D. B. A. Epstein and R. C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Dierential Geom. 27 (1988), 67-80.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....volume is an e ective invariant for distinguishing knots. It distinguishes nearly all hyperbolic knots with up to 10 crossings ( AHW91] A noncompact nite volume hyperbolic 3 manifold (e.g. a knot complement) can be canonically decomposed into a nite number of ideal hyperbolic polyhedra ( EP88, Wee93] Ideal tetrahedra form natural building blocks from which to construct hyperbolic 3 manifolds. In [CHW] it is shown that there are exactly 6075 noncompact hyperbolic 3 manifolds which can be obtained by gluing the faces of at most seven ideal hyperbolic tetrahedra. We will refer to this ....

D. B. A. Epstein and R. C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Dierential Geom. 27 (1988), 67-80.

....element in C via the cross ratio. This induces a well defined homomorphism H 3 (PSL(2; C ) Z) Gamma B(C ) Although the cross ratio depends on the particular choice of the base point, the above homomorphism is independent of this choice (see [6, 4. 10) In the cusped case, we note that by [7], each cusped hyperbolic 3 manifold M has an ideal triangulation M = Delta 1 [ Delta Delta Delta [ Delta n : Let z i be the cross ratio of any ordering of the vertices of Delta i consistent with its orientation. The following relation was first discovered by W. Thurston (unpublished ....

D. B. A. Epstein, R. Penner, Euclidean decompositions of non-compact hyperbolic manifolds, J. Diff. Geom. 27 (1988), 67-80.

....H 3 M is the projection. In [40] Thurston shows that any compact hyperbolic 3 manifold has degree one ideal triangulations with jY j M . Ideal triangulations also arise in practice (e.g. in the program SNAPPEA for exploring hyperbolic manifolds [42] as follows. Epstein and Penner in [15] show that any non compact M has a genuine ideal triangulation, that is, one for which f is arbitrarily closely deformable to a homeomorphism (they actually give an ideal polyhedral subdivision; to subdivide these polyhedra into ideal tetrahedra it is conceivable that one may need flat ideal ....

....this vanishes on the extra relations (2) that define P(K) from P 0 (K) we can replace P 0 (K) by P(K) in the above diagram, and the rest of the argument carries through as in the case K = C . Similarly, the argument in the compact case also carries through. Now if M has cusps, then by [15], M has a genuine decomposition into convex ideal polyhedra with vertices at the cusps. We can further subdivide these polyhedra into ideal tetrahedra. These subdivisions may not agree on common faces of the ideal polyhedra, in which case we must add some flat ideal tetrahedra to mediate between ....

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D. B. A. Epstein, R. Penner, Euclidean decompositions of non-compact hyperbolic manifolds, J. Diff. Geom. 27 (1988), 67--80.

.... for deciding whether or not two cusped hyperbolic three manifolds are isometric [Hildebrand and Weeks 1989] and for computing the symmetry group of a cusped hyperbolic threemanifold [Henry and Weeks 1992] The canonical cell decomposition depends on a convex hull construction in Minkowski space [Epstein and Penner 1988; Weeks 1993; Sakuma and Weeks 1995] That construction makes essential use of a manifold s cusps, and does not generalize directly to closed manifolds. Fortunately we can transfer questions about closed manifolds to questions about cusped manifolds by looking at the complement of a set of ....

D. B. A. Epstein and R. Penner, "Euclidean decompositions of noncompact hyperbolic manifolds", J. Diff. Geom. 27 (1988), 67-- 80.

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