  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.
