Explicit formulae for Chern-Simons invariants of the hyperbolic orbifolds of the knot with Conway's notation $C(2n, 3)$

TL;DR

This paper derives explicit formulas for the Chern-Simons invariants of hyperbolic orbifolds associated with a family of knots denoted by $C(2n, 3)$, using the Schl"afli formula and extending existing methods.

Contribution

It provides the first explicit formulas for these invariants and extends previous techniques to a broader class of cone-manifold structures.

Findings

Explicit formulas for Chern-Simons invariants of $C(2n, 3)$ orbifolds.

Application to cyclic coverings of these orbifolds.

Extension of existing methods to new knot families.

Abstract

We calculate the Chern-Simons invariants of the hyperbolic orbifolds of the knot with Conway's notation $C(2n, 3)$ using the Schl\"{a}fli formula for the generalized Chern-Simons function on the family of $C(2n,3)$ cone-manifold structures. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano, and Montesinos-Amilibia and extend the Ham and Lee's methods. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic $C(2n,3)$ orbifolds.