Heisenberg action in skein theory and external edge condition

TL;DR

This paper provides an explicit matrix representation of a Heisenberg-type action in skein theory using ribbon graphs, introduces the external edge condition, and generalizes trace formulas to colored surfaces.

Contribution

It offers a new explicit matrix formula for the Heisenberg action in skein theory, incorporating the external edge condition and extending known results to colored surfaces.

Findings

Explicit matrix representation of the Heisenberg action.

Introduction of the external edge condition for ribbon graphs.

Generalized trace formula using Verlinde formula.

Abstract

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.