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

TL;DR

This paper provides explicit formulas for Chern-Simons invariants of twist knot orbifolds and explores their relation to A-polynomials and edge polynomials, offering new computational tools and insights into knot invariants.

Contribution

It introduces concrete formulas for Chern-Simons invariants of twist knot orbifolds and links these to A-polynomials and edge polynomials, advancing computational methods in knot theory.

Findings

Explicit Chern-Simons invariants for twist knot orbifolds derived.

Connection established between A-polynomials and edge polynomials.

Number of boundary components for certain surfaces identified as two.

Abstract

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 $2$.