On some ternary operations in knot theory

TL;DR

This paper introduces a novel ternary operation-based coloring method for knot diagrams, establishing a new class of knot invariants and generalizing classical group presentations to broader algebraic structures.

Contribution

It presents a new ternary operation framework for coloring knot diagrams, linking to knot and core groups, and extends classical presentations to loops and Moufang loops.

Findings

Colorings are knot invariants.

Relations from knot and core groups are derived.

Generalizations to loops and Moufang loops are achieved.

Abstract

We introduce a way to color the regions of a classical knot diagram using ternary operations, so that the number of colorings is a knot invariant. By choosing appropriate substitutions in the algebras that we assign to diagrams, one obtains the relations from the knot group, and from the core group. Using the ternary operator approach, we generalize the Dehn presentation of the knot group to extra loops, and a similar presentation for the core group to the variety of Moufang loops.