Non-characterizing slopes for hyperbolic knots

TL;DR

This paper demonstrates that certain hyperbolic knots in $S^3$, including the knot $8_6$, have infinitely many non-characterizing slopes, answering a question about the nature of large slopes in knot theory.

Contribution

It provides the first example of a hyperbolic knot with infinitely many non-characterizing slopes, challenging previous assumptions.

Findings

The hyperbolic knot $8_6$ has no integral characterizing slopes.

There exist hyperbolic knots with infinitely many non-characterizing slopes.

The paper answers negatively a question posed by Ni and Zhang.

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 $K'$ is orientation preservingly homeomorphic to the result of $r$-surgery on $K$, then $K'$ 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.