Partial duality and Bollobas and Riordan's ribbon graph polynomial

TL;DR

This paper links the duality of ribbon graph polynomials to knot theory, providing a new proof using the HOMFLY polynomial, thus bridging graph theory and knot invariants.

Contribution

It offers a novel interpretation of ribbon graph polynomial duality through knot theory and presents a simplified proof leveraging the HOMFLY polynomial.

Findings

Duality relation can be understood via knot theory.

A simple proof of the relation is provided.

Connects ribbon graph polynomials with knot invariants.

Abstract

Recently S. Chmutov introduced a generalization of the dual of a ribbon (or embedded) graph and proved a relation between Bollobas and Riordan's ribbon graph polynomial of a ribbon graph and its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the relation via the homfly polynomial of a knot.