TWO-SIDED BOUNDS FOR THE COMPLEXITY OF CYCLIC BRANCHED COVERINGS OF TWO-BRIDGE LINKS

We consider closed orientable 3-dimensional hyperbolic manifolds which are cyclic branched coverings of the 3-sphere, with branching set being a two-bridge knot (or link). We establish two-sided linear bounds depending on the order of the covering for the Matveev complexity of the covering manifold. The lower estimate uses the hyperbolic volume and results of Cao-Meyerhoff, Gueritaud-Futer (who recently improved previous work of Lackenby), and Futer-Kalfigianni-Purcell, and it comes in two versions: a weaker general form and a shaper form. The upper estimate is based on an explicit triangulation, which also allows us to give a bound on the Delzant T-invariant of the fundamental group of the manifold.