A non-commutative generalisation of Thurston's gluing equations

TL;DR

This paper extends Thurston's classical gluing equations into a non-commutative framework, demonstrating their applicability to all knot complements in homology spheres and exploring additional examples.

Contribution

It generalizes the deformation variety to a non-commutative setting and proves the isomorphism with Kashaev's ring for all knot complements in homology spheres.

Findings

The non-commutative ring matches the classical deformation variety for all knot complements in homology spheres.

The approach applies to various examples beyond the initial cases.

Provides a new perspective on the algebraic structures underlying 3-manifold topology.

Abstract

In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a non-commutative ring from H-triangulations of 3-manifolds and observed that for trefoil and figure-eight knot complements the abelianization of this ring is isomorphic to the ring of regular functions on the deformation variety, \cite{Kashaev-definition_delta_groupoid, def_anneau, Kashaev-Delta-groupoids_and_ideal_triangulations}. In this paper, we prove that this is true for any knot complement in a homology sphere. We also analyse some examples on other manifolds.