skip to main content

Title: Heegaard Floer homology and cosmetic surgeries in $S^3$
If a knot K in S^3 admits a pair of truly cosmetic surgeries, we show that the surgery slopes are either ±2 or ±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 ±1 and ±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.  more » « less
Award ID(s):
1812527 2105501
Author(s) / Creator(s):
Date Published:
Journal Name:
Journal of the European Mathematical Society
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. In this paper generalizes the work of the second author and prove a grading shifting property, in sutured monopole and instanton Floer theories, for general balanced sutured manifolds. This result has a few consequences. First, we offer an algorithm that computes the Floer homologies of a family of sutured handle-bodies. Second, we obtain a canonical decomposition of sutured monopole and instanton Floer homologies and build polytopes for these two theories, which was initially achieved by Juhász for sutured (Heegaard) Floer homology. Third, we establish a Thurston-norm detection result for monopole and instanton knot Floer homologies, which were introduced by Kronheimer and Mrowka. The same result was originally proved by Ozsváth and Szabó for link Floer homology. Last, we generalize the construction of minus versions of monopole and instanton knot Floer homology, which was initially done for knots by the second author, to the case of links. Along with the construction of polytopes, we also proved that, for a balanced sutured manifold with vanishing second homology, the rank of the sutured monopole or instanton Floer homology bounds the depth of the balanced sutured manifold. As a corollary, we obtain an independent proof that monopole and instanton knot Floer homologies, as mentioned above, both detect fibred knots in S3. This result was originally achieved by Kronheimer and Mrowka. 
    more » « less
  2. In this paper, we study the Riley polynomial of double twist knots with higher genus. Using the root of the Riley polynomial, we compute the range of rational slope [Formula: see text] such that [Formula: see text]-filling of the knot complement has left-orderable fundamental group. Further more, we make a conjecture about left-orderable surgery slopes of two-bridge knots. 
    more » « less
  3. Gay, David ; Wu, Weiwei (Ed.)
    We give new obstructions to the module structures arising in Heegaard Floer homology. As a corollary, we characterize the possible modules arising as the Heegaard Floer homology of an integer homology sphere with one-dimensional reduced Floer homology. Up to absolute grading shifts, there are only two. We use this corollary to show that the chain complex depicted by Ozsváth, Stipsicz, and Szabó to argue that there is no algebraic obstruction to the existence of knots with trivial epsilon invariant and non-trivial upsilon invariant cannot be realized as the knot Floer complex of a knot. 
    more » « less
  4. null (Ed.)
    Abstract We show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology of rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology. 
    more » « less
  5. We use deep neural networks to machine learn correlations betweenknot invariants in various dimensions. The three-dimensional invariantof interest is the Jones polynomial J(q) J ( q ) ,and the four-dimensional invariants are the Khovanov polynomial \text{Kh}(q,t) Kh ( q , t ) ,smooth slice genus g g ,and Rasmussen’s s s -invariant.We find that a two-layer feed-forward neural network can predict s s from \text{Kh}(q,-q^{-4}) Kh ( q , − q − 4 ) with greater than 99% 99 % accuracy. A theoretical explanation for this performance exists in knottheory via the now disproven knight move conjecture, which is obeyed byall knots in our dataset. More surprisingly, we find similar performancefor the prediction of s s from \text{Kh}(q,-q^{-2}) Kh ( q , − q − 2 ) ,which suggests a novel relationship between the Khovanov and Leehomology theories of a knot. The network predicts g g from \text{Kh}(q,t) Kh ( q , t ) with similarly high accuracy, and we discuss the extent to which themachine is learning s s as opposed to g g ,since there is a general inequality |s| ≤2g | s | ≤ 2 g .The Jones polynomial, as a three-dimensional invariant, is not obviouslyrelated to s s or g g ,but the network achieves greater than 95% 95 % accuracy in predicting either from J(q) J ( q ) .Moreover, similar accuracy can be achieved by evaluating J(q) J ( q ) at roots of unity. This suggests a relationship with SU(2) S U ( 2 ) Chern—Simons theory, and we review the gauge theory construction ofKhovanov homology which may be relevant for explaining the network’sperformance. 
    more » « less