Volumes of double twist knot cone-manifolds

TL;DR

This paper derives explicit volume formulas for hyperbolic cone-manifolds associated with double twist knots, a specific class of two-bridge knots, and investigates related Riley polynomials of certain one-relator groups.

Contribution

It provides explicit volume formulas for a broad class of double twist knot cone-manifolds and analyzes their Riley polynomials, advancing understanding of their geometric and algebraic properties.

Findings

Explicit volume formulas for double twist knot cone-manifolds.

Analysis of Riley polynomials for related one-relator groups.

Identification of properties for specific subclasses like twist knots.

Abstract

We give explicit formulae for the volumes of hyperbolic cone-manifolds of double twist knots, a class of two-bridge knots which includes twist knots and two-bridge knots with Conway notation $C(2n,3)$. We also study the Riley polynomial of a class of one-relator groups which includes two-bridge knot groups.