The Tutte polynomial ${T}_G(X,Y)$ of a graph $G$ 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 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.
@article{10_37236_4162,
author = {Vyacheslav Krushkal and David Renardy},
title = {A polynomial invariant and duality for triangulations},
journal = {The electronic journal of combinatorics},
year = {2014},
volume = {21},
number = {3},
doi = {10.37236/4162},
zbl = {1301.05381},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4162/}
}
TY - JOUR
AU - Vyacheslav Krushkal
AU - David Renardy
TI - A polynomial invariant and duality for triangulations
JO - The electronic journal of combinatorics
PY - 2014
VL - 21
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/4162/
DO - 10.37236/4162
ID - 10_37236_4162
ER -
%0 Journal Article
%A Vyacheslav Krushkal
%A David Renardy
%T A polynomial invariant and duality for triangulations
%J The electronic journal of combinatorics
%D 2014
%V 21
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/4162/
%R 10.37236/4162
%F 10_37236_4162
Vyacheslav Krushkal; David Renardy. A polynomial invariant and duality for triangulations. The electronic journal of combinatorics, Tome 21 (2014) no. 3. doi: 10.37236/4162
