Volume 15, issue 4 (2011)

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Asymptotics of the colored Jones function of a knot

Stavros Garoufalidis and Thang T Q Lê

Geometry & Topology 15 (2011) 2135–2180
Abstract

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.

 Dedicated to Louis Kauffman on the occasion of his 60th birthday
Keywords
hyperbolic volume conjecture, colored Jones function, Jones polynomial, $R$–matrices, regular ideal octahedron, weave, hyperbolic geometry, Catalan's constant, Borromean rings, cyclotomic expansion, loop expansion, asymptotic expansion, WKB, $q$–difference equations, perturbation theory, Kontsevich integral
Primary: 57N10
Secondary: 57M25