Knot-theoretic flocks

TL;DR

This paper explores the algebraic structures called knot-theoretic flocks, characterizing their properties, classifying them up to order 64, and enhancing knot invariants using group actions on their colorings.

Contribution

It introduces a detailed characterization and classification of knot-theoretic flocks and improves knot invariants through group actions on flock colorings.

Findings

Characterized para-associative ternary quasigroups (flocks) for knot theory.

Enumerated all such structures up to order 64.

Enhanced cocycle invariants using group actions.

Abstract

We characterize the para-associative ternary quasigroups (flocks) applicable to knot theory, and show which of these structures are isomorphic. We enumerate them up to order 64. We note that the operation used in knot-theoretic flocks has its non-associative version in extra loops. We use a group action on the set of flock colorings to improve the cocycle invariant associated with the knot-theoretic flock (co)homology.