On the Tutte-Krushkal-Renardy polynomial for cell complexes

Details On the Tutte-Krushkal-Renardy polynomial for cell complexes On the Tutte-Krushkal-Renardy polynomial for cell complexes

2012-04-16

arxiv.org/abs/1204.3563v1

Mathematics

Combinatorics

9 pages, 2 figures

Scientific paper

Recently V.Krushkal and D.Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A.Duval, C.Klivans, and J.Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decomposition of a sphere, this modified polynomial satisfies the same duality identity as before. We find that evaluating the Tutte-Krushkal-Renardy along a certain line is the Bott polynomial. Finally we prove skein relations for the Tutte-Krushkal-Renardy polynomial.

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