A polynomial invariant and duality for triangulations

TL;DR

This paper introduces a new 4-variable polynomial invariant for triangulations of manifolds, extending the Tutte polynomial's duality property through topological dualities like Alexander and Poincare duality.

Contribution

It generalizes the Tutte polynomial to higher-dimensional manifolds, establishing duality properties based on topological dualities.

Findings

Polynomial duality follows from Alexander and Poincare duality.

Specializes to known invariants in 2D cases.

Provides examples and evaluations of the polynomial.

Abstract

The Tutte polynomial is a classical invariant, important in combinatorics and statistical mechanics. An essential feature of the Tutte polynomial is the duality for planar graphs G, $T_G(X,Y)\; =\; {T}_{G^*}(Y,X)$ where $G^*$ denotes the dual graph. We examine this property from the perspective of manifold topology, formulating polynomial invariants for higher-dimensional simplicial complexes. 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 Poincare duality on manifolds. In dimension 2 these invariants specialize to the well-known polynomial invariants of ribbon graphs defined by B. Bollobas and O. Riordan. Examples and specific evaluations of the polynomials are discussed.