Heisenberg action in skein theory and external edge condition

Details Heisenberg action in skein theory and external edge condition Heisenberg action in skein theory and external edge condition

2011-09-26

arxiv.org/abs/1109.5548v1

Kyushu Journal of Mathematics Volume 64, 59-79, No.1 (2010)

Mathematics

Geometric Topology

19 pages, 9 figures. This is the improved version of arXiv:0708.3309v1

Scientific paper

In this article we give an explicit description of the representation matrix of a Heisenberg type action constructed by Blanchet, Habegger, Masbaum and Vogel. We give the matrix in terms of a ribbon graph and its admissible colorings. We show that components of the representation matrix satisfies the {\it external edge condition}, which is a natural combinatorial/geometric condition for maps from the first homology of the graph. We give the explicit formula of the trace of the action in the case of surfaces with colored structure using the external edge condition, the Verlinde formula and elementary counting arguments. Our formula is a generalization of the results for a surface without colored structure, which are already known.

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