Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs

TL;DR

This paper introduces 2-decompositions for ribbon graphs, providing formulas for their Bollobas-Riordan polynomial, generalizing known results, and applying these to knot theory and mutation in knot diagrams.

Contribution

It defines 2-decompositions of ribbon graphs and derives formulas for their Bollobas-Riordan polynomial, extending classical results to a broader class of graph operations.

Findings

Derived formulas for Bollobas-Riordan polynomial of 2-decompositions

Generalized Brylawski formula for tensor products

Applied results to mutation in knot diagrams

Abstract

We define 2-decompositions of ribbon graphs, which generalise 2-sums and tensor products of graphs. We give formulae for the Bollobas-Riordan polynomial of such a 2-decomposition, and derive the classical Brylawski formula for the Tutte polynomial of a tensor product as a (very) special case. This study was initially motivated from knot theory, and we include an application of our formulae to mutation in knot diagrams.