Geodesic
Three-dimensional manifolds with poor spines
Informatics and Automation, 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 $n$. 
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A. Yu. Vesnin; V. G. Turaev; E. A. Fominykh. Three-dimensional manifolds with poor spines. Informatics and Automation, Geometry, topology, and applications, Tome 288 (2015), pp. 38-48. http://geodesic.mathdoc.fr/item/TRSPY_2015_288_a2/

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