  This is a survey talk on one of the best known quantum knot invariants, the
colored Jones polynomial of a knot, and its relation to the algebraic/geometric
topology and hyperbolic geometry of the knot complement. We review several
aspects of the colored Jones polynomial, emphasizing modularity, stability and
effective computations. The talk was given in the Mathematische Arbeitstagung
June 24-July 1, 2011. Updated the bibliography.
