Strongly invertible knots, rational-fold branched coverings and hyperbolic spatial graphs

A construction of a spatial graph from a strongly invertible knot was developed by the second author, and a necessary and sufficient condition for the given spatial graph to be hyperbolic was provided as well. The condition is improved in this paper. This enable us to show that certain classes of knots can yield hyperbolic spatial graphs via the construction.Comment: 18 pages, 3 figures; proofs of several results, including Proposition 2 and Lemma 6 has been improved. Mail theorem is also improve

Hyperbolic spatial graphs arising from strongly invertible knots

AbstractSpatial graphs in the three-dimensional sphere are constructed from strongly invertible knots. Such a graph is proved to be hyperbolic, which means that its exterior admits a hyperbolic structure with totally geodesic boundary, if the exterior has no equivalent essential torus, or a pair of tori, with respect to the involution

Generic fundamental polygons for Fuchsian groups

A Dirichlet fundamental polygon for a Fuchsian group is said to be generic if its combinatorial shape is stable under any small permutation of the center of the polygon. Almost all points in the hyperbolic plane are known to be centers of generic fundamental polygons. We prove that the same property holds for points in the boundary of the hyperbolic plane. © 2011 by Pacific Journal of Mathematics

The Tilt Formula for Generalized Simplices in Hyperbolic Space

Abstract. For a simplex in Lorentzian space whose vertices are in the positive light cone, Weeks defined the ``tilt\u27\u27 relative to each face. It gave us an efficient tool for deciding whether or not the dihedral angle between two simplices holding a face in common is convex. He also provided an efficient formula, called the ``tilt formula,\u27\u27 to obtain tilts from the intrinsic hyperbolic structure of the simplex when its dimension is two or three. Sakuma and Weeks generalized it to general dimensions. In this paper we generalize the concept of the tilt and the tilt formula to the case where not all vertices are in the positive light cone. A key to our generalization is to give a correspondence between points and hyperplanes (or half-spaces) in Lorentzian space