Three-dimensional manifolds with poor spines

A special spine of a $3$-manifold is said to be poor if it does not contain proper simple subpolyhedra. Using the Turaev–Viro invariants, we establish that every compact $3$-manifold $M$ with connected nonempty boundary has a finite number of poor special spines. Moreover, all poor special spines of the manifold $M$ have the same number of true vertices. We prove that the complexity of a compact hyperbolic $3$-manifold with totally geodesic boundary that has a poor special spine with two $2$-components and $n$ true vertices is equal to $n$. Such manifolds are constructed for infinitely many values of $n$.