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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

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 eαn, when α is a fixed complex number and n tends to infinity. We analyze this asymptotic behavior to all orders in 1n when α 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 α is near 2πi. 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

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
Mathematical Subject Classification 2000
Primary: 57N10
Secondary: 57M25
Received: 27 September 2007
Revised: 31 August 2011
Accepted: 4 October 2011
Published: 28 October 2011
Proposed: Vaughan Jones
Seconded: Tom Mrowka, Joan Birman
Stavros Garoufalidis
School of Mathematics
Georgia Institute of Technology
Atlanta GA 30332-0160
Thang T Q Lê
School of Mathematics
Georgia Institute of Technology
Atlanta GA 30332-0160