Heegaard Floer homology and cosmetic surgeries in $S^3$

Abstract

If a knot K in S 3 admits a pair of truly cosmetic surgeries, we show that the surgery slopes are ±2 or ±1/q for some q. Moreover, in the former case the genus of K must be two, and in the latter case there is an upper bound on q which depends on 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 two. 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.

Alternating odd pretzel knots and chirally cosmetic surgeries

“…This can be seen, for instance, from the fact that of the signatures and v 3 of these knots are nonzero(see section 2.1 below). By [2], alternating knots that admit purely cosmetic surgeries must have zero signature. Alternatively, by [4], knots admitting purely cosmetic surgeries must have v 3 = 0.…”

A quantum obstruction to purely cosmetic surgeries

“…In that case, hypothetical alternating counterexamples of the conjecture have been shown to have a very special form. Indeed, Hanselman showed also that in [9,Theorem 3] that an alternating knot K (or, more generally, a thin knot, which has th(K) = 0) with a purely cosmetic surgery must have signature 0 and Alexander polynomial ∆ K (t) = nt 2 − 4nt + (6n + 1) − 4nt −1 + nt −2 for some n ∈ Z. As a corollary of our main result, we can put an extra condition on the Jones polynomial of K: Corollary 1.6.…”

“…With their method they proved the conjecture for knots with less than 15 crossings. Using a new method to express the Heegaard-Floer homology of surgeries, Hanselman put further restrictions on the surgery slopes [9]. Similarly to Futer, Purcell and Schleimer's result, those conditions restrict the set of possible slopes to a finite set.…”

“…It reduces the conjecture to the case of prime knots. We will now describe some of Hanselman's main theorem from [9] in more detail below, as our results will build up on his work.…”

“…But for any knot K, the Turaev genus g T (K) is bounded above by c(K)/2 where c(K) is the crossing number of K. (see for example [1,Proposition 2.4]). Thus a knot with c(K) 31 crossings has thickness at most 15, and can not have a purely cosmetic surgery pair if J K (e Let us note that according to Hanselman's computations [9] a prime knot with at most 16 crossings has thickness at most 2, so the inequality th(K) c(K) 2…”