Tribracket Polynomials

TL;DR

This paper introduces a new six-variable polynomial invariant for Niebrzydowski tribrackets, enhancing knot and link invariants by incorporating subtribracket polynomials to distinguish topological structures more effectively.

Contribution

It develops a novel polynomial invariant for tribrackets and establishes conditions for subtribracket polynomial equivalence, improving knot and link classification methods.

Findings

Defined a six-variable polynomial invariant for tribrackets

Established conditions for subtribracket polynomial equivalence

Enhanced the tribracket counting invariant for knots and links

Abstract

We introduce a six-variable polynomial invariant of Niebrzydowski tribrackets analogous to quandle,rack and biquandle polynomials. Using the subtribrackets of a tribracket, we additionally define subtribracket polynomials and establish a sufficient condition for isomorphic subtribrackets to have the same polynomial regardless of their embedding in the ambient tribracket. As an application, we enhance the tribracket counting invariant of knots and links using subtribracket polynomials and provide examples to demonstrate that this enhancement is proper.