An algorithm producing hyperbolicity equations for a link complement in S 3

Abstract

We describe a constructive and effective method for decomposing the complement of an alternating link in the three-sphere into tetrahedra with identifications and vertices removed. Consequently we obtain an algorithm for writing down the byperbolicity equations associated to such decomposition.According to the combined results of [Thl], ] and [Ep.-Pe] (see also [Ne-Za] and [Be-Pc, Chapter El, in the latter of which one could find a selfcontained exposition of the subject) an orientable non-compact hyperbolic three-manifold of finite volume can be decomposed in a finite number of tetrahedra with glued faces and vertices removed; these tetrahedra are isometric to ideal tetrahedra in the hyperbolic three-space and they are parametrized by complex numbers satisfying certain rational equations depending on the combinatorics of the gluings (see Section 3 for definitions and precise statements). Moreover it is well known that such a manifold is the interior of a compact manifold whose boundary consists of tori. The natural question, arising from this topological description, whether the complement of a link in S a can be endowed with a hyperbolic structure, was faced and given important answers in [Th3], [Adl], [Ad2] and [Me2], the results being purely existential as based on Thurston's hyperbolization theorem ([Th2] and [Th4]). On the other hand one can produce examples using the fact that for any three-manifold having a topological decomposition into tetrahedra as the one sketched above the equations can be written down and a solution, if any, actually defines a hyperbolic structure on the manifold.As suggested by these remarks in this paper we provide an algorithm for writing down hyperbolicity equations (associated to a decoraposition into tetrahedra) for the complement of a link in S 3 (not an arbitrary one, but quite a general one; moreover we prove that it is always possible to add components to a link in order to get one to which the algorithm applies).Some basic examples are studied in [Thl] (Thurston's notes are actually the main source for this subject); in [Mel] a description of a more general procedure is partially carried out but in a quite rough way (for instance the conditions under which the procedure works are not explicitly stated; on the contrary we collect those we need in step 1 of our algorithm). We actually Geometriae Dedicata 44: 67-104, 1992.

Determining isotopy classes of crossing arcs in alternating links

“…W. Thurston noticed that if we choose a decomposition of a hyperbolic link complement into two polyhedra, where the edges are crossing arcs of a link diagram, then we often obtain an ideal geodesic triangulation from it by subdividing the polyhedra ( [15]). The method was generalized by Menasco for alternating links ( [10]), and by Petronio beyond alternating ( [12]). While this suggests that the arcs are often isotopic to geodesics, the procedure fails to provide an ideal geodesic triangulation with positive-volume tetrahedra in general.…”

AN INFINITE FAMILY OF HYPERBOLIC GRAPH COMPLEMENTS IN S3

“…7. Such diagrams encode the combinatorial rule which defines the face-pairing of P g+1 described in the previous paragraph (see [6] for the details). This implies Proposition 2.1.…”

“…Constructing the ideal triangulation In this paragraph we sketch the proof of Proposition 2.1. To this aim we apply (a slight generalization of) the algorithm producing ideal triangulations for link complements in S 3 described in [6]. Such an algorithm can be easily modified in order to work with graphs rather than with links.…”