Properties of Euclidean Whitehead link cone-manifolds
The author considers cone-manifolds \(W(m,n)\) with underlying space the 3-sphere and singular set the Whitehead link with cone-angles \(2\pi/m\) and \(2\pi/n\). By using results of Mednykh and Rasskazov, a fundamental polyhedron is constructed for \(\pi_1(W(m,n))\) in Euclidean space. Values of \(m\) and \(n\) for which \(W(m,n)\) admits a Euclidean geometric structure are found. Also some relations for lengths of the singular geodesics in \(W(m,n)\) and the cone angles are obtained (a tangent rule and a law of sines). These formulas are analogous to those in the hyperbolic case obtained by \textit{A. D. Mednykh} in [Knots in Hellas '98. Proceedings of the international conference on knot theory and its ramifications, European Cultural Centre of Delphi, Greece, August 7--15, 1998. Singapore: World Scientific. Ser. Knots Everything. 24, 290--305 (2000; Zbl 0983.53027)].
scientific article; zbMATH DE number 2102653