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Shortest vertical geodesics in surgeries on one cusp of the Borromean rings
by
Barbara E. Nimershiem
Franklin & Marshall College
Thurston showed that almost all knots and links in S3 have a complete hyperbolic structure on their complements. An even more surprising result (also by Thurston) is that all but finitely-many Dehn surgeries on a hyperbolic knot or link have a complete hyperbolic structure. This talk will examine, in a very explicit way, these statements for the Borromean rings. Specifically, I will present the standard description of the complete hyperbolic structure on the complement of the Borromean rings. I will also describe an explicit parameterization of the complete structures on the hyperbolic 3-manifolds resulting from Dehn surgeries on one component of the Borromean rings. This parameterization allows us to determine explicitly the shortest geodesic(s) connecting one of the remaining ends to itself. Knowledge of such geodesics enables us to obtain interesting results about lengths of curves in the boundary of a maximal cusp.
Date received: December 5, 2000
Copyright © 2000 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cafy-10.