A polynomial invariant and duality for triangulations

Abstract

The Tutte polynomial TG(X, Y) of a graph G is a classical invariant, impor- tant in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs G, TG(X, Y) = TG∗(Y,X) where G∗ denotes the dual graph. We examine this property from the perspective of manifold topology, formulating polynomial invariants for higher-dimensional simplicial com- plexes. Polynomial duality for triangulations of a sphere follows as a consequence of Alexander duality. The main goal of this paper is to introduce and begin the study of a more general 4-variable polynomial for triangulations and handle decompositions of orientable manifolds. Polynomial duality in this case is a consequence of Poincaré duality on manifolds. In dimension 2 these invariants specialize to the well-known polynomial invariants of ribbon graphs defined by B. Bollobás and O. Riordan. Examples and specific evaluations of the polynomials are discussed.