Bollobas-Riordan and relative Tutte polynomials

TL;DR

This paper explores the connections between Bollobás-Riordan and relative Tutte polynomials, providing new formulas and relations that link ribbon graphs, plane graphs, and virtual links, advancing combinatorial and topological graph theory.

Contribution

It establishes a novel relation between Bollobás-Riordan and relative Tutte polynomials, introduces a duality formula for the latter, and connects virtual link invariants to these polynomials.

Findings

Derived a relation between Bollobás-Riordan and relative Tutte polynomials.

Provided a duality formula for the relative Tutte polynomial of dual plane graphs.

Expressed the Kauffman bracket of virtual links as a specialization of the relative Tutte polynomial.

Abstract

We establish a relation between the Bollobas-Riordan polynomial of a ribbon graph with the relative Tutte polynomial of a plane graph obtained from the ribbon graph using its projection to the plane in a nontrivial way. Also we give a duality formula for the relative Tutte polynomial of dual plane graphs and an expression of the Kauffman bracket of a virtual link as a specialization of the relative Tutte polynomial.