Abstract:
In this report, we study hyperbolic, spherical and Euclidean structures
on a cone manifold whose underlying space is the three-dimensional sphere
and the singular locus is a given knot or link.
We present trigonometric identities relating the lengths of singular
geodesics to the cone angles of such manifolds. Quite curious things were
discovered: for such lengths and angles, analogues of the school theorems
of sines and cosines are valid. This made it possible to find a completely
new approach to solving Schläfli's differential equations, relating the
volume of a manifold to the lengths of singular geodesics and its cone
angles. As a result, it is possible to find the volumes of knots in
hyperbolic, spherical and, the most difficult, Euclidean geometry.
In the Euclidean case, it is necessary to take the length of the knot
itself as the unit of length. Then it is possible to prove that the
Euclidean volume calculated in this way is the root of an algebraic
equation with integer coefficients. This result can be regarded as a
profound generalization of the Sabitov-Gaifullin theorem on the volumes
of Euclidean polyhedra.