The Kauffman bracket of virtual links and the Bollob\'as-Riordan polynomial

TL;DR

This paper establishes a connection between the Kauffman bracket of checkerboard colorable virtual links and the Bollobás-Riordan polynomial of associated ribbon graphs, extending classical results to virtual links.

Contribution

It generalizes Thistlethwaite's theorem by relating the Kauffman bracket of virtual links to the Bollobás-Riordan polynomial of ribbon graphs.

Findings

Kauffman bracket of virtual links can be evaluated via Bollobás-Riordan polynomial.

Extension of classical graph polynomial relations to virtual link theory.

Provides a new tool for analyzing virtual links using ribbon graph invariants.

Abstract

We show that the Kauffman bracket $[L]$ of a checkerboard colorable virtual link $L$ is an evaluation of the Bollob\'as-Riordan polynomial $R_{G_L}$ of a ribbon graph associated with $L$. This result generalizes Thistlethwaite's celebrated theorem relating the Kauffman bracket with the Tutte polynomial of planar graphs.