Finite Dehn surgeries on knots in $S^3$

TL;DR

This paper investigates the limitations on finite Dehn surgeries on hyperbolic knots in $S^3$, establishing bounds on slope distances, classifying specific cases, and providing explicit bounds and characterizations for certain surgeries.

Contribution

It proves bounds on the distances between finite surgery slopes, classifies knots with multiple finite surgeries, and improves finiteness results for D-type surgeries, including explicit bounds and characterizations.

Findings

Maximum distance between finite surgery slopes is two.

At most three nontrivial finite surgeries can occur.

The pretzel knot P(-2,3,7) admits three nontrivial finite surgeries.

Abstract

We show that on a hyperbolic knot $K$ in $S^3$, the distance between any two finite surgery slopes is at most two and consequently there are at most three nontrivial finite surgeries. Moreover in case that $K$ admits three nontrivial finite surgeries, $K$ must be the pretzel knot $P(-2,3,7)$. In case that $K$ admits two noncyclic finite surgeries or two finite surgeries at distance two, the two surgery slopes must be one of ten or seventeen specific pairs respectively. For $D$-type finite surgeries, we improve a finiteness theorem due to Doig by giving an explicit bound on the possible resulting prism manifolds, and also prove that $4m$ and $4m+4$ are characterizing slopes for the torus knot $T(2m+1,2)$ for each $m\geq 1$.