A note on the recursive computation of the bracket polynomial for closed 4-tangles

TL;DR

This paper presents a recursive method to compute the bracket polynomial of knots formed by concatenating a 4-tangle shadow n times and closing it without crossings, utilizing a states matrix in the Kauffman 4-strand diagram monoid.

Contribution

It introduces a recursive approach for calculating the bracket polynomial of knots derived from repeated 4-tangle concatenations using a states matrix framework.

Findings

Provides a recursive formula for bracket polynomial computation.

Utilizes the Kauffman 4-strand diagram monoid basis.

Simplifies calculations for complex tangle concatenations.

Abstract

Given a 4-tangle shadow, we concatenate it with itself n times and form a knot by applying a closure operation that connects each top endpoint to the corresponding bottom endpoint on the same side without introducing any crossings. We then compute the Kauffman bracket polynomial for the resulting knot using a states matrix defined with respect to the basis of the Kauffman 4-strand diagram monoid.