Asymptotic aspects of the Teichmüller TQFT

Abstract

We calculate the knot invariant coming from the Teichmüller TQFT [AK1]. Specifically we calculate the knot invariant for the complement of the knot $6_1$ both in the original [AK1] and the new formulation of the Teichmüller TQFT [AK2] for the one-vertex H-triangulation of $(S^3,6_1)$. We show that the two formulations give equivalent answers. Furthermore we apply a formal stationary phase analysis and arrive at the Andersen- Kashaev volume conjecture as stated in [AK1, Conj. 1]. Furthermore we calculate the first examples of knot complements in the new formulation showing that the new formulation is equivalent to the original one in all the special cases considered. Finally, we provide an explicit isomorphism between the Teichmüller TQFT representation of the mapping class group of a once punctured torus and a representation of this mapping class group on the space of Schwartz class functions on the real line.

To appear in Travaux Mathematiques

References in corpus (5)

• The Volume Conjecture, Perturbative Knot Invariants, and Recursion Relations for Topological Strings
• Generalized Volume Conjecture and the A-Polynomials -- the Neumann-Zagier Potential Function as a Classical Limit of Quantum Invariant
• Complex Quantum Chern-Simons
• Asymptotics of Toeplitz operators and applications in TQFT
• The Hitchin-Witten Connection and Complex Quantum Chern-Simons Theory