The volume of hyperbolic cone-manifolds of the knot with Conway's notation $C(2n, 3)$

TL;DR

This paper computes explicit volume formulas for hyperbolic cone-manifolds of the knot family $C(2n, 3)$, extending existing methods and applying them to cyclic coverings, with implications for knot group presentations.

Contribution

It provides explicit volume formulas for $C(2n, 3)$ cone-manifolds and confirms the derivation of their fundamental group presentations from Schubert's diagrams.

Findings

Explicit volume formulas for $C(2n, 3)$ cone-manifolds.

Volumes of cyclic coverings over these knots.

Confirmation of the group presentation derivation from Schubert diagrams.

Abstract

Let $C(2n, 3)$ be the family of two bridge knots of slope $(4n+1)/(6n+1)$. We calculate the volumes of the $C(2n, 3)$ cone-manifolds using the Schl\"{a}fli formula. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano, and Montesinos-Amilibia and extend the Ham, Mednykh, and Petrov's methods. As an application, we give the volumes of the cyclic coverings over those knots. For the fundamental group of $C( 2n, 3)$, 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.