Ideal Turaev–viro Invariants

A Turaev–Viro invariant is a state sum, i.e., a polynomial that can be read off from a special spine or a triangulation of a compact 3-manifold. If the polynomial is evaluated at the solution of a certain system of polynomial equations (Biedenharn–Elliott equations) then the result is a homeomorphism invariant of the manifold (" numerical Turaev-Viro invariant "). The equation system defines an ideal, and actually the coset of the polynomial with respect to that ideal is a homeomorphism invariant as well (" ideal Turaev–Viro invariant "). It is clear that ideal Turaev–Viro invariants are at least as strong as numerical Turaev–Viro invariants, and we show that there is reason to expect that they are strictly stronger. They offer a more unified approach, since many numerical Turaev–Viro invariants can be captured in a singly ideal Turaev–Viro invariant. Using computer algebra, we obtain computational results on some examples of ideal Turaev–Viro invariants.