The search for alternating surgeries

TL;DR

This paper investigates alternating surgeries on knots in $S^3$, providing an algorithmic approach to determine possible surgery slopes, calculating examples, and establishing bounds related to knot genus.

Contribution

It introduces an algorithm to compute alternating surgery slopes and analyzes their structure, including bounds and specific examples.

Findings

Set of alternating surgery slopes is algorithmically computable.

Calculated slopes for many knots, including all hyperbolic knots in the SnapPy census.

Established bounds on surgery slopes related to knot genus.

Abstract

Surgery on a knot in $S^3$ is said to be an alternating surgery if it yields the double branched cover of an alternating link. The main theoretical contribution is to show that the set of alternating surgery slopes is algorithmically computable and to establish several structural results. Furthermore, we calculate the set of alternating surgery slopes for many examples of knots, including all hyperbolic knots in the SnapPy census. These examples exhibit several interesting phenomena including strongly invertible knots with a unique alternating surgery and asymmetric knots with two alternating surgery slopes. We also establish upper bounds on the set of alternating surgeries, showing that an alternating surgery slope on a hyperbolic knot satisfies $|p/q| \leq 3g(K)+4$. Notably, this bound applies to lens space surgeries, thereby strengthening the known genus bounds from the conjecture of Goda and Teragaito.