Skein-theoretical derivation of some formulas of Habiro

Abstract

We use skein theory to compute the coefficients of certain power series considered by Habiro in his theory of ${\mathfrak{s}\mathfrak{l}}_2$ invariants of integral homology $3$–spheres. Habiro originally derived these formulas using the quantum group $U_q{\mathfrak{s}\mathfrak{l}}_2$. As an application, we give a formula for the colored Jones polynomial of twist knots, generalizing formulas of Habiro and Le for the trefoil and the figure eight knot.