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

We calculate the Chern-Simons invariants of the hyperbolic $J(2n,-2m)$ knot orbifolds using the Schl\"{a}fli 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.Comment: 9 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1601.00723, arXiv:1607.0804

An explicit formula for the AA-polynomial of the knot with Conway's notation C(2n,4)C(2n, 4)

An explicit formula for the $A$-polynomial of the knot having Conway's notation $C(2n,4)$ is computed up to repeated factors. Our polynomial contains exactly the same irreducible factors as the $A$-polynomial defined in~\cite{CCGLS1}.Comment: 17 pages, 2 figures, To appear in Topology and its Application