Quasi-alternating links with small determinant

TL;DR

This paper classifies quasi-alternating links with small determinants, showing they are connected sums of two-bridge links for determinants up to 7, by analyzing their branched covers and lens space surgeries.

Contribution

It extends the classification of small determinant quasi-alternating links, demonstrating they are connected sums of two-bridge links for determinants up to 7.

Findings

All quasi-alternating links with determinant ≤ 7 are connected sums of two-bridge links.

Characterization of small-order lens space surgeries and formal L-spaces.

Identification of the boundary where quasi-alternating links are no longer all connected sums of two-bridge links.

Abstract

Quasi-alternating links of determinant 1, 2, 3, and 5 were previously classified by Greene and Teragaito, who showed that the only such links are two-bridge. In this paper, we extend this result by showing that all quasi-alternating links of determinant at most 7 are connected sums of two-bridge links, which is optimal since there are quasi-alternating links not of this form for all larger determinants. We achieve this by studying their branched double covers and characterizing distance-one surgeries between lens spaces of small order, leading to a classification of formal L-spaces with order at most 7.