Knot state asymptotics II, Witten conjecture and irreducible representations

Details Knot state asymptotics II, Witten conjecture and irreducible representations Knot state asymptotics II, Witten conjecture and irreducible representations

2011-07-08

arxiv.org/abs/1107.1646v1

Mathematics

Geometric Topology

45 pages, 2 figures

Scientific paper

This article pursues the study of the knot state asymptotics in the large level limit initiated in "Knot sate Asymptotics I". As a main result, we prove the Witten asymptotic expansion conjecture for the Dehn fillings of the figure eight knot. The state of a knot is defined in the realm of Chern-Simons topological quantum field theory as a holomorphic section on the SU(2)-character manifold of the peripheral torus. In the previous paper, we conjectured that the knot state concentrates on the character variety of the knot with a given asymptotic behavior on the neighborhood of the abelian representations. In the present paper we study the neighborhood of irreducible representations. We conjecture that the knot state is Lagrangian with a phase and a symbol given respectively by the Chern-Simons and Reidemeister torsion invariants. We show that under some mild assumptions, these conjectures imply the Witten conjecture on the asymptotic expansion of WRT invariants of the Dehn fillings of the knot. Using microlocal techniques, we show that the figure eight knot state satisfies our conjecture starting from q-differential relations verified by the colored Jones polynomials. The proof relies on a differential equation satisfied by the Reidemeister torsion along the branches of the character variety, a phenomenon which has not been observed previously as far as we know.

Affiliated with

Also associated with

No associations

Rating

Knot state asymptotics II, Witten conjecture and irreducible representations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Knot state asymptotics II, Witten conjecture and irreducible representations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Knot state asymptotics II, Witten conjecture and irreducible representations will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-222429