Relative Tutte polynomials of tensor products of colored graphs

TL;DR

This paper extends the formula for computing the Tutte polynomial of tensor products to the realm of relative Tutte polynomials for colored graphs with zero edges, with applications to virtual knot invariants.

Contribution

It generalizes the colored tensor product formula to relative Tutte polynomials, incorporating zero edges and broadening computational tools for virtual knot theory.

Findings

Derived a generalized tensor product formula for relative Tutte polynomials.

Extended the applicability of Tutte polynomial computations to graphs with zero edges.

Facilitated calculations of the Jones polynomial for virtual knots.

Abstract

The tensor product $(G_1,G_2)$ of a graph $G_1$ and a pointed graph $G_2$ (containing one distinguished edge) is obtained by identifying each edge of $G_1$ with the distinguished edge of a separate copy of $G_2$, and then removing the identified edges. A formula to compute the Tutte polynomial of a tensor product of graphs was originally given by Brylawski. This formula was recently generalized to colored graphs and the generalized Tutte polynomial introduced by Bollob\'as and Riordan. In this paper we generalize the colored tensor product formula to relative Tutte polynomials of relative graphs, containing zero edges to which the usual deletion-contraction rules do not apply. As we have shown in a recent paper, relative Tutte polynomials may be used to compute the Jones polynomial of a virtual knot.