Asymptotics of the colored Jones function of a knot

Stavros Garoufalidis; Thang T. Q. Le

doi:10.2140/gt.2011.15.2135

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Asymptotics of the colored Jones function of a knot

Stavros Garoufalidis·Thang T. Q. Le

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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 $e^{\a/n}$, when $\a$ is a fixed complex number and $n$ tends to infinity. We analyze this asymptotic behavior to all orders in $1/n$ when $\a$ 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-W.Thurston implies that our bounds are asymptotically optimal. Moreover, we give results for the Generalized Volume Conjecture when $\a$ is near $2 πi$. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.

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BibTeX RIS

Stavros Garoufalidis, Thang T. Q. Le. 2011-08-31. Asymptotics of the colored Jones function of a knot. https://doi.org/10.2140/gt.2011.15.2135

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