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Ideal Turaev-Viro invariants. (English) Zbl 1114.57011
For a special spine (with at least \(2\) vertices) \(P\) of a compact \(3\)-manifold \(M\) the Turaev-Viro state sum is a polynomial whose sums correspond to different colorings of the \(2\)-cells and edges of \(P\). Any two special spines of the same \(3\)-manifold are related by a finite sequence of local \(T\) moves.
The author shows that the changes of the summands of the Turaev-Viro state sum under the \(T\) moves correspond to certain Biedenharn-Elliott equations and he defines the Turaev-Viro ideal in the ring of polynomials on (equivalence classes) of colorings to be the ideal generated by the Biedenharn-Elliott polynomials. He then obtains an invariant of \(M\), the ideal Turaev-Viro invariant, as the Turaev-Viro state sum modulo the Turaev-Viro ideal.
Numerical Turaev-Viro invariants of \(M\) are obtained by evaluating the state sum at any solution of the Biedenharn-Elliott equations. The author shows that the ideal Turaev-Viro invariant is stronger than the numerical Turaev-Viro invariants. Finally he computes (using computer algebra) several examples of ideal Turaev-Viro invariants for all closed orientable irreducible \(3\)-manifolds of Matveev-complexity at most \(9\).

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
57-04 Software, source code, etc. for problems pertaining to manifolds and cell complexes
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