Δ-groupoids in knot theory

A Δ-groupoid is an algebraic structure which axiomatizes the combinatorics of a truncated tetrahedron. It is shown that there are relations of Δ-groupoids to rings, group pairs, and (ideal) triangulations of three-manifolds. In particular, we describe a class of representations of group pairs \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H\subset G}$$\end{document} into the group of upper triangular two-by-two matrices over an arbitrary ring R, and associate to that group pair a universal ring so that any representation of that class factorizes through a respective ring homomorphism. These constructions are illustrated by two examples coming from knot theory, namely the trefoil and the figure-eight knots. It is also shown that one can associate a Δ-groupoid to ideal triangulations of knot complements, and a homology of Δ-groupoids is defined.