Geometric methods to solving Schläfli differential equations

The classical Schläfli equation relates the volume differential of a non-Euclidean polyhedron to the lengths of its sides and the differentials of its two-sided angles. In the case when the Gaussian curvature of a non-Euclidean space is constant and equal to K, it has the form
$$KdV=\sum_i l_{a_i} da_i$$
where the sum is taken over all edges with lengths $l_{a_i}$, and dihedral angles $a_i$. A similar formula was applied for the volumes of knots and links modeled in non-Euclidean geometry. The report examines polyhedra, knots and links modeled in hyperbolic, spherical and Euclidean geometries. We present the trigonometric identities obtained in [1]-[4] that connect the lengths of edges and angles of these manifolds. Quite interesting things have been discovered: for such lengths and angles, analogs of the school theorems of sines, cosines and tangents are valid. This allowed us to find a completely new geometric approach to solving Schläfli differential equations. As a result, the volumes of various families of polygons, knots and links in hyperbolic and spherical geometries are calculated in quadratures. If the knot is modeled in Euclidean geometry, then the length of the knot itself must be taken as the unit of length. Then it is possible to prove that the Euclidean volume is the root of an algebraic equation with integer coefficients.