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Ideal Turaev-Viro invariants. (English) Zbl 1117.57010
Summary: Turaev-Viro invariants are defined via state sum polynomials associated to special spines of a 3-manifold. Their evaluation at solutions of certain polynomial equations yield 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 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 compute various ideal Turaev-Viro invariants for closed orientable irreducible manifolds of complexity up to 9. This is an outline of the author’s paper [Topology Appl. 154, No. 6, 1141–1156 (2007; Zbl 1114.57011)].
MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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