Characterizing slopes for torus knots

TL;DR

This paper identifies large slopes that uniquely determine torus knots in 3-spheres, using advanced topological tools like Heegaard Floer homology and the Agol–Lackenby theorem.

Contribution

It establishes explicit bounds for characterizing slopes of torus knots and refines results for specific cases such as T_{5,2}.

Findings

Slopes larger than a specific bound are characterizing for torus knots.

Heegaard Floer homology techniques are effective in studying knot characterizations.

More precise characterizing slopes are obtained for T_{5,2}.

Abstract

A slope $\frac pq$ is called a characterizing slope for a given knot $K_0$ in $S^3$ if whenever the $\frac pq$-surgery on a knot $K$ in $S^3$ is homeomorphic to the $\frac pq$-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 $\frac pq$ 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.