Turaev torsion, definite 4-manifolds, and quasi-alternating knots

TL;DR

This paper constructs an infinite family of hyperbolic, homologically thin knots that are not quasi-alternating by analyzing their branched double-covers and using Heegaard Floer homology correction terms related to Turaev torsion.

Contribution

It introduces a method to distinguish non-quasi-alternating knots using Turaev torsion and Heegaard Floer correction terms, expanding understanding of knot properties.

Findings

Identified an infinite family of non-quasi-alternating hyperbolic knots.

Established a link between Turaev torsion and Heegaard Floer correction terms.

Showed that certain branched double-covers do not bound negative definite 4-manifolds.

Abstract

We construct an infinite family of hyperbolic, homologically thin knots that are not quasi-alternating. To establish the latter, we argue that the branched double-cover of each knot in the family does not bound a negative definite 4-manifold with trivial first homology and bounded second betti number. This fact depends in turn on information from the correction terms in Heegaard Floer homology, which we establish by way of a relationship to, and calculation of, the Turaev torsion.