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 $5_2$ 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 $5_2$ knot. As coefficients of the expansion, we obtain a series of new invariants of the $5_2$ knot.