Characterizing slopes for torus knots, II

Abstract

A slope $\frac{p}{q}$ is called a characterizing slope for a given knot $K_0\subset S^3$ if whenever the $\frac{p}{q}$--surgery on a knot $K\subset S^3$ is homeomorphic to the $\frac{p}{q}$--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 $\frac{p}{q}$ is a characterizing slope for $T_{5,2}$ if $\frac{p}{q}>-1$ and $\frac{p}{q}\notin \{0,1,\pm \frac{1}{2},\pm \frac{1}{3}\}$. In particular, every nontrivial L-space slope of $T_{5,2}$ is characterizing for $T_{5,2}$. As a consequence, if a nontrivial $\frac{p}{q}$-surgery on a non-torus knot in $S^3$ yields a manifold of finite fundamental group, then $|p|>9$.

7 pages, 2 figures

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