Delta-groupoids in knot theory

TL;DR

This paper introduces Delta-groupoids, an algebraic framework linked to three-manifold triangulations and knot complements, and explores their relations to rings, group pairs, and homology theories, with applications in knot theory.

Contribution

It defines Delta-groupoids, connects them to various mathematical structures, and demonstrates their use in knot theory and three-manifold topology.

Findings

Delta-groupoids relate to rings, group pairs, and triangulations.

They can be associated with ideal triangulations of knot complements.

A homology theory for Delta-groupoids is developed.

Abstract

A Delta-groupoid is an algebraic structure which axiomitizes the combinatorics of a truncated tetrahedron. It is shown that there are relations of Delta-groupoids to rings, group pairs, and (ideal) triangulations of three-manifolds. In particular, one can associate a Delta-groupoid to ideal triangulations of knot complements. It is also possible to define a homology theory of Delta-groupoids. The constructions are illustrated by examples coming from knot theory.