A noncommutative generalization of Thurston’s gluing equations

In his famous Princeton Notes, Thurston introduced the so-called gluing equations defining the deformation variety. Later, Kashaev defined a noncommutative 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, Kashaev, [Formula: see text]-groupoids in knot theory, Geom. Dedicata 150(1) (2010) 105–130; Kashaev, Noncommutative teichmüller spaces and deformation varieties of knot completeness; Kashaev, Delta-groupoids and ideal triangulation in Chern–Simons gauge theory: 20 Years After, AMS/IP Studies Advanced Mathematics, Vol. 50 (American Mathematical Society, RI, 2011). In this paper, we prove that this is true for any knot complement in a homology sphere. We also analyze some examples on other manifolds.