The invariant of n-punctured ball tangles

TL;DR

This paper introduces a new invariant for n-punctured ball tangles based on the Kauffman bracket, providing formulas for their composition and operations, and explores the algebraic structure of elementary operations.

Contribution

It defines a novel invariant for n-punctured ball tangles using the Kauffman bracket and establishes its algebraic properties and relations to Coxeter groups.

Findings

The invariant takes values in a set of matrices over integers modulo scalar multiplication.

Formulas are provided for computing invariants of composed tangles and their connect sums.

The elementary operations generate a Coxeter group structure.

Abstract

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.