# Thistlethwaite's theorem for virtual links

## TL;DR

This paper extends Thistlethwaite's theorem, linking the Jones polynomial of virtual links to the Bollobás-Riordan polynomial of graphs embedded on higher genus surfaces, broadening the theorem's applicability.

## Contribution

It generalizes Thistlethwaite's theorem to virtual links by incorporating graphs on higher genus surfaces and utilizing the Bollobás-Riordan polynomial.

## Key findings

- Established a relation between Jones polynomial and Bollobás-Riordan polynomial for virtual links.
- Extended classical graph invariants to virtual link theory.
- Provided a framework for analyzing virtual links via surface-embedded graphs.

## Abstract

The celebrated Thistlethwaite theorem relates the Jones polynomial of a link with the Tutte polynomial of the corresponding planar graph. We give a generalization of this theorem to virtual links. In this case, the graph will be embedded into a (higher genus) surface. For such graphs we use the generalization of the Tutte polynomial discovered by B.Bollobas and O.Riordan.