$\Delta$-Tribrackets and Link Homotopy

TL;DR

This paper introduces $ abla$-tribrackets, a new algebraic structure, and demonstrates their invariance under link-homotopy, while analyzing classes with trivial invariants and providing examples for future exploration.

Contribution

It defines $ abla$-tribrackets and proves their counting invariants are link-homotopy invariants, expanding the algebraic tools in knot theory.

Findings

$ abla$-tribrackets are invariants of link-homotopy.

Certain classes of tribrackets have trivial invariants for classical knots and links.

Examples and open questions for future research are provided.

Abstract

We define a type of Niebrzydowski tribracket we call $\Delta$-tribrackets and show that their counting invariants are invariants of link-homotopy. We further identify several classes of tribrackets whose counting invariants for oriented classical knots and links are trivial, including vertical tribrackets satisfying the center-involutory condition and horizontal tribrackets satisfying the late-commutativity condition. We provide examples and end with questions for future research.