Handle numbers of guts of sutured manifolds and nearly fibered knots

Abstract

Extending Haken's Theorem to product annuli and disks for Heegaard splittings of sutured manifolds, we show that the handle number of an irreducible sutured manifold equals the handle number of its guts. We further show that reduced sutured manifolds with torus boundary contained in $S^3$ fall in to three types that generalize the three models of guts of knots that are nearly fibered in the instanton or Heegaard Floer sense. In conjunction with these results and another concerning uniqueness of incompressible Seifert surfaces, we show that while many nearly fibered knots have handle number $2$ and a unique incompressible Seifert surface, some have handle number $4$ and others have extra incompressible Seifert surfaces. Examples of nearly fibered knots with non-isotopic incompressible Seifert surfaces are exhibited.