Ideal Turaev–Viro invariants

Turaev–Viro invariants are defined via state sum polynomials associated to a special spine or a triangulation of a compact 3-manifold. By evaluation of the state sum at any solution of the so-called Biedenharn–Elliott equations, one obtains a homeomorphism invariant of the manifold (“numerical Turaev–Viro invariant”). The Biedenharn–Elliott equations define a polynomial ideal. The key observation of this paper is that the coset of the state sum polynomial with respect to that ideal is a homeomorphism invariant of the manifold (“ideal Turaev–Viro invariant”), stronger than the numerical Turaev–Viro invariants. Using computer algebra, we obtain computational results on several examples of ideal Turaev–Viro invariants, for all closed orientable irreducible manifolds of complexity at most 9.