Relative Tutte Polynomials for Colored Graphs and Virtual Knot Theory

TL;DR

This paper introduces a new relative Tutte polynomial for colored graphs and demonstrates its application in virtual knot theory, providing an alternative computational method for the Jones polynomial.

Contribution

It defines the relative Tutte polynomial for colored graphs and applies it to compute virtual knot invariants, offering a new approach distinct from ribbon graph methods.

Findings

The relative Tutte polynomial can be computed via a spanning tree expansion.

It enables calculation of the Kauffman bracket and Jones polynomials from face graphs.

Provides an alternative method to ribbon graph approaches in virtual knot theory.

Abstract

We introduce the concept of a relative Tutte polynomial of colored graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly.