A counterexample to Question 1 of "A survey on the Turaev genus of knots"

TL;DR

This paper provides a counterexample showing that the polynomial counting quasi-trees of the all-A ribbon graph is not an invariant of the knot, answering an open question from prior survey work.

Contribution

It demonstrates that the polynomial in question is not an invariant, using specific diagrams of the knot 8_{21} as a counterexample.

Findings

The polynomial varies between different diagrams of the same knot.

Counterexample derived from diagrams of knot 8_{21}.

Answers an open question negatively.

Abstract

In "A survey on the Turaev genus of knots," Champanerkar and Kofman propose several open questions. The first asks whether the polynomial whose coefficients count the number of quasi-trees of the all-A ribbon graph obtained from a diagram with minimal Turaev genus is an invariant of the knot. We answer negatively by showing a counterexample obtained from the two diagrams of $8_{21}$ on the KnotAtlas and KnotScape.