Levels of knotting of spatial handlebodies (Q2839390)

A spatial handlebody (often called a handlebody knot) is an embedding of a handlebody into \(S^{3}\). For a spatial handlebody, especially for the genus two case, the authors introduce several levels of knotting phenomena by looking at linking of its spines and constituent links. They compare the strength of knotting levels and show various (non-)implications from one to other knotting levels. They also relate the levels of knotting to the topology of the complement, in particular by using a (non-)existence of cut-systems having some special properties. Here a cut system is a family of disjoint properly embedded surfaces whose complement is connected. In the proof of their results, quandle cocycle and coloring invariants and Alexander ideals are used effectively.