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Cosmetic operations and Khovanov multicurves
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Abstract
We prove an equivariant version of the Cosmetic Surgery Conjecture for strongly invertible knots. Our proof combines a recent result of Hanselman with the Khovanov multicurve invariants and . We apply the same techniques to reprove a result of Wang about the Cosmetic Crossing Conjecture and split links. Along the way, we show that and detect if a Conway tangle is split.
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When restricted to alternating links, both Heegaard Floer and Khovanov homology concentrate along a single diagonal -grading. This leads to the broader class of thin links that one would like to characterize without reference to the invariant in question. We provide a relative version of thinness for tangles and use this to characterize thinness via tangle decompositions along Conway spheres. These results bear a strong resemblance to the L-space gluing theorem for three-manifolds with torus boundary. Our results are based on certain immersed curve invariants for Conway tangles, namely the Heegaard Floer invariant and the Khovanov invariant that were developed by the authors in previous works.
We show that two Dehn surgeries on a knot K never yield manifolds that are homeomorphic as oriented manifolds if or . As an application, we verify the cosmetic surgery conjecture for all knots with no more than 11 crossings except for three 10-crossing knots and five 11-crossing knots. We also compute the finite type invariant of order 3 for two-bridge knots and Whitehead doubles, from which we prove several nonexistence results of purely cosmetic surgery.
We study cosmetic crossings in knots of genus one and obtain obstructions to
such crossings in terms of knot invariants determined by Seifert matrices. In
particular, we prove that for genus one knots the Alexander polynomial and the
homology of the double cover branching over the knot provide obstructions to
cosmetic crossings. As an application we prove the nugatory crossing conjecture
for twisted Whitehead doubles of non-cable knots. We also verify the conjecture
for several families of pretzel knots and all genus one knots with up to 12
crossings.
We prove the nugatory crossing conjecture for fibered knots. We also show
that if a knot K is n-adjacent to a fibered knot , for some , then
either the genus of K is larger than that of or K is isotopic to .
If a knot K in S^3 admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either \pm 2 or \pm 1/q for some value of q that is explicitly determined by the knot Floer homology of K . Moreover, in the former case the genus of K must be 2 , and in the latter case there is a bound relating q to the genus and the Heegaard Floer thickness of K . As a consequence, we show that the cosmetic crossing conjecture holds for alternating knots (or more generally, Heegaard Floer thin knots) with genus not equal to 2 . We also show that the conjecture holds for any knot K for which each prime summand of K has at most 16 crossings; our techniques rule out cosmetic surgeries in this setting except for slopes \pm 1 and \pm 2 on a small number of knots, and these remaining examples can be checked by comparing hyperbolic invariants. These results make use of the surgery formula for Heegaard Floer homology, which has already proved to be a powerful tool for obstructing cosmetic surgeries; we get stronger obstructions than previously known by considering the full graded theory. We make use of a new graphical interpretation of knot Floer homology and the surgery formula in terms of immersed curves, which makes the grading information we need easier to access.
In this paper, we introduce the annular instanton Floer homology which is defined for links in a thickened annulus. It is an analogue of the annular Khovanov homology. A spectral sequence whose second page is the annular Khovanov homology and which converges to the annular instanton Floer homology is constructed. As an application of this spectral sequence, we prove that the annular Khovanov homology detects the unlink in the thickened annulus (assuming all the components are null-homologous). Another application is a new proof of Grigsby and Ni's result that tangle Khovanov homology distinguishes braids from other tangles.
A pair of surgeries on a knot is called purely cosmetic if the
pair of resulting 3-manifolds are homeomorphic as oriented manifolds. An outstanding
conjecture predicts that no nontrivial knots admit any purely cosmetic
surgeries. Recent work of Hanselman [5] uses Heegaard Floer homology to obtain
new obstructions for the existence of such surgeries. In this work, we apply
those obstructions to show that (nontrivial) knots which arise as the closure of a
3-stranded braid do not admit any purely cosmetic surgeries.
Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that non-trivial knots in do not admit purely cosmetic surgeries. In this article, we confirm this conjecture for cable knots.
The cosmetic crossing conjecture (also known as the "nugatory crossing
conjecture") asserts that the only crossing changes that preserve the oriented
isotopy class of a knot in the 3-sphere are nugatory. We use the Dehn surgery
characterization of the unknot to prove this conjecture for knots in integer
homology spheres whose branched double covers are L-spaces satisfying a
homological condition. This includes as a special case all alternating and
quasi-alternating knots with square-free determinant. As an application, we
prove the cosmetic crossing conjecture holds for all knots with at most nine
crossings and provide new examples of knots, including pretzel knots,
non-arborescent knots and symmetric unions for which the conjecture holds.
Many three-dimensional manifolds are two-fold branched covers of the three-dimensional sphere. However, there are some that are not. This paper includes exposition about two-fold branched covers and includes many examples. It shows that there are three-dimensional homology spheres that do not two-fold branched cover any manifold, ones that only two-fold branched cover the three-dimensional sphere, ones that just two-fold branched cover a non-trivial manifold, and ones that two-fold branched cover both the sphere and non-trivial manifolds. When a manifold is surgery on a knot, the possible quotients via involutions generically correspond to quotients of the knot. There can, however, be a finite number of surgeries for which there are exceptional additional symmetries. The included proof of this result follows the proof of Thurston’s Dehn surgery theorem. The paper also includes examples of such exceptional symmetries. Since the quotients follow the behavior of knots, a census of the behavior for knots with less than 11 crossings is included.
A complete classification is given for equivariant surgery on incompressible tori with respect to involutions with possible 1- or 2-dimensional fixed sets.
BY“hyperbolic 3-manifold” we will mean an orientable complete hyperbolic 3-manifold M of finite volume. By Mostow rigidity the volume of M is a topological invariant, indeed a homotopy invariant, of the manifold M. There is in fact a purely topological definition of this invariant, due to Gromov. The set of all possible volumes of hyperbolic 3-manifolds is known to be a well-ordered subset of the real numbers and is of considerable interest (for number theoretic aspects see, for instance, [2], [13]) but remarkably little is known about it: the smallest element is not known even approximately, and it is not known whether any element of this set is rational or whether any element is irrational. For more details see Thurston’s Notes [73. In this paper we prove a result which, among other things, gives some metric or analytic information about the set of hyperbolic volumes. Given a hyperbolic 3-manifold M with h cusps, one can form the manifold MK = M(P1.41. . , Ph4J obtained by doing a (pi, q,)-Dehn surgery on the i-th cusp, where (pi, qi) is a coprime pair of integers, or the symbol 03 if the cusp is left unsurgered. This notation is well defined only after choosing a basis mi, di for the homology HI (Q, where Z is a torus cross section of the i-th cusp. Then (pi, q,)-Dehn surgery means: cut off the i-th cusp and paste in a solid torus to kill Pimi+qiei.
We study nugatory crossings for knots in S3. In particular, we have results on nugatory crossings for composite knots and two-bridge knots.
Two Dehn surgeries on a knot are called {\it purely cosmetic}, if they yield
manifolds that are homeomorphic as oriented manifolds. Suppose there exist
purely cosmetic surgeries on a knot in , we show that the two surgery
slopes must be the opposite of each other. One ingredient of our proof is a
Dehn surgery formula for correction terms in Heegaard Floer homology.
s Amer. Math. Soc.]. Part II: The skein module based on the Homflypt skein relation. Let M be an oriented 3-manifold, L the set of all oriented links in M up to ambient isotopy of M , R = Z[v ±1 , z ±1 ], RL the free R-module generated by L, and M 3 the submodule of RL generated by the skein expressions v -1 L+ - vL- - zL 0 . For convenience we allow the empty knot, #, and add the relation v -1 # - v# - zT 1 , where T 1 denotes the trivial knot. 73 + L - L L 0 Figure 1.92.1. Then the third skein module of M is defined to be: S 3 (M) = S 3 (M ; Z[v ±1 , z ±1 ], v -1 L+ - vL- - zL 0 ) = RL/M 3 . (B) Conjecture: If M is a rational homology 3-sphere or a compact, connected 3-dimensional submanifold of a rational homology sphere, then S 3 (M) is isomorphic as an R-module to the symmetric tensor algebra S(R# 0 ), where # 0 = # - {1}, and # is the set of conjugacy classes of the fundamental group # = # 1 (M). Remarks: The Conjecture holds for S 3 , for then it is ...
For a 3-manifold with torus boundary admitting an appropriate involution, we
show that Khovanov homology provides obstructions to certain exceptional Dehn
fillings. For example, given a strongly invertible knot in S^3, we give
obstructions to lens space surgeries, as well as obstructions to surgeries with
finite fundamental group. These obstructions are based on homological width in
Khovanov homology, and in the case of finite fundamental group depend on a
calculation of the homological width for a family of Montesinos links.
In this paper, we prove that there are no truly cosmetic surgeries on genus one classical knots. If the two surgery slopes have the same sign, we give the only possibilities of reflectively cosmetic surgeries. The result is an application of Heegaard Floer theory and number theory.
Let K be a rationally null-homologous knot in a three-manifold Y. We construct a version of knot Floer homology in this context, including a description of the Floer homology of a three-manifold obtained as Morse surgery on the knot K. As an application, we express the Heegaard Floer homology of rational surgeries on Y along a null-homologous knot K in terms of the filtered homotopy type of the knot invariant for K. This has applications to Dehn surgery problems for knots in . In a different direction, we use the techniques developed here to calculate the Heegaard Floer homology of an arbitrary Seifert fibered three-manifold.
We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. And then a simple application of an appropriate functor (a `TQFT') to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants.
Khovanov homology is a recently introduced invariant of oriented links in S^3. Its (graded) Euler characteristic is a version of the Jones polynomial of the link. In this paper we study torsion of the Khovanov homology. Based on our calculations, we conjecture that * every link except the trivial knot, the Hopf link, their connected sums and disjoint unions has torsion of order 2; * no link has torsion of odd order; * all homologically thin links are torsion thin, in particular, their torsion is completely determined by the free part of the Khovanov homology; * a knot is torsion rich if and only if its reduced Khovanov homology has torsion; * two knots have the same ranks of the Khovanov homology if and only if they have the same torsion. We prove the first two conjectures for all non-split alternating links. We also prove a weakened version of the third conjecture, namely, that all homology slim links are weakly torsion thin.
We show that the Mahler measures of the Jones polynomial and of the colored Jones polynomials converge under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume under Dehn surgery. For pretzel links P(a_1,...,a_n), we show that the Mahler measure of the Jones polynomial converges if all a_i tend to infinity, and approaches infinity for a_i = constant if n tend to infinity, just as hyperbolic volume. We also show that after sufficiently many twists, the coefficient vector of the Jones polynomial and of any colored Jones polynomial decomposes into fixed blocks according to the number of strands twisted.
Immersed curves in Khovanov homology
- Artem Kotelskiy
- Liam Watson
- Claudius Zibrowius
Artem Kotelskiy, Liam Watson, and Claudius Zibrowius. Immersed curves in Khovanov homology,
2019. ArXiv preprint 1910.14584v2.
Khovanov homology and strong inversions, 2021. To appear in The Open Book Series
- Artem Kotelskiy
- Liam Watson
- Claudius Zibrowius
Artem Kotelskiy, Liam Watson, and Claudius Zibrowius. Khovanov homology and strong inversions,
2021. To appear in The Open Book Series. ArXiv preprint 2104.13592v1.
Khovanov homology also detects split links
- Robert Lipshitz
- Sucharit Sarkar
Robert Lipshitz and Sucharit Sarkar. Khovanov homology also detects split links, 2019. ArXiv preprint
1910.04246.
Connected sums of knots do not admit purely cosmetic surgeries
- Ran Tao
Ran Tao. Connected sums of knots do not admit purely cosmetic surgeries, 2019. ArXiv: 1909.05048v1.