A Tutte polynomial for signed graphs

Abstract

This paper introduces a generalization of the Tutte polynomial [14] that is defined for signed graphs. A signed graph is a graph whose edges are each labelled with a sign (+l or 1). The generalized polynomial will be denoted Q[G] = Q[G](A, B, d). Here G is the signed graph, and the letters A, B, d denote three independent polynomial variables. The polynomial Q[G] can be specialized to the Tutte polynomial, and it satisfies a spanning tree expansion analogous to the spanning tree expansion for the original Tutte polynomial. Planar signed graphs are, by a medial construction, in one-to-one correspondence with diagrams for knots and links. By this correspondence, the polynomial Q[G] specializes to the Kauffman bracket polynomial [5-S] and hence (with a normalization) to the Jones polynomial invariant [3] for knots and links. The Jones polynomial is an important invariant in knot theory. One purpose of this paper is to provide a link between knot theory and graph theory, and to explore a context embracing both subjects. Since the relationship with knots and knot diagrams is the primary motivation for our polynomial, we will explain this connection early in the paper. The first two sections provide graph theoretic and topological background. The reader may wish to begin reading directly in Section 4 and then turn to Section 2 and Section 3 for this background. On the other hand, a direct reading of the sections in order will give an account of the genesis of the polynomial Q[G]. Section 2 discusses chromatic, dichromatic and Tutte polynomials. Section 3 explains the medial graph construction and the relation to the bracket polynomial for unoriented link diagrams. Section 3 also contains a result of independent interest: a reformulation of the definitions of activities in maximal trees (if the graph is disconnected, one should properly refer to maximalforests to denote disjoint collections of trees; we shall speak of trees and ask the reader to read forest for tree when the graphs are disconnected) of a planar graph in terms of properties of Euler trails