The invariant of n-punctured ball tangles

Based on the Kauffman bracket at $A=e^{i \pi/4}$, we defined an invariant for a special type of $n$-punctured ball tangles. The invariant $F^n$ takes values in the set $PM_{2\times2^n}(\mathbb Z)$ of $2\times 2^n$ matrices over $\mathbb Z$ modulo the scalar multiplication of $\pm1$. We provide the formula to compute the invariant of the $k_1 + ... + k_n$-punctured ball tangle composed of given $n,k_1,...,k_n$-punctured ball tangles. Also, we define the horizontal and the vertical connect sums of punctured ball tangles and provide the formulas for their invariants from those of given punctured ball tangles. In addition, we introduce the elementary operations on the class $\textbf{\textit{ST}}$ of 1-punctured ball tangles, called spherical tangles. The elementary operations on $\textbf{\textit{ST}}$ induce the operations on $PM_{2\times2}(\mathbb Z)$, also called the elementary operations. We show that the group generated by the elementary operations on $PM_{2\times2}(\mathbb Z)$ is isomorphic to a Coxeter group.