The volume conjecture states that a certain limit of the colored Jones polynomial of a knot in
the three-dimensional sphere would give the volume of the knot complement. Here the colored
Jones polynomial is a generalization of the celebrated Jones polynomial and is defined by using
a so-called R-matrix that is associated with the N-dimensional representation of the Lie
algebra sl(2 C). The volume conjecture was first stated by R. Kashaev in terms of his own
invariant defined by using the quantum dilogarithm. Later H. Murakami and J. Murakami proved
that Kashaev's invariant is nothing but the N-dimensional colored Jones polynomial evaluated at
the Nth root of unity. Then the volume conjecture turns out to be a conjecture that relates an
algebraic object the colored Jones polynomial with a geometric object the volume. In this
book we start with the definition of the colored Jones polynomial by using braid presentations
of knots. Then we state the volume conjecture and give a very elementary proof of the
conjecture for the figure-eight knot following T. Ekholm. We then give a rough idea of the
proof that is we show why we think the conjecture is true at least in the case of hyperbolic
knots by showing how the summation formula for the colored Jones polynomial looks like the
hyperbolicity equations of the knot complement. We also describe a generalization of the volume
conjecture that corresponds to a deformation of the complete hyperbolic structure of a knot
complement. This generalization would relate the colored Jones polynomial of a knot to the
volume and the Chern-Simons invariant of a certain representation of the fundamental group of
the knot complement to the Lie group SL(2 C). We finish by mentioning further generalizations
of the volume conjecture.
