Asymptotics of the Colored Jones Function of a Knot

To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose nth term is the nth colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the nth colored Jones polynomial at eα/n, when α is a fixed complex number and n tends to infinity. We analyze this asymptotic behavior to all orders in 1/n when α is a sufficiently small complex number. In addition, we give upper bounds for the coefficients and degree of the nth colored Jones polynomial, with applications to upper bounds in the Generalized Volume Conjecture. Work of Agol-Dunfield-Storm-W.Thurston implies that our bounds are asymptotically optimal. Moreover, we give results for the Generalized Volume Conjecture when α is near 2πi. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.