Construction and properties of the $t$-invariant

The $t$-invariant is a new invariant of a compact 3-manifold. We construct this invariant by means of special spine theory. Behavior of the $t$-invariant under connected sum and under boundary connected sum is described. One of the Turaev–Viro invariants is expressed through the $t$-invariant. We show that the $t$-invariant fits into the conception of TQFT. We present the values of the $t$-invariant for all closed irreducible orientable 3-manifolds of complexity $\le6$, and for all lens spaces. Also some upper estimate for the number of values of the $t$-invariant of a Seifert manifold over a given closed surface with $n$ exceptional fibers is obtained.