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ON n-PUNCTURED BALL TANGLES

JAE-WOOK CHUNG

Department of Mathematics, University of California, Riverside, CA 92521, USA

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and

XIAO-SONG LIN

Department of Mathematics, University of California, Riverside, CA 92521, USA

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https://doi.org/10.1142/S0218216506004695Cited by:1 (Source: Crossref)

Abstract

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 = e^iπ/4, an invariant for a special type of n-punctured ball tangles is defined. The invariant F takes values in PM_2×2ⁿ(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 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 PM_2×2ⁿ(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.

Dedicated to Louis H. Kauffman on the occasion of his 60th birthday

Keywords:

AMSC: 57M27

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