HEISENBERG ACTION IN SKEIN THEORY AND EXTERNAL EDGE CONDITION

In this paper we give an explicit description of the representation matrix of a Heisenberg type action previously constructed by Blanchet et al. Our construction is based on the combinatorial part of Yoshida' s construction. We give the matrix in terms of a ribbon graph and its admissible colorings. We show that the components of the representation matrix satisfy the 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.