Invariants for Turaev genus one links

TL;DR

This paper characterizes Turaev genus one links by their signature, Jones polynomial coefficients, and all-A state components, revealing structural invariants that distinguish them from other knots.

Contribution

It establishes new invariants for Turaev genus one links, linking signature and Jones polynomial properties to their all-A state configurations.

Findings

Signature determined by all-A state components, positive crossings, and determinant.

Leading or trailing Jones polynomial coefficient has absolute value one for Turaev genus one links.

Turaev genus zero knots are exactly the alternating knots.

Abstract

The Turaev genus defines a natural filtration on knots where Turaev genus zero knots are precisely the alternating knots. We show that the signature of a Turaev genus one knot is determined by the number of components in its all-A Kauffman state, the number of positive crossings, and its determinant. We also show that either the leading or trailing coefficient of the Jones polynomial of a Turaev genus one link (or an almost alternating link) has absolute value one.