Construction and properties of the \(t\)-invariant
Special spine theory is a well-known method for representing 3-manifolds. It is known that any compact 3-manifold possesses a special spine and is uniquely recovered from its special spine. Any manifold possesses many different special spines and any two special spines of the same manifold are connected by a sequence of standard moves [see \textit{S. V. Matveev}, Math. USSR, Izv. 31, 423--434 (1988); translation from Izv. Akad. Nauk SSSR 51, 1104--1116 (1987; Zbl 0676.67002), and \textit{R. Piergallini}, Topology, 3rd Natl. Meet., Trieste/Italy 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 18, 391--414 (1988; Zbl 0672.57004)]. In the paper under review, special spine theory is used for constructing a new invariant of compact 3-manifolds: the \(t\)-invariant. This is done by assigning to each special polyhedron a number that is preserved under elementary moves. Therefore, the number assigned to a special spine of a 3-manifold \(M\) is a well-defined invariant of \(M\). It is proved that if \(M\) is a connected sum of 3-manifolds \(M_1\) and \(M_2\), then \(t(M)=t(M_1)t(M_2)\), and that one of the Turaev-Viro invariants [see \textit{V. G. Turaev} and \textit{O. Y. Viro}, Topology 31, 865--902 (1992; Zbl 0779.57009)] can be expressed in terms of the \(t\)-invariant. Turaev-Viro invariants fit into the conception of 3-dimensional topological quantum field theory (TQFT), which is a functor from the 3-dimensional bordism category to the category of vector spaces; for Turaev-Viro invariants instead of the bordism category we must consider a category whose objects are surfaces with a fixed 1-dimensional special spine and the morphisms of the category are 3-manifolds with boundary. In this paper, the \(t\)-invariant admits a similar interpretation. It is also proved that the set of values of the \(t\)-invariant on Seifert manifolds with fixed base and fixed number of singular fibers is finite. A table with the computation of the \(t\)-invariant for all closed irreducible orientable 3-manifolds of complexity at most six completes the paper.
scientific article; zbMATH DE number 1969371