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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 \(C=\log_5((\sqrt 5+1)/2)^2 \sim 0.598\).

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