Geodesic
Three-dimensional manifolds with poor spines
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and applications, Tome 288 (2015), pp. 38-48

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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 $n$. 
@article{TM_2015_288_a2,
     author = {A. Yu. Vesnin and V. G. Turaev and E. A. Fominykh},
     title = {Three-dimensional manifolds with poor spines},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {38--48},
     publisher = {mathdoc},
     volume = {288},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2015_288_a2/}
}
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A. Yu. Vesnin; V. G. Turaev; E. A. Fominykh. Three-dimensional manifolds with poor spines. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and applications, Tome 288 (2015), pp. 38-48. http://geodesic.mathdoc.fr/item/TM_2015_288_a2/