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Exact values of complexity for an infinite number of 3-manifolds. (English) Zbl 1107.57008
The notion of complexity for a 3-dimensional manifold $$M$$ was introduced by S. Matveev in [Acta Appl. Math. 19, 101–130 (1990; Zbl 0724.57012)], as the minimal possible number of vertices of an almost simple spine of $$M$$. In the paper under review the author finds the values of complexity for an infinite class of 3-manifolds $$N_n$$ described as the total space of the punctured torus bundle over the circle $$\mathbb S^1$$ with monodromy $$\left(\begin{smallmatrix} 2 & 1 \\ 1 & 1 \end{smallmatrix}\right) ^n$$, that is, the manifolds $$N_n$$ are $$n$$-fold covers of the figure eight knot complement. By calculating hyperbolic volumes the author proves that the complexity $$c(N_n)$$ of $$N_n$$ is exactly $$2n$$. Then, applying a result of S. V. Matveev and L. Pervova [Dokl. Akad. Nauk 378, No. 2, 151–152 (2001; Zbl 1048.57013)] he shows that if $$M_n$$ is a compact 3-manifold described as the total space of the torus boundle over $$\mathbb S^1$$ with the same monodromy as $$N_n$$, then $$c(M_n)\geq 2Cn$$, with $$C=\log_5((\sqrt 5+1)/2)^2 \sim 0.598$$.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57Q15 Triangulating manifolds