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Blow-ups of ideal triangulations and applications to knot theory
by
William Jaco
Oklahoma State University, USA
Suppose T is an ideal triangulation of a compact 3-manifold M with nonempty boundary a torus. The ideal vertex has a vertex-linking torus (normal with respect to the ideal triangulation). A framing for the vertex-linking torus is a minimum (number of edges) among the collection of spines in the 1-skeleton of the induced triangulation of the vertex-linking torus. There is a construction, depending on the framing, which ``blows-up" the ideal triangulation T to a one-vertex triangulation T* of the 3-manifold M. The one-vertex triangulation T* itself has a natural ``crushing" that gives back the original ideal triangulation T with the torus boundary of M crushing to the ideal vertex. We define this construction and its inverse operation of crushing. Applications are given to Dehn-fillings and exceptional blow-ups are described. The exceptional blow-ups corresponding to the two tetrahedron ideal triangulation of the exterior of the Figure Eight knot are shown to correspond to the exceptional Dehn-fillings of the Figure Eight knot. These constructions can be generalized to ideal triangulations of link exteriors and to ideal triangulations having vertex-linking surfaces of genus greater than one.
Date received: January 31, 2001
Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagh-27.