Turaev genus and alternating decompositions

TL;DR

This paper establishes a relationship between the Turaev surface genus of a link diagram and a specific graph structure, providing classifications for low-genus cases and extending to all genera.

Contribution

It introduces a graph-based method to determine Turaev surface genus and classifies diagrams for genus one and two, with generalizations to higher genera.

Findings

Genus of Turaev surface is determined by boundary component graphs.

Classified diagrams with Turaev genus one or two.

Proved classification theorems for all genera.

Abstract

We prove that the genus of the Turaev surface of a link diagram is determined by a graph whose vertices correspond to the boundary components of the maximal alternating regions of the link diagram. Furthermore, we use these graphs to classify link diagrams whose Turaev surface has genus one or two, and we prove that similar classification theorems exist for all genera.