On the volume of double twist link cone-manifolds

TL;DR

This paper derives formulas for the volume and A-polynomial of hyperbolic cone-manifolds associated with double twist links, specifically focusing on cases where the link parameters are equal, expanding understanding of their geometric structures.

Contribution

It provides explicit formulas for the volume and A-polynomial of hyperbolic cone-manifolds of double twist links with equal parameters, a case previously not fully characterized.

Findings

Formulas for the volume of hyperbolic cone-manifolds of $J(2m+1, 2m+1)$.

Explicit A-polynomial 2-tuple for the canonical component of the character variety.

Characterization of the cases with reducible nonabelian $SL_2(C)$-representations.

Abstract

We consider the double twist link $J(2m+1, 2n+1)$ which is the two-bridge link corresponding to the continued fraction $(2m+1)-1/(2n+1)$. It is known that $J(2m+1, 2n+1)$ has reducible nonabelian $SL_2(\mathbb{C})$-character variety if and only if $m=n$. In this paper we give a formula for the volume of hyperbolic cone-manifolds of $J(2m+1,2m+1)$. We also give a formula for the A-polynomial 2-tuple corresponding to the canonical component of the character variety of $J(2m+1,2m+1)$.