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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 $$9$$.

##### MSC:
 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
##### Keywords:
Turaev-Viro invariant; Gröbner basis; special spine
##### Software:
Maple; primdec; SINGULAR; slimgb
Full Text:
##### References:
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