2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131854One can define the complexity of a smooth 4-manifold as the minimal sum of the number of disks, strands and crossings in a Kirby diagram. Martelli proved that the number of homeomorphism classes of complexity less than n grows as $n^2$. In this paper we prove that the number of diffeomorphism classes grows at least as fast as $n^{c\sqrt[3]{n}}$. Along the way we construct complete kirby diagrams for a large family of knot surgery manifolds.Schematics of Kirby diagrams for the knot surgery manifolds were replaced with actual Kirby diagrams, and minor errors were fixedGeometric Topologytext