Explicit formulae for Chern-Simons invariants of the hyperbolic $J(2n,-2m)$ knot orbifolds

Abstract

We calculate the Chern-Simons invariants of the hyperbolic J(2n, −2m) knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of J(2n, −2m) knot. 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 to a bi-infinite family. We dealt with even slopes just as easily as odd ones. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic J(2n, −2m) knot orbifolds. For the fundamental group of J(2n, −2m) knot, we take and tailor Hoste and Shanahan's. As a byproduct, we give an affirmative answer for their question whether their presentation is actually derived from Schubert's canonical 2-bridge diagram or not.