Levels of knotting of spatial handlebodies

Abstract

Abstract. Given a (genus 2) cube-with-holes M = S 3 \ H, where H is a handlebody, we relate intrinsic properties of M (like its cut number) with extrinsic features depending on the way the handlebody H is knotted in S 3 . Starting from a first level of knotting that requires the non-existence of a planar spine for H, we define several instances of knotting of H in terms of the non-existence of spines with special properties. Some of these instances are implied by an intrinsic counterpart in terms of the non-existence of special cut-systems for M . We study a natural partial order on these instances of knotting, as well as its intrinsic counterpart, and the relations between them. To this aim, we recognize a few effective "obstructions" 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 M , and on the very classical use of Alexander elementary ideals of the fundamental group of M . Our treatment of the matter also allows us to revisit a few old-fashioned beautiful themes of 3-dimensional geometric topology.

Alexander Quandle Lower Bounds for Link Genera

“…1 is now an easy consequence of Proposition 2.6. In [1] we have used Ishii's quandle coloring invariants of graphs (only exploiting the dihedral case) in order to detect different level of knottings of spatial handlebodies.…”

Dehn surgery and Seifert surface system

“…If W is a handlebody (i.e. W = E(M )), then the theorem follows Lemma 2.2 and Lemma 2.5 (1). Otherwise as g(∂W ) < g(∂M ), by the assumption of the induction, there exists a null-homologous reflexive link L ′ in each component of E(M ) − int W such that handlebodies can be obtained from the component by a 1/Z-Dehn surgery along L ′ .…”

Automorphisms of the 3-sphere that preserve spatial graphs and handlebody-knots

“…Otherwise, (S 3 , V ) is said to be irreducible. It is proved in Benedetti-Frigerio [2] that a handlebody-knot (S 3 , V ) of genus two is irreducible if and only if its exterior E(V ) is boundary-irreducible, i.e. ∂E(V ) is incompressible in E(V ).…”