Tribracket Modules

TL;DR

This paper introduces tribracket modules, an algebraic structure that enhances knot invariants derived from Niebrzydowski tribrackets, providing more powerful tools for distinguishing knots and links.

Contribution

The paper defines tribracket modules and demonstrates how they improve the tribracket counting invariant for knots and links.

Findings

Tribracket modules can be computed explicitly for given knots.

The enhancement distinguishes knots that the original invariant cannot.

The method is effective and proper, improving knot invariants.

Abstract

Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link diagram is an invariant of oriented knots and links. We introduce tribracket modules analogous to quandle/biquandle/rack modules and use these structures to enhance the tribracket counting invariant. We provide examples to illustrate the computation of the invariant and show that the enhancement is proper.