Relative Tutte Polynomials for Coloured Graphs and Virtual Knot Theory

Relative Tutte Polynomials of Tensor Products of Coloured Graphs

The tensor product (G1, G2) of a graph G1 and a pointed graph G2 (containing one distinguished edge) is obtained by identifying each edge of G1 with the distinguished edge of a separate copy of G2, 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 gra...

Tutte Polynomials of Tensor Products of Signed Graphs and Their Applications in Knot Theory

It is well-known that the Jones polynomial of an alternating knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. Relying on the results of Bollobás and Riordan, we introduce a generalization of Kauffman’s Tutte polynomial of signed graphs for which describing the effect of taking a signed tensor product of signed graphs is very simp...

A Subset Expansion of the Coloured Tutte Polynomial

Bollobás and Riordan introduce a Tutte polynomial for coloured graphs and matroids in [3]. We observe that this polynomial has an expansion as a sum indexed by the subsets of the ground-set of a coloured matroid, generalizing the subset expansion of the Tutte polynomial. We also discuss similar expansions of other contraction–deletion invariants of graphs and matroids. 1. The coloured Tutte pol...

Tutte Polynomials of Flower Graphs

Tutte Polynomials of Signed Graphs and Jones Polynomials of Some Large Knots

It is well-known that the Jones polynomial of a knot is closely related to the Tutte polynomial of a special graph obtained from a regular projection of the knot. In this paper, we study the Tutte polynomials for signed graphs. We show that if a signed graph is constructed from a simpler graph via k-thickening or k-stretching, then its Tutte polynomial can be expressed in terms of the Tutte pol...