Noncharacterizing slopes for hyperbolic knots

Abstract

A nontrivial slope $r$ on a knot $K$ in $S^3$ is called a characterizing slope if whenever the result of $r$–surgery on a knot $K^{\prime }$ is orientation-preservingly homeomorphic to the result of $r$–surgery on $K$, then $K^{\prime }$ is isotopic to $K$. Ni and Zhang ask: for any hyperbolic knot $K$, is a slope $r=p∕q$ with $\left|p\right|+\left|q\right|$ sufficiently large a characterizing slope? In this article, we prove that if we can take an unknot $c$ so that $\left(0,0\right)$–surgery on $K\cup c$ results in $S^3$ and $c$ is not a meridian of $K$, then $K$ has infinitely many noncharacterizing slopes. As the simplest known example, the hyperbolic, two-bridge knot $8_6$ has no integral characterizing slopes. This answers the above question in the negative. We also prove that any L-space knot never admits such an unknot $c$.