A polynomial invariant and duality for triangulations

Details A polynomial invariant and duality for triangulations A polynomial invariant and duality for triangulations

2010-12-06

arxiv.org/abs/1012.1310v3

Mathematics

Combinatorics

19 pages. v3: minor changes, references added

Scientific paper

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. This result may be formulated in the context of matroid theory, and in fact matroid duality then precisely corresponds to topological 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.

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