Asymptotics of the colored Jones function of a knot

作者：Garoufalidis Stavros^*; Le Thang T Q

来源：Geometry and Topology, 2011, 15(4): 2135-2180.

DOI：10.2140/gt.2011.15.2135

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

• 出版日期2011