Narrow Results

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.

Based on the Kauffman bracket at A = e^iπ/4, we define an invariant for a special type of n-punctured ball tangles. The invariant Fⁿ takes values in the set PM_{2 × 2ⁿ}(ℤ) of 2 × 2ⁿ matrices over ℤ modulo the scalar multiplication of ±1. We provide the formula to compute the invariant of the k₁ + ⋯ + k_n-punctured ball tangle composed of given n, k₁, …, k_n-punctured ball tangles. Also, we define the horizontal and the vertical connect sums of punctured ball tangles and provide the formulae for their invariants from those of given punctured ball tangles. In addition, we introduce the elementary operations on the class ST of 1-punctured ball tangles, called spherical tangles. The elementary operations on ST induce the operations on PM_{2 × 2}(ℤ), also called the elementary operations. We show that the group generated by the elementary operations on PM_{2 × 2}(ℤ) is isomorphic to a Coxeter group.