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3-Manifolds, Tangles and Persistent Invariant
by
Daniel S. Silver
University of South Alabama
Coauthors: Jozef H. Przytycki (GWU), Susan G. Williams (University of South Alabama)
Given a 2n-tangle t embedded in a link l, it is natural to ask which invariants of t necessarily persist as invariants of l. In his 1997 Ph.D. dissertation D. Krebes considered the case of a 4-tangle. He proved that if d divides the determinants of both the numerator closure and the denominator closure of the tangle, then d also divides the determinant of l. Since then two other proofs of Krebes's theorem have been given, one by D. Ruberman using classical algebraic topology and the other by Krebes, Silver and Williams using Temperley-Lieb algebra. Ruberman exploited a well-known relationship between the determinant of a link and the first homology group of its 2-fold cyclic branched cover. From his perspective, Krebes's theorem is a result about invariants of compact, oriented 3-manifolds that persist as invariants of rational homology 3-spheres in which they embed. We extend Ruberman's techniques in order to prove a generalization of Krebes's theorem for 2n-tangles. We also discuss results for the category of virtual links and tangles.
Date received: December 10, 2001
Copyright © 2001 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 # caip-04.