On the 3D-index of finite cyclic covers of hyperbolic knot complements

Abstract

The 3D-index is an invariant of a 3-manifold with cusps, which would be related to the volume conjecture, and it would be useful to study properties of this invariant. In this paper, we calculate the 3D-index of the nth cyclic covers of hyperbolic knot complements, and show that the dth coefficient of this 3D-index is equal to a polynomial in n of degree ≤ 2d for any sufficiently large n. In particular, we calculate these polynomials concretely for lower degrees for the 4₁, 5₂, 6₁ knots.