Euclidean volumes of hyperbolic knots

TL;DR

This paper demonstrates that the normalized Euclidean volume associated with deformations of hyperbolic cone-manifolds into Euclidean structures is always an algebraic number, contrasting with the complex nature of hyperbolic volumes.

Contribution

It establishes the algebraic nature of Euclidean volumes in the context of hyperbolic cone-manifold deformations, a novel insight in geometric topology.

Findings

Normalized Euclidean volume is always algebraic.

Contrasts with the complex number-theoretic nature of hyperbolic volumes.

Provides new understanding of volume invariants in geometric structures.

Abstract

The hyperbolic structure on a 3-dimensional cone-manifold with a knot as singularity can often be deformed into a limiting Euclidean structure. In the present paper we show that the respective normalised Euclidean volume is always an algebraic number. This stands in contrast to hyperbolic volumes whose number-theoretic nature is usually quite complicated.