Seminar for Geometry, education and visualization with applications

PROGRAM

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Četvrtak, 24.09.2026. u 17:15, Sala 301f, MI SANU, Kneza Mihaila 36 i On-line
Evgeny Fominykh, St. Petersburg State University
TETRAHEDRAL AND ALMOST TETRAHEDRAL 3-MANIFOLDS
Thurston observed that ideal triangulations are a useful tool to study the geometry and topology of cusped hyperbolic 3-manifolds. Minimal triangulations, that is, topological ideal triangulations using the least number of ideal 3-simplices, are used in census enumeration, and as a platform to study the topology of the manifold using normal surface theory. We define the triangulation complexity c(M) of a 3-manifold M to be the minimum number of tetrahedra in a semi-simplicial triangulation of M. Determining the complexity of a given manifold is a hard problem in general. Upper bounds usually arise from the explicit construction of triangulations, and the difficulty lies in closing the gap between upper and lower bounds. Let v_3 denote the volume of a regular ideal hyperbolic tetrahedron; this is approximately 1.0149. An argument due to Thurston shows that c(M)>=Vol(M)/v_3. In particular, any geometric triangulation only involving regular ideal hyperbolic tetrahedra is minimal. The figure-eight knot complement has such a triangulation. By taking finite sheeted coverings leads to infinitely many minimal triangulations that are geometric and for which the above inequality is an equality. A census of such manifolds, that are called tetrahedral, decomposed into at most 25 ideal regular ideal hyperbolic tetrahedra was given by Fominykh, Garoufalidis, Goerner, Tarkaev and Vesnin. We will define almost tetrahedral manifolds and establish the complexity of any almost tetrahedral manifold. We will also present a method for constructing infinite series of almost tetrahedral manifolds. In particular, this will allow us to construct some infinite series of link complements in the 3-sphere of known triangulation complexity.