Quasi-alternating surgeries

TL;DR

This paper investigates quasi-alternating surgeries on knots, showing their relation to L-space knots, classifying such surgeries on various knots, and exploring their properties and slopes, thereby advancing understanding of knot surgeries and L-spaces.

Contribution

It classifies quasi-alternating surgeries on census L-space knots, completes Dunfield's classification, and analyzes slopes on asymmetric L-space knots and torus knots.

Findings

Most SnapPy census L-space knots admit quasi-alternating surgeries.

Asymmetric census L-space knots have exactly two quasi-alternating slopes, which are consecutive integers.

The set of formal L-space slopes on torus knots is either empty or infinite.

Abstract

In this article, we explore phenomena relating to quasi-alternating surgeries on knots, where a quasi-alternating surgery on a knot is a Dehn surgery yielding the double branched cover of a quasi-alternating link. Since the double branched cover of a quasi-alternating link is an L-space, quasi-alternating surgeries are special examples of L-space surgeries. We show that all SnapPy census L-space knots admit quasi-alternating surgeries except for the knots t09847 and o9_30634, neither of which have any quasi-alternating surgeries. In particular, this finishes Dunfield's classification of the L-space knots among all SnapPy census knots. In addition, we show that all asymmetric census L-space knots have exactly two quasi-alternating slopes and that these are consecutive integers. Similar behavior is observed for some of the Baker-Luecke asymmetric L-space knots. We also classify the quasi-alternating surgeries on torus knots and show that the set of formal L-space slopes is either empty or infinite This allows us to give examples of asymmetric formal L-spaces.