3-braid knots do not admit purely cosmetic surgeries
33-braid knots do not admit purely cosmeticsurgeries
Konstantinos VarvarezosMay 25, 2020
Abstract
A pair of surgeries on a knot is called purely cosmetic if the pair ofresulting 3-manifolds are homeomorphic as oriented manifolds. Usingrecent work of Hanselman, we show that (nontrivial) knots which ariseas the closure of a 3-stranded braid do not admit any purely cosmeticsurgeries.
Given a knot K in S and a pair of coprime integers p, q , we denote the Dehn surgery on K with slope p/q by S p/q ( K ) . Surgeries on K along distinctslopes r and r (cid:48) are called cosmetic if S r ( K ) and S r (cid:48) ( K ) are homeomorphicmanifolds. Furthermore, a pair of such surgeries is said to be purely cosmetic if S r ( K ) and S r (cid:48) ( K ) are homeomorphic as oriented manifolds, whereas ifthey are homeomorphic but have opposite orientations, the pair of surgeriesis called chirally cosmetic .No purely cosmetic surgeries are have been found on nontrivial knotsin S ; indeed, the Cosmetic Surgery Conjecture predicts that none exist(compare also Problem 1.81(A) in [8]). On the other hand, there are examplesof chirally cosmetic surgeries. For instance, S r ( K ) ∼ = − S − r ( K ) whenever K is an amphicheiral knot. Also, (2 , n )-torus knots are known to admit chirallycosmetic surgeries; see [6, 11].In this work, we verify the Cosmetic Surgery Conjecture for 3-braid knots.1 a r X i v : . [ m a t h . G T ] M a y a a Figure 1: The three generators of the band presentation of the 3-braid group.
Theorem 1.1.
Suppose K is a nontrivial knot which is the closure of a3-braid. Then K admits no purely cosmetic surgeries. Acknowledgements
The author would like to thank Professor Zolt´an Szab´o for suggesting work onthis problem as well as many helpful conversations. The author also thanksProfessor Kazuhiro Ichihara for useful comments and corrections. This workwas supported by the NSF RTG grant DMS-1502424.
Here we recall some useful results about the genera of 3-braid knots. We shallmake use of the so-called “band presentation” of the 3-braid group, studiedfor braid groups in general by Birman, Ko, Lee [2] and which has proveduseful in the study of surfaces bounded by braid knots; see, for instance, [18,21, 9]. For 3-braids, the presentation is as follows: (cid:104) a , a , a | a a = a a = a a (cid:105) Here a and a correspond to the standard Artin generators σ and σ re-spectively, and a corresponds to σ σ σ − ; see Figure 1 for an illustration.A word in the band generators very naturally gives rise to a Seifert surfacefor the corresponding closed braid, which we shall call the banded surface a a − a .associated to the word. In particular, starting with three disks, which onecan imagine as being vertically “stacked”, one attaches a half-twisted bandbetween two of the disks for each instance of a ± j : for a between the bottomand middle disks, for a between the middle and top, and for a between thetop and bottom, twisted appropriately depending on the sign of the exponent;see Figure 2 for the banded surface corresponding to the braid a a − a .Now suppose that a 3-braid in the band generators has a knot as its braid-closure. A straightforward computation of the Euler characteristic revealsthat the banded surface produced with the above method has genus g = (cid:96) −
22 (1)where (cid:96) is the length of the braid word (that is, the number of instances ofthe a ± j s). Moreover, in the case of 3-braids, a theorem of Bennequin’s saysthat a minimal Seifert surface can always be produced in this way (see also[3]): Theorem 2.1 (Proposition 3 of [1]) . If K is a the closure of a 3-braid, thenthe Seifert genus of K can be realized by a banded surface. K is a knot which is the closure of a 3-braid expressed as a wordin the band generators. By Bennequin’s theorem, we may take this word tobe minimal so that the associated banded surface achieves the Seifert genusof the knot g ( K ). Notice that each instance of a ± and a ± contributes acrossing to the diagram of the braid, while each instance of a ± contributes3 crossings. Let A j denote the total number of instances of a ± j in the givenword for K . Then K admits a diagram with A + A +3 A crossings. Further,observe that K is isotopic to the closure of the braid obtained by cyclicallypermuting the generators: a → a → a → a (indeed, one can even realizethis as an isotopy of a banded surface like the one in Figure 2 by “flipping” thetop disk across the banded areas so that it becomes the bottom disk). Thus,we may assume that A is the least of the A j s; that is, A ≤ ( A + A + A ).On the other hand, we have that the length of the word (cid:96) = A + A + A .Combining this with equation (1), we see that K admits a diagram withcrossing number: c = A + A + 3 A = (cid:96) + 2 A ≤ (cid:96) + 23 (cid:96) = 103 ( g ( K ) + 1) . Therefore, we have shown the following lemma:
Lemma 2.2.
Let K be a 3-braid knot. If g ( K ) denotes the Seifert genus of K , then K admits a diagram with no more than ( g ( K ) + 1) crossings.Remark. In fact, we may additionally require that K admit a diagram withan even number of crossings less than or equal to ( g ( K ) + 1). This followsfrom the fact that, in order for the closure of a 3-braid to result in a knot,the permutation of the strands induced by the braid must be even (indeed itmust be one of the cycles of order 3 in S , the symmetric group on the threestrands). Knot Floer Homology is a knot invariant introduced by Ozsv´ath and Szab´o[16] and also independently by Rasmussen [17]. It is related to Heegaard Floer4omology, a 3-manifold invariant also developed by Ozsv´ath and Szab´o, andit has been used with much success to obtain general obstructions for cosmeticsurgeries; see, for instance, [20, 15, 12]. In this work, we shall make use ofobstructions coming from recent work of Hanselman [5].Given a knot K , we consider the so-called “hat” version of Knot FloerHomology, denoted (cid:92) HF K ( K ), which is the homology of a bigraded chaincomplex denoted (cid:92) CF K ( K ). The two gradings are called the Alexander andMaslow gradings. In an algebraic re-interpretation of Knot Floer Homology,Ozsv´ath and Szab´o showed [14, 13] that the generators of (cid:92) CF K ( K ) may beidentified with so-called Kauffman states of a diagram for K . A Kauffmanstate consists of a choice of one of the four “corners” for each crossing ap-pearing in the diagram for K , such that all but two (chosen once and for all)of the “regions” that make up the complement of the diagram have exactlyone marking (the two “distinguished” regions never receive any markings).We denote the set of all Kauffman states by S (note that this depends on thechoice of diagram for K ). The Alexander and Maslow grading of each suchgenerator is the sum of the local grading contributions at each crossing, asshown in Figure 3. The delta grading of a generator of (cid:92) CF K ( K ) is defined tobe the difference between its Alexander and Maslow gradings; hence, it toomay be computed as a sum of local delta-gradings, also displayed in Figure3. The (homological) thickness of a knot K , which we shall denote th ( K ), isdefined to be the difference between the maximal and minimal delta gradingsof the support of (cid:92) HF K ( K ). Using the Kauffaman state interpretation, weobtain the following bound on the thickness: th ( K ) ≤ max s ∈ S δ ( s ) − min s ∈ S δ ( s )= max s ∈ S (cid:88) c ∈ s δ loc ( c ) − min s ∈ S (cid:88) c ∈ s δ loc ( c ) ≤ (cid:88) c ∈ s positive − (cid:88) c ∈ s negative −
12= 12 ( n + + n − )where a sum over c ∈ s means summing over all the (marked) crossings c ofthe Kauffman state s , and where n + and n − denote the number of positiveand negative crossings appearing in the diagram, respectively. Hence, the5 ½-½½½ -110 0000 ½½-½-½A loc M loc δ loc Figure 3: The local Alexander, Maslow, and delta grading contributions ata crossing. The top row shows a positive crossing and the bottom shows anegative crossing.thickness of a knot K is at most one-half the number of crossings appearingin any diagram for K . Combining this with Lemma 2.2, we obtain: Lemma 2.3. If K is a knot which is the closure of a 3-braid, then th ( K ) ≤
53 ( g ( K ) + 1) We are now ready to prove our main result. We shall make use of the follow-ing obstruction obtained by Hanselman for the existence of purely cosmeticsurgeries:
Theorem 3.1 (Theorem 2 of [5]) . Let K ⊂ S be a nontrivial knot andsuppose that S r and S r (cid:48) are homeomorphic as oriented manifolds for r, r (cid:48) distinct rational numbers. Then { r, r (cid:48) } = {± } or {± q } for some positiveinteger q . Moreover:i) If { r, r (cid:48) } = {± } then g ( K ) = 26 i) If { r, r (cid:48) } = {± q } then q ≤ th ( K ) + 2 g ( K )2 g ( K )( g ( K ) − Remark.
Notice that if for a knot K with genus greater than 2 the fractionon the right hand side of the inequality in ii) is less than one, then K admitsno purely comsetic surgeries. Theorem 1.1.
Suppose K is a nontrivial knot which is the closure of a3-braid. Then K admits no purely cosmetic surgeries.Proof. First, consider a 3-braid knot K with genus g ( K ) = g ≥
4. By Lemma2.3, th ( K ) ≤ ( g + 1). Hence, th ( K ) + 2 g ( K )2 g ( K )( g ( K ) − ≤ g + g ( g − g − (cid:18)
116 + 56 g (cid:19) ≤ (cid:18)
116 + 524 (cid:19) = 4972 < K has genus at least 4 was used in the last step.By the remark following Theorem 3.1, K admits no purely cosmetic surgeries.So we are left with the case g ( K ) ≤
3. By Lemma 2.2 along with theremark following it, any such knot has crossing number at most 12. As com-posite knots were shown not to admit any purely cosmetic surgeries by Tao[19], we may restrict our attention to prime knots. On the other hand, if g ( K ) ≤
2, it has crossing number at most 10. In [7], Ito showed that theCosmetic Surgery Conjecture holds for all prime knots of crossing number atmost 11, with the possible exception of 10
18. According to KnotInfo [10],this knot is a 3-braid knot of genus 4, and so admits no purely cosmetic surg-eries by the above computation. Hence, the remaining unchecked knots havegenus 3 and crossing number 12. A search through KnotInfo reveals threeknots satisfying these conditions (with bridge index 3). KnotInfo also givesthe Alexander polynomials of these knots, from which we may compute theinvariant a ( K ) = ∆ (cid:48)(cid:48) K (1), where ∆ K ( t ) denotes the symmetrized Alexanderpolynomial of K . The results are shown in the following table:7 ∆ K ( t ) a ( K )12 n t − − t − + t − − t − t + t n
750 2 t − − t − + 7 − t + 2 t n
830 2 t − − t − + 5 − t + 2 t K admits purely cosmeticsurgeries then a ( K ) = 0. Hence the three remaining knots are ruled out aswell. References [1] Daniel Bennequin. “Entranclements et ´equations de Pfaff”. In:
Ast´erisque
Ad-vances in Mathematics issn : 0001-8708. doi : https://doi.org/10.1006/aima.1998.1761 .[3] Joan S. Birman and William W. Menasco. “Studying links via closedbraids II: On a theorem of Bennequin”. In: Topology and its Applica-tions issn : 0166-8641. doi : https://doi.org/10.1016/0166-8641(91)90059-U .[4] Steven Boyer and Daniel Lines. “Surgery formulae for Casson’s in-variant and extensions to homology lens spaces.” In: Journal fr diereine und angewandte Mathematik doi : .[5] Jonathan Hanselman. Heegaard Floer homology and cosmetic surgeriesin S . 2019. arXiv: .[6] Kazuhiro Ichihara, Tetsuya Ito, and Toshio Saito. “Chirally cosmeticsurgeries and Casson invariants”. In: Tokyo J. Math.
On LMO invariant constraints for cosmetic surgery andother surgery problems for knots in S . 2020. doi : . 88] Rob Kirby, editor. “Problems in Low-Dimensional Topology”. In: Ge-ometric Topology : 1993 Georgia International Topology Conference,August 2-13, 1993, University of Georgia, Athens, Georgia . Ed. byWilliam H. Kazez. Vol. 2. AMS/IP Studies in Advanced Mathemat-ics. Providence, R.I.: American Mathematical Society, 1996, pp. 35–473.[9] Eon-Kyung Lee and Sang-Jin Lee. “Unkotting number and genus of 3-braid knots”. In:
Journal of Knot Theory and Its Ramifications doi : .[10] C. Livingston and A. H. Moore. KnotInfo: Table of Knot Invariants . url : .[11] Yves Mathieu. “Closed 3-manifolds unchanged by Dehn surgery”. In: Journal of Knot Theory and Its Ramifications doi : .[12] Yi Ni and Zhongtao Wu. “Cosmetic surgeries on knots in S ”. In: Jour-nal fr die reine und angewandte Mathematik doi : .[13] Peter Ozsv´ath and Zolt´an Szab´o. Algebras with matchings and knotFloer homology . 2019. arXiv: .[14] Peter Ozsv´ath and Zolt´an Szab´o. “Kauffman states, bordered alge-bras, and a bigraded knot invariant”. In:
Advances in Mathematics
328 (2018), pp. 1088–1198. issn : 0001-8708. doi : https://doi.org/10.1016/j.aim.2018.02.017 .[15] Peter S Ozsv´ath and Zolt´an Szab´o. “Knot Floer homology and rationalsurgeries”. In: Algebraic & Geometric Topology
11 (1 2011), pp. 1–68. doi : .[16] Peter Ozsvth and Zoltn Szab. “Holomorphic disks and knot invariants”.In: Advances in Mathematics issn : 0001-8708. doi : https://doi.org/10.1016/j.aim.2003.05.001 .[17] Jacob Rasmussen. “Floer homology and knot complements”. PhD the-sis. Harvard University, 2003. arXiv: math/0306378 [math.GT] .[18] Lee Rudolph. “Braided surfaces and Seifert ribbons for closed braids.”In: Commentarii mathematici Helvetici
58 (1983), pp. 1–37. doi : . 919] Ran Tao. Connected sums of knots do not admit purely cosmetic surg-eries . 2019. arXiv: .[20] Jiajun Wang. “Cosmetic surgeries on genus one knots”. In:
Algebraic& Geometric Topology doi : .[21] Peijun Xu. “The genus of closed 3-braids”. In: Journal of Knot The-ory and Its Ramifications doi :10.1142/S0218216592000185