Entropic Niebrzydowski Tribrackets

TL;DR

This paper introduces entropic Niebrzydowski tribrackets, a new algebraic structure analogous to known medial quandles, and explores their properties and potential applications in link invariants.

Contribution

It defines entropic tribrackets, studies their algebraic properties, and proposes their use in distinguishing links with identical counting invariants.

Findings

Defined entropic tribrackets and their properties.

Showed that homsets inherit entropic tribracket structures.

Computed operation tables for small cardinalities.

Abstract

We introduce the notion of entropic Niebrzydowski tribrackets or just entropic tribrackets, analogous to entropic (also known as abelian or medial ) quandles and biquandles. We show that if X is a finite entropic tribracket then for any tribracket T , the homset Hom(T, X) (and in particular, for any oriented link L, the homset Hom(T (L), X)) also has the structure of an entropic tribracket. This operation yields a product on the category of entropic tribrackets; we compute the operation table for entropic tribrackets of small cardinality and prove a few results. We conjecture that this structure can be used to distinguish links which have the same counting invariant with respect to a chosen entropic coloring tribracket X.