THE INVARIANT OF n-PUNCTURED BALL TANGLES

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.