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

Abstract

We give presentations of the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In particular, we show the volume conjecture for these knots, which states that the leading terms of the expansions present the hyperbolic volume and the Chern-Simons invariant of the complements of the knots. As higher coefficients of the expansions, we obtain a new series of invariants of these knots.A non-trivial part of the proof is to apply the saddle point method to calculate the asymptotic expansion of an integral which presents the Kashaev invariant. A key step of this part is to give a concrete homotopy of the (real 3-dimensional) domain of the integral in C 3 in such a way that the boundary of the domain always stays in a certain domain in C 3 given by the potential function of the hyperbolic structure. Classification (2010). Primary: 57M27. Secondary: 57M25, 57M50. Mathematics Subject(3) directly. We give such a homotopy concretely in Sections 3.5, 4.5, 5.5 for the 6 1 , 6 2 , 6 3 knots respectively. This part of the proof is quite non-trivial; in fact, we note that we can not make such a homotopy for the 7 2 knot as shown in [21].By the method of this paper, the asymptotic behavior of the Kashaev invariant is discussed for the hyperbolic knots with 7 crossings in [21] and for some hyperbolic knots with 8 crossings in [24].

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

“…Hence, we obtain (27) from this formula similarly as in the proof of Proposition 3.2. The above expansion is concretely calculated as…”

“…to a neighborhood of 0 ∈ C n is homotopy equivalent to S n−1 . Let D be an oriented n-ball embedded in C n such that ∂D is included in the domain (26) whose inclusion is homotopic to a homotopy equivalence to the above S n−1 in the domain (26).…”

“…By the method of this paper, the asymptotic behavior of the Kashaev invariant is discussed for the hyperbolic knots with up to 7 crossings in [27,26] and for some hyperbolic knots with 8 crossings in [34].…”

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 Kashaev invariant and the twisted Reidemeister torsion of two-bridge knots

“…As for rigorous proofs for other hyperbolic knots, it is shown by the first author in [17; 18] and the first author and Yokota [20] that for any hyperbolic knot K with up to 7 crossings, the asymptotic expansions of the Kashaev invariant of K is represented by (2)…”