P P ν. Let us observe that f u 0, u S 2 C belong to the nteror of the same olygon P k then µ C (u 0, ) = µ C (u, ), V C. To see ths we choose a contnuous ath u : [0, ] P k such that u(0) = u 0, u() = u. We set l t := l u(t), we consder a vertex of C and we denote by and ts neghbors. Snce the vector u(t) s nongenerate the quanttes d t = l t ( ) l() and d t = l t ( ) l t () are nonzero for any t [0, ]. In artcular, the sgns of these quanttes are ndeendent of t. Observe that s a local mnmum for l 0 f and only f both d 0 and d 0 are ostve, that s, f and only f d and d are ostve. Thus s a local mnmum for l 0 f and only f t s a local mnmum for l,.e., µ C (u 0, ) = µ C (u, ). Ths shows that the functon µ C s constant and fnte on each of the regons P k and n artcular, t s ntegrable. Now defne κ(c) := area (S 2 µ C (u) da u = µ C (u) da u, ) S 2 4π S 2 where da denotes the area element on S 2. In other words κ(c) s the average number of local mnma of the collecton of functon { lu : C R; u S 2 }. We have the followng beautful result due to Mlnor [2] Theorem 2.. For any olygonal curve C R 3 we have K(C) = κ(c). Proof. The roof s based on one of the oldest trcks n the book, namely, changng the order of summaton (or ntegraton) n a double sum (or ntegral). We have κ(c) = µ C (u) da u = ( ) µ C (u, ) da u = µ C (u, ) da u. 4π S 2 4π S 2 4π V S 2 C V C

4 4 LIVIU I. NICOLAESCU Let V C = {,..., n }. We want to comute the ntegral S 2 µ C (u, ) da u. Above, for almost all u we have µ C (u, ) = 0,. Note that s a local mnmum of l u f and only f u belongs to the lune L S 2 defned as follows. - β π π - + L β FIGURE 2. A lanar secton of a dhedral angle and the assocated lune wth oenng β =. Consder the lanes π and π erendcular to the lnes + and resectvely ; see Fgure 2. The lanes π and π determne four dhedral angles. Let let D denote the dhedral angle characterzed by the nequaltes u D u u, u +. Then L = D S 2. The area of the lune L s twce the measure β of the dhedral angle D (can you argue why?) and uon nsectng Fgure 2 we see that β = Hence µ C (u, 4π ) da u = S 2 4π area (L ) =. Hence κ(c) = n = K(C). = Remark 2.2. For a dfferent robablstc nterretaton of K(C) we refer to the aer of Istvan Fáry []. 3. THE TOTAL CURVATURE OF A SMOOTH CLOSED CURVE Suose now that C s a C 2 closed curve n R 3 wthout self-ntersectons. In other words we can fnd a twce contnuously dfferentable ma r : R R 3, t r(t) that s -erodc, ts restrcton to [0, ) s njectve, and r(t + n) = r(t), t R, n Z, ṙ(t) 0, t R, where the dot ndcates a t-dervatve, such that C concdes wth the mage of r. The arametrzaton r nduces an orentaton on C. We set 0 = r(0).

5 KNOTS AND THEIR CURVATURES 5 For every = r(t) C we denote by γ C () the unt vector tangent to C at and ontng n the same drecton as the velocty vector ṙ(t) at. More formally, We the resultng C -ma γ C () = ṙ(t) ṙ(t). γ C : C S 2 s called the Gauss ma of the orented closed curve C. Its mage σ C s a C curve on S 2 called the gaussan mage of C. We denote by ds the arclength element along C, ds = ṙ(t) dt so that L C := length(c) = ds = ṙ(t) dt. For every C \ { 0 } we denote by s() the length of the arc of C connectng 0 to followng the orentaton gven by r. Set s( 0 ) = 0. We can use the quantty s to ndcate the oston of a ont on C. Thus we can vew r as a functon of s, r = r(s). Note that dr ds =, C 0 dr ds = γ(s). We aroxmate C by a sequence of nscrbed olygonal curves C n, obtaned nductvely as follows. The olygonal curve C has 2 k vertces 0,,..., 2 k, 2 k = 0 orented followng the orentaton of C, and s( ) s( ) = L C 2 k. V Cn V Cn+ and new vertces of C n+ are the mdonts of the arcs of C formed by the consecutve vertces of C n. Observe that the set can be dentfed wth the dense subset of [0, L C ] V = n V Cn V = { s [0, L], s = m 2 n L C; m, n Z 0, n k, m 2 n }. Note that f V, then V Cn for all n. Denote by,n the vertex of C n that concdes wth, and by +,n ts succesor. We set Note that so that γ Cn () = s,n := s(,n ), s +,n := s( +,n ). ( r(s+,n ) r(s,n ) ), r(s +,n ) r(s,n ) lm γ ( n C n () = lm r(s+,n ) r(s,n ) ) γ C (). n r(s +,n ) r(s,n ) ( = lm r(s+,n ) r(s,n ) ) = γ C (). n s +,n s,n Thus the gaussan mages of C n are curves convergng to the gaussan mage of C, so we could exect that lm length(σ C n n ) = length(σ C ).

6 6 LIVIU I. NICOLAESCU In fact somethng more recse s true. We set K(C) = length σ C = C dγ ds ds. (3.) The quanttty K(C) s called the total curvature of C and t s a measure of the total bendng of C. Theorem 3.. (a) K(C n ) K(C n+ ), n and lm K(C n) = K(C). n (b) There exsts n 0 > 0 such that for any n n 0 and any u S 2 we have Moreover µ Cn (u) µ Cn+ (u) = µ C (u) := the number of local mnmal of L u C. lm n µ Cn (u) da u = S 2 µ C (u) da u. S 2 The roof s not very hard, but t s rather techncal and we refer for detals to [2]. In artcular we deduce that for any closed C 2 curve we have K(C) = κ(c), (3.2) where the left-hand sde s the bendng measure (3.) and t s a urely geometrc quantty, whle κ(c) s a robablstc quantty κ(c) = µ C (u) da u. (3.3) 4π S 2 4. TOTAL CURVATURE AND KNOTTING The toologsts refer to closed C 2 curves n R 3 as knots. In the 40s K Borsuk ask the followng queston Is t true that f a knot C s not too bent, then t s not really knotted? More recsely, he sked to rove that f K(C) 2 then C s not knotted. In 949, whle an undergraduate at Prnceton, J. Mlnor gave a roof to ths conjecture n the beautful aer [2] that served as nsraton for ths talk. At about the same tme, n Euroe, I. Fáry gave a dfferent but related roof of ths fact. We want to rove a slghtly weaker result. Theorem 4. (Mlnor-Fáry). If C s a knot and K(C) < 2, then C s not knotted. Proof. Here s brefly Mlnor s strategy. He ntroduced an nvarant m(c) of a knot C, called crookedness and he showed that f m(c) = then C s not knotted. A smle argument based on (3.2) then shows that K(C) 2 mles that m(c) =. The crookedness m(c) s the nteger Lemma 4.2. If m(c) = then C s not knotted. m(c) := mn u S 2 µ C(u). Proof of the lemma. Snce m(c) = there exsts u S 2 such that the functon L u C has a unque local mnmum, whch has to be a global mnmum. In artcular ths functon must have a unque local maxmum, because between two local maxma there must be a local mnmum. By a sutable choce of coordnates we can assume that u s the basc vector k, so that L u (x + y + zk) = z

7 KNOTS AND THEIR CURVATURES 7.e., L u s the alttude functon. k FIGURE 3. Unknottng a curve wth small crookedness. By removng two small cas,.e., small connected neghborhoods of the mnmum and the maxmum onts we obtan two dsjont arcs n R 3 as dected n Fgure 3-2. The restrcton of the alttude along each of these arcs s a contnuous njectve functon. These two arcs start at the same alttude z 0 and end at the same alttude z > z 0. For t [z 0, z ] these two arcs ntersect the horzontal lane {z = t} n two onts t and q t. Denote by S t lne segment connectng t to q t. The unon of these segments sans a rbbon between the two arcs whch shows that they can be untwsted, as n Fgure 3-3,4,5. To unknot C we let the boundary of the cas follow the boundares of the two arcs as they are untwsted. We can now comlete the roof of Theorem 4.. We observe that K(C) = µ C (u) da u m(c) da u = m(c). 4π S 2 4π S 2 Thus f K(C) < 2 then the ostve nteger m(c) s strctly less than 2 so that m(c) =. From Lemma 4.2 we deduce that C s not knotted. REFERENCES [] I. Fáry: Sur la courbure totale d une courbe gauche fasant un noed, Bull. Soc. Math. France, 77(949), [2] J.W. Mlnor: On the total curvature of knots, Ann. Math., 52(950), DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NOTRE DAME, NOTRE DAME, IN E-mal address: ncolaescu.@nd.edu URL: htt:// lncolae/