The research project is divided into several parts and includes scientific collaborations with A. Cattaneo (Zurich), R. Kashaev (Geneva), I. Korepanov (Chelyabinsk), and J. Porti (Barcelona).
- Connections between the volume conjecture and the volume form Rinat Kashaev and I are studying the connections between the non-abelian Reidemeister torsion for knot exteriors and the volume conjecture. The volume conjecture is currently a very active field of research and combines quantum invariants with hyperbolic geometry. The main objectives are the following. We are interested in obtaining new asymptotic expansions of certain specialization of the colored Jones polynomial for torus knots in terms of certain integrals with respect to the volume form constructed in my Ph.D. thesis. We next want to understand the geometrical meaning of the non-abelian Reidemeister torsion in such expansions. The case of hyperbolic knots will also be explored as a next step.
- Reidemeister torsion of geometric origin I am also beginning a collaboration with Igor Korepanov. The principal aim of this collaboration is to study connections between two different constructions of a knot invariant (and more generally for 3-dimensional manifolds) based on the theory of Reidemeister torsion. The main purpose of this collaboration is to establish some connections between Korepanov's invariant and a certain specialization of the torsion form that I define in my Ph.D. thesis. Moreover, we expect to obtain an algorithm (implemented on Maple) to explicitly compute this invariant for specific knots (e.g. 2-bridge knots).
- Integrability and integration of the volume form. We are interested in the computation of the integral of the volume form on the manifold of conjugacy classes of regular representations of the knot group. The objectives are to study the existence and to find methods of integration. This problem seems to be connected to the volume conjecture.
Call for proposal
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