Hitoshi Murakami
An Introduction to the Volume Conjecture.
Tue, 02/02/2010 - 16:01 — SomNambulAuthors: Hitoshi Murakami
Subjects: Geometric Topology
AbstractThis is an introduction to the Volume Conjecture and its generalizations for
nonexperts. The Volume Conjecture states that a certain limit of the colored
Jones polynomial of a knot would give the volume of its complement. If we
deform the parameter of the colored Jones polynomial we also conjecture that it
would also give the volume and the Chern-Simons invariant of a three-manifold
obtained by Dehn surgery determined by the parameter. I start with a definition
of the colored Jones polynomial and include elementary examples and short
description of elementary hyperbolic geometry.Representations and the colored Jones polynomial of a torus knot.
Mon, 01/18/2010 - 16:01 — SomNambulAuthors: Kazuhiro Hikami, Hitoshi Murakami
Subjects: Geometric Topology
AbstractWe show that for a torus knot the SL(2;C) Chern-Simons invariants and the
SL(2;C) twisted Reidemeister torsions appear in an asymptotic expansion of the
colored Jones polynomial. This suggests a generalization of the volume
conjecture that relates the asymptotic behavior of the colored Jones polynomial
of a knot to the volume of the knot complement.
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