Asymptotics of the colored Jones function of a knot

Garoufalidis, Stavros¹ ; Lê, Thang T Q¹

¹ School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332-0160, USA

Geometry & topology, Tome 15 (2011) no. 4, pp. 2135-2180.

Résumé

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.

Détail

DOI : 10.2140/gt.2011.15.2135

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

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