A surgical perspective on quasi-alternating links

TL;DR

This paper explores the relationship between surgeries on L-space knots and quasi-alternating links, showing that many large surgeries produce links that are two-fold branched covers of quasi-alternating links, with applications to Montesinos links.

Contribution

It establishes a new connection between L-space surgeries and quasi-alternating links, providing an iterative construction for certain Montesinos links.

Findings

Many large surgeries on L-space knots yield two-fold branched covers of quasi-alternating links.

A method to realize these links as branched covers, linking L-spaces and quasi-alternating links.

An iterative construction for quasi-alternating Montesinos links using Seifert fibered spaces.

Abstract

We show that quasi-alternating links arise naturally when considering surgery on a strongly invertible L-space knot (that is, a knot that yields an L-space for some Dehn surgery). In particular, we show that for many known classes of L-space knots, every sufficiently large surgery may be realized as the two-fold branched cover of a quasi-alternating link. Consequently, there is considerable overlap between L-spaces obtained by surgery on $S^3$, and L-spaces resulting as two-fold branched covers of quasi-alternating links. By adapting this approach to certain Seifert fibered spaces, it is possible to give an iterative construction for quasi-alternating Montesinos links.