Explicit formulae for Chern-Simons invariants of the twist knot orbifolds and Edge polynomials of twist knots

We calculate the Chern-Simons invariants of the twist knot orbifolds using the Schl\"{a}fli formula for the generalized Chern-Simons function on the family of the twist knot cone-manifold structures. Following the general instruction of Hilden, Lozano, and Montesinos-Amilibia, we here present the concrete formulae and calculations. We use the Pythagorean Theorem \cite{HMP} to relate the complex length of the longitude and the complex distance between the two axes fixed by two generators. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic twist knot orbifolds. We also derive some interesting results. The $A$-polynomials of twist knots are obtained from the complex distance polynomials. Hence the edge polynomials corresponding to the edges of the Newton polygons of A-polynomials of twist knots can be obtained. In particular, the number of boundary components of every incompressible surface corresponding to slope $-4n+2$ appear to be $2$.