Expansions for the Bollobas-Riordan polynomial of separable ribbon graphs

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.

Version 2 has minor changes. To appear in Annals of Combinatorics

References in corpus (4)

• The Jones polynomial and graphs on surfaces
• Inapproximability of the Tutte polynomial
• The Kauffman bracket of virtual links and the Bollobás-Riordan polynomial
• Knot invariants and the Bollobas-Riordan polynomial of embedded graphs

Cited by in corpus (5)

• The multivariate signed Bollobas-Riordan polynomial
• Partial duality and Bollobas and Riordan's ribbon graph polynomial
• Unsigned state models for the Jones polynomial
• Non-orientable quasi-trees for the Bollobas-Riordan polynomial
• The expansion of polynomial invariants for $2$-decompositions of generalized graphs