Geometry & Topology
- Geom. Topol.
- Volume 15, Number 4 (2011), 2135-2180.
Asymptotics of the colored Jones function of a knot
To a knot in –space, one can associate a sequence of Laurent polynomials, whose –th term is the –th colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the –th colored Jones polynomial at , when is a fixed complex number and tends to infinity. We analyze this asymptotic behavior to all orders in when is a sufficiently small complex number. In addition, we give upper bounds for the coefficients and degree of the –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 is near . Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.
Geom. Topol., Volume 15, Number 4 (2011), 2135-2180.
Received: 27 September 2007
Revised: 31 August 2011
Accepted: 4 October 2011
First available in Project Euclid: 20 December 2017
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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
Garoufalidis, Stavros; Lê, Thang T Q. Asymptotics of the colored Jones function of a knot. Geom. Topol. 15 (2011), no. 4, 2135--2180. doi:10.2140/gt.2011.15.2135. https://projecteuclid.org/euclid.gt/1513732364