Characterizing slopes for torus knots

Abstract

A slope $\frac{p}{q}$ is called a characterizing slope for a given knot $K_0$ in $S^3$ if whenever the $\frac{p}{q}$-surgery on a knot $K$ in $S^3$ is homeomorphic to the $\frac{p}{q}$-surgery on $K_0$ via an orientation preserving homeomorphism, then $K=K_0$. In this paper we try to find characterizing slopes for torus knots $T_{r,s}$. We show that any slope $\frac{p}{q}$ which is larger than the number $\frac{30(r^2-1)(s^2-1)}{67}$ is a characterizing slope for $T_{r,s}$. The proof uses Heegaard Floer homology and Agol--Lackenby's 6--Theorem. In the case of $T_{5,2}$, we obtain more specific information about its set of characterizing slopes by applying more Heegaard Floer homology techniques.

Version 2: 19 pages. This is a major revision. The title of the first version was "Towards a Dehn surgery characterization of $T_{5,2}$". We extended the result in the first version to general torus knots. We also fixed a gap in the first version, so our result for $T_{5,2}$ is slightly weaker than the originally claimed one

References in corpus (6)

• Ricci flow with surgery on three-manifolds
• Floer homology and knot complements
• Knot Floer homology detects fibred knots
• Lens space surgeries and L-space homology spheres
• Characterizing slopes for torus knots
• Small genus knots in lens spaces have small bridge number

Cited by in corpus (18)

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• Non-characterizing slopes for hyperbolic knots
• Finite knot surgeries and Heegaard Floer homology
• Finite Dehn surgeries on knots in $S^3$
• Mapping cone formula in Heegaard Floer homology and Dehn surgery on knots in $S^3$
• An enhanced Euler characteristic of sutured instanton homology
• Khovanov homology and the cinquefoil
• Surgeries, sharp 4-manifolds and the Alexander polynomial
• Heegaard Floer homology and knots determined by their complements
• Characterizing slopes for $5_2$
• Twist families of L-space knots, their genera, and Seifert surgeries
• Stein traces and characterizing slopes
• Characterising slopes for hyperbolic knots and Whitehead doubles
• Alternating Surgeries
• Knots with infinitely many non-characterizing slopes
• Cables of the figure-eight knot via real Frøyshov invariants
• Only finitely many alternating knots can yield a given manifold by surgery
• Computation of the knot Floer complex of knots of thickness one