Euclidean volumes of hyperbolic knots

Abstract

The hyperbolic structure on a 3 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, which is reminiscent of Sabitov’s theorem (the Bellows Conjecture). This fact also stands in contrast to hyperbolic volumes whose number–theoretic nature is usually quite complicated.