ON n-PUNCTURED BALL TANGLES

Abstract

We consider a class of topological objects in the 3-sphere S3which will be called n-punctured ball tangles. Using the Kauffman bracket at A = eiπ/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 × 2nmatrices 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 S3disjointly. 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.