an:05132976
Zbl 1114.57011
King, Simon A.
Ideal Turaev-Viro invariants
EN
Topology Appl. 154, No. 6, 1141-1156 (2007).
0166-8641
2007
j
57M27 13P10 57-04
Turaev-Viro invariant; GrÃ¶bner basis; special spine
For a special spine (with at least \(2\) vertices) \(P\) of a compact \(3\)-manifold \(M\) the \textit{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 \textit{Biedenharn-Elliott equations} and he defines the \textit{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 \textit{ideal Turaev-Viro invariant}, as the Turaev-Viro state sum modulo the Turaev-Viro ideal.
\textit{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 \(9\).
Wolfgang Heil (Tallahassee)