Non-characterizing slopes for hyperbolic knots

Abstract

A non-trivial 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 $|p|+|q|$ sufficiently large a characterizing slope? In this article we answer this question in the negative by demonstrating that there is a hyperbolic knot $K$ in $S^3$ which has infinitely many non-characterizing slopes. As the simplest known example, the hyperbolic knot $8_6$ has no integral characterizing slopes.

13 pages, 7 figures

References in corpus (1)

• Characterizing slopes for torus knots

Cited by in corpus (6)

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