Evaluations of topological Tutte polynomials

Details Evaluations of topological Tutte polynomials Evaluations of topological Tutte polynomials

2011-08-16

arxiv.org/abs/1108.3321v1

Mathematics

Combinatorics

Scientific paper

We find a number of new combinatorial identities for, and interpretations of evaluations of, the topological Tutte polynomials of Las Vergnas, $L(G)$, and of and Bollob\'as and Riordan, $R(G)$, as well as for the classical Tutte polynomial $T(G)$. For example, we express $R(G)$ and $T(G)$ as a sum of chromatic polynomials, show that $R(G)$ counts non-crossing graph states and $k$-valuations, and reformulate the Four Colour Theorem in terms of $R(G)$. Our main approach is to apply identities for the topological transition polynomial, one involving twisted duals, and one involving doubling the edges of a graph. These identities for the transition polynomial allow us to show that the Penrose polynomial $P(G)$ can be recovered from $R(G)$, a fact that we use to obtain identities and interpretations for $R(G)$. We also consider enumeration of circuits in medial graphs and use this to relate $R(G)$ and $L(G)$ for graphs embedded in low genus surfaces.

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