Abstract
Habiro has shown using quantum groups that the colored Jones polynomial
of a knot has a so-called cyclotomic expansion. This expansion is useful
for studying the volume conjecture. It is also an important tool for
constructing Habiro's universal $fsl_2$ invariant of integral homology
$3$-spheres. Explicit computations of this expansion, however, existed only
for a few knots.
In this talk I will explain how to use skein theory to compute the cyclotomic
expansion for twist knots. This is an infinite series of knots including
the trefoil and the figure eight knot, and my formulas generalize previously
known formulas of Habiro and Le for those two knots.
Contact person:Jørgen Ellegaard Andersen.