Knot-theoretic ternary groups

TL;DR

This paper explores special ternary groups inspired by knot theory, characterizing their properties and using them to create invariants for curves on surfaces, linking algebraic structures with topological features.

Contribution

It introduces new properties and characterizations of ternary groups satisfying specific axioms from knot theory, and constructs a novel invariant for curves on surfaces.

Findings

Ternary groups satisfying Reidemeister move axioms are characterized.

Semi-commutativity of these groups is established.

A new invariant for curves on compact surfaces is constructed.

Abstract

We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special attributes of such ternary groups, such as semi-commutativity, we construct a ternary invariant of curves immersed in compact surfaces, considered up to flat Reidemeister moves.