Stavros Garoufalidis and TTQ Le
Abstract: Given 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 $2 \pi i$. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.
Key words: 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, asymptotics, perturbation theory, Kontsevich integral.
Notes: 27 pages, 12 figures.