REVIEW OF: "Fominykh Evgeny - Wiest Bert, Upper bounds for the complexity of torus knot complements, J. Knot Theory Ramifications 22, No. 10, Article ID 1350053, 19 p. (2013)". [DE062240723]

The definition of Matveev complexity c(M) of a compact 3-manifold with nonempty boundary M is based on the existence of an almost simple spine for M: see [Acta Appl. Math. 19 (2), 101130 (1990; Zbl 0724.57012)]. The complexity of a given 3-manifold is generally hard to compute from the theoretical point of view, leaving aside the concrete enumeration of its spines: see, for example, [ACM monogr. 9 (2003; Zbl 1048.57001)] and [Algebr. Geom. Topol. 11 (3), 1257-1265 (2011; Zbl. 1229.57010)], together with their references. In the paper under discussion the authors establish an upper bound for the Matveev complexity of any Seifert fibered 3-manifold with nonempty boundary M, by realizing M as an assembling of several copies of five particular building blocks, whose skeleta contain a known number of true vertices. As a consequence, they obtain potentially sharp bounds on the Matveev complexity of torus knot complements.