Trigonometrical identities and inequalities for knots and links
by
Alexander Mednykh
Sobolev Institute of Mathematics, Novosibirsk-90, 630090, Russia
Coauthors: Ruslan Shmatkov (Sobolev Institute of Mathematics, Novosibirsk-90, 630090, Russia)

A 3-dimensional Euclidean cone-manifold is a metric space obtained as the quotient space of a disjoint union of a collection of geodesic 3-simplices in the 3-dimensional Euclidean space E³ by an isometric pairing of faces in such a combinatorial fashion that the underlying topological spa+BEE-e is a manifold. Hyperbolic and spherical cone-manifolds are defined similarly.

Such a space possesses a Riemannian metric of constant sectional curvature on the union of the top-dimensional cells and the dimension-2 cells. On each dimension-1 cell, the structure is completely described by an angle, which is the sum of the dihedral angles around all of the dimension-1 simplicial faces which are identified to give the cell. The singular set (or singular geodesics) of a cone-manifold is the closure of all the dimension-1 cells for which this angle, called the cone angle, is not 2+AFw-pi (the Riemannian metric may be extended smoothly over all cells whose angle is 2+AFw-pi).

Cone-manifolds related with elementary knots and links are investigated in the+AH4-papers [1], [2], [3], [4].

In this report trigonometrical identities are established relating cone angles and singular geodesics lengthes of cone-manifolds with the following singular sets: the figure-eight knot, the Whitehead link, the Borromean rings and others knots and links.

Moreover, some geometrical inequalities for spherical, Euclidean and hyperbolic cone-manifolds volumes are obtained. These inequalities can be considered in some sense as 3-dimensional analogues of Toponogov's Theorem about the+AH4-angles.

This report is supported by the RFBR (grant 99-01-00630).

References

[1] W. Thurston, Three-dimensional Geometry and Topology, Princeton University Press, Princeton 1997.

[2] A. Mednykh, A. Rasskazov, On the structure of the canonical fundamental set for the 2-bridge link orbifolds, Universitaet Bielefeld, Sonderforschungsbereich 343, "Discrete Structuren in der Matematik", Preprint, 98-062.

[3] A.D. Mednykh, On the remarkable properties of the hyperbolic Whitehead link cone-manifold, Universitaet Bielefeld, Sonderforschungsbereich 343, "Discrete Structuren in der Matematik", Preprint, 99-039.

[4] R. Shmatkov, On a cone-manifold with the Euclidean structure on the Whitehead link, Universitaet Bielefeld, Sonderforschungsbereich 343, "Discrete Structuren in der Matematik", Preprint, 98-061

Date received: February 15, 2000