  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.
