A one-variable bracket polynomial for some Turk's head knots

TL;DR

This paper introduces a method to compute the Kauffman bracket polynomial for specific Turk's head knots by analyzing their construction through repeated concatenation of a 3-tangle and using the Kauffman monoid diagram elements.

Contribution

It provides a novel approach to calculating the bracket polynomial for certain complex knots using 3-tangle concatenation and monoid diagrams.

Findings

Computed bracket polynomials for three-lead Turk's head, chain sinnet, and figure-eight chain shadow diagrams.

Demonstrated the use of Kauffman monoid diagrams in evaluating knot invariants.

Established a systematic method for analyzing knots constructed from repeated 3-tangles.

Abstract

We compute the Kauffman bracket polynomial of the three-lead Turk's head, the chain sinnet and the figure-eight chain shadow diagrams. Each of these knots can in fact be constructed by repeatedly concatenating the same 3-tangle, respectively, then taking the closure. The bracket is then evaluated by expressing the state diagrams of the concerned 3-tangle by means of the Kauffman monoid diagram's elements.