Delta-groupoids and ideal triangulations

TL;DR

This paper introduces Delta-groupoids, algebraic structures modeling truncated tetrahedron combinatorics, and demonstrates their application to ideal triangulations of three-manifolds with detailed ring constructions.

Contribution

It defines Delta-groupoids, connects them to ideal triangulations, and explores associated rings through examples from knot theory.

Findings

Delta-groupoids model tetrahedral combinatorics

Application to ideal triangulations of 3-manifolds

Explicit ring constructions for examples from knot theory

Abstract

A Delta-groupoid is an algebraic structure which axiomatizes the combinatorics of a truncated tetrahedron. By considering two simplest examples coming from knot theory, we illustrate how can one associate a Delta-groupoid to an ideal triangulation of a three-manifold. We also describe in detail the rings associated with the Delta-groupoids of these examples.