Ideal Turaev–Viro invariants

Turaev–Viro invariants are defined via state sum polynomials associated to special spines of a $3$-manifold. Its evaluation at solutions of certain polynomial equations yields a homeomorphism invariant of the manifold, called a numerical Turaev–Viro invariant. The coset of the state sum modulo the ideal generated by the equations also is a homeomorphism invariant of compact $3$-manifolds, called an { it ideal Turaev–Viro invariant}. Ideal Turaev–Viro invariants are at least as strong as numerical ones, without the need to compute any explicit solution of the equations. We computed various ideal Turaev–Viro invariants for closed orientable irreducible manifolds of complexity up to $9$. This is an outline of [5].