April 28, 2022. A. D. Mednykh (Sobolev Institute of Mathematics; NSU). Volumes of knots and links in spaces of constant curvature.

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.

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