Euclidean volumes of hyperbolic knots

Nikolay Abrosimov, Alexander Kolpakov, Alexander Mednykh

Published 2021-07-07Version 1

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.