Relative Tutte Polynomials for Coloured Graphs and Virtual Knot Theory

Abstract

We introduce the concept of a relative Tutte polynomial of coloured 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 (and 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.

Bollobás–Riordan and Relative Tutte Polynomials

“…In Sect. 6 we obtain the Kauffman bracket of a virtual link in terms of the relative Tutte polynomial, improving the theorem of Diao and Hetyei (2010). Section 7 places our relation between the Bollobás-Riordan polynomial and relative Tutte polynomial within the context of other polynomial invariants of graphs on surfaces.…”

“…One (Chmutov 2009;Chmutov and Pak 2007;Chmutov and Voltz 2008;Dasbach et al 2008;Moffatt 2010) involves graphs on surfaces and a topological version of the Tutte polynomial due to Bollobás and Riordan (2002). Another generalization is based on a relative version of the Tutte polynomial found by Diao and Hetyei (2010). In this paper we establish a direct relation between the Bollobás-Riordan and relative Tutte polynomials that explains how these two generalizations are connected.…”

Reduced Relative Tutte, Kauffman Bracket and Jones Polynomials of Virtual Link Families

“…The relative Tutte polynomial of colored graphs is introduced in the paper [10] by Y. Diao and G. Hetei. An alternative approach to the computation of Bollobás-Riordan polynomial and Kauffman bracket polynomial of virtual KLs, via ribbon graphs, is given by S. Chmutov and I. Pak [11], and S. Chmutov [12].…”

“…First we restrict our attention to graphs corresponding only to alternating virtual KLs, hence we consider graphs corresponding to virtual KLs with all edges labeled by + or 0 (see [10], Sections 2, 5), with variables X + = X, Y + = Y , x + = x, y + = y, X 0 = 1, Y 0 = 1, x 0 = x, y 0 = y. In this setting, we have the following recursive formula for computing the relative Tutte polynomial:…”

“…According to the Theorem 5.4 [10], the relative Tutte polynomial T H (G) (see Section 5.1 [10]) corresponding to a virtual link diagram K gives the Kauffman bracket through the following variable substitution:…”