On the asymptotic expansion of the Kashaev invariant of the $5_2$ knot

Abstract

We give a presentation of the asymptotic expansion of the Kashaev invariant of the 52 knot. As the volume conjecture states, the leading term of the expansion presents the hyperbolic volume and the Chern-Simons invariant of the complement of the 52 knot. Further, we obtain a method to compute the full Poincare asymptotics to all orders of the Kashaev invariant of the 52 knot.

All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern–Simons Theory

“…For a knot/link complement, there are several developed state-integral models [29,30,31]. For our purpose, in particular, we will use the state-integral model developed in [30,32] (see also [33,34,35,36] for discussion of higher order terms for knot complements). This result was motivated from complex Chern-Simons theory [1,2]; in our context this is natural since Jones polynomial is nothing but the vacuum expectation value of the Wilson line in Chern-Simons theory [4] and an interpretation for the volume conjecture (for a link complement) is provided in [37].…”

On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings

“…We put q = exp(2π √ −1/N ), and put (x) n = (1 − x)(1 − x 2 ) · · · (1 − x n ) for n ≥ 0. It is known [18] (see also [20]) that for any n, m with n ≤ m, (q) n (q) N −n−1 = N, (4) ∑ n≤k≤m 1 (q) m−k (q) k−n = 1.…”

“…More precisely, as for the convergence of 1 N φ(t) as N → ∞, we recall the following proposition. [20]). We fix any sufficiently small δ > 0 and any M > 0.…”

“…The volume conjecture has been rigorously proved for some particular knots and links such as torus knots [15] (see also [4] 1 ), the figure-eight knot (by Ekholm, see also [1] 2 ), Whitehead doubles of (2, p)-torus knots [36], positive iterated torus knots [27], the 5 2 knot [16,20], and some links [8,11,26,27,28,36]; for details see e.g. [17].…”

“…A non-trivial part of the proof is to apply the saddle point method, whose concrete procedure is quite different from the proof for the 5 2 knot in [20]. In this part, we need to calculate the asymptotic behavior of an integral of the following form as N → ∞, ∫…”