Characterizing slopes for torus knots, II

TL;DR

This paper refines the understanding of which slopes uniquely determine the torus knot T_{5,2} through Dehn surgery, showing that most slopes greater than -1 are characterizing, especially for L-space slopes.

Contribution

It improves previous results by identifying a broader class of slopes that are characterizing for T_{5,2} using recent advances in the field.

Findings

Nontrivial slopes greater than -1 are characterizing for T_{5,2}.

All nontrivial L-space slopes of T_{5,2} are characterizing.

If a non-torus knot yields a finite fundamental group after surgery, then |p|>9.

Abstract

A slope $\frac pq$ is called a characterizing slope for a given knot $K_0\subset S^3$ if whenever the $\frac pq$--surgery on a knot $K\subset S^3$ is homeomorphic to the $\frac pq$--surgery on $K_0$ via an orientation preserving homeomorphism, then $K=K_0$. In a previous paper, we showed that, outside a certain finite set of slopes, only the negative integers could possibly be non-characterizing slopes for the torus knot $T_{5,2}$. Applying recent work of Baldwin--Hu--Sivek, we improve our result by showing that a nontrivial slope $\frac pq$ is a characterizing slope for $T_{5,2}$ if $\frac pq>-1$ and $\frac pq\notin \{0,1, \pm\frac12,\pm\frac13\}$. In particular, every nontrivial L-space slope of $T_{5,2}$ is characterizing for $T_{5,2}$. As a consequence, if a nontrivial $\frac pq$-surgery on a non-torus knot in $S^3$ yields a manifold of finite fundamental group, then $|p|>9$.