A survey on the Turaev genus of knots

TL;DR

This survey reviews the Turaev genus of knots, a topological measure of non-alternating complexity, and explores its relationships with various knot invariants like the Jones polynomial and homology theories.

Contribution

It provides a comprehensive overview of the Turaev genus and its connections to other important knot invariants, highlighting recent research developments.

Findings

Turaev genus measures how far a knot is from being alternating.

Connections between Turaev genus and Jones polynomial are established.

Relationships with knot homology theories and ribbon-graph invariants are discussed.

Abstract

The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other knot invariants, including the Jones polynomial, knot homology theories, and ribbon-graph polynomial invariants.