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

The triangulation of manifolds

A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the questions.Comment: 26 pages, 2 figures. version 2: spellings corrected, analytic speculations in 4.8.2 sharpene