A characterization of adequate Turaev genus one links

TL;DR

This paper characterizes links with Turaev genus one that are adequate, showing they are precisely those whose Jones polynomial span is one less than their crossing number.

Contribution

It provides a complete characterization of adequate Turaev genus one links based on Jones polynomial span and crossing number relationship.

Findings

Links with Turaev genus one and adequacy have a specific Jones polynomial span property.

The span of the Jones polynomial is exactly one less than the crossing number for these links.

The characterization helps identify such links through polynomial and diagrammatic properties.

Abstract

We prove that a link is adequate and has Turaev genus one if and only if the span of its Jones polynomial is one less than its crossing number.