Levels of knotting of spatial handlebodies

If H is a spatial handlebody, i.e. a handlebody embedded in the 3-sphere, a spine of H is a graph Γ ⊂ S 3 such that H is a regular neighbour-hood of Γ. Usually, H is said to be unknotted if it admits a planar spine. This suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties. Building on this remark, we define several levels of knotting of spatial handlebodies, and we provide a complete description of the relationships between these levels, focusing our attention on the case of genus 2. We also relate the knotting level of a spatial handlebody H to classical topological properties of its complement M = S 3 \ H, such as its cut number. More precisely, we show that if H is not highly knotted, then M admits special cut systems for M , and we discuss the extent to which the converse implication holds. Along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies. These obstructions are based on recent quandle–coloring invariants for spatial handlebodies, on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links, on the analysis of appropriate coverings of handlebody complements, and on the study of the classical Alexander elementary ideals of their fundamental groups.