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On n-punctured ball tangles
by
Xiao-Song Lin
University of California, Riverside
Coauthors: Jae-Wook Chung
We consider a class of topological objects in the 3-sphere S3 which will be called n-punctured ball tangles. Using the Kauffman bracket at A=ei pi /4, an invariant for a special type of n-punctured ball tangles is defined. The invariant F takes values in PM2×2n(Z), that is the set of 2×2n matrices over Z modulo the scalar multiplication of ±1. This invariant leads to a generalization of a theorem of D. Krebes which gives a necessary condition for a given collection of tangles to be embedded in a link in S3 disjointly. We also address the question of whether the invariant F is surjective onto PM2×2n(Z). We will show that the invariant F is surjective when n=0. When n=1, n-punctured ball tangles will also be called spherical tangles. We show that det F(S)=0 or 1 mod 4 for every spherical tangle S. Thus F is not surjective when n=1.
Date received: February 4, 2005
Copyright © 2005 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 # capo-18.