Exact volume of hyperbolic 2--bridge links

TL;DR

This paper provides explicit formulas for calculating the exact hyperbolic volume of 2-bridge links directly from their diagrams, improving precision over previous approximation methods.

Contribution

It introduces a novel method to derive exact volumes of hyperbolic 2-bridge links using polynomial equations from link diagrams, advancing beyond approximation techniques.

Findings

Exact volume formulas for hyperbolic 2-bridge links

Polynomial equations derived from link diagrams

Analytic volume computation from roots of polynomials

Abstract

W. Thurston suggested a method for computing hyperbolic volume of hyperbolic 3-manifolds, based on a triangulation of the manifold. The method was implemented by J. Weeks in the program SnapPea, which produces a decimal approximation as a result. For hyperbolic 2-bridge links, we give formulae that allow one to find the exact volume, i.e. to construct a polynomial and to find volume as an analytic function of one of its roots. The computation is performed directly from a reduced, alternating link diagram.