Abstract. The concept of adequacy of a link was introduced to determine its crossing number. We give crossing number estimates for the (much larger) class of semiadequate links, in particular improving the previously known estimates for positive links. We also give examples showing that semiadequacy may not be attained in minimal crossing number diagrams. Our approach uses the description of links of a given canonical genus, and is related to skein invariants, the signature, the new concordance invariants, and hyperbolic volume. It allows to settle the A/B-adequacy status of many examples, including all knots up to 10 crossings. As two applications, we describe generic alternating and positive knots of given genus, determining the asymptotical behaviour of the number of such knots, and classify Seifert fibered, and therewith also hyperbolic, Montesinos links.