Levels of knotting of spatial handlebodies

TL;DR

This paper investigates various levels of knotting in spatial handlebodies of genus 2, relating intrinsic properties of their complements to how they are knotted in three-dimensional space, using modern invariants and classical tools.

Contribution

It introduces new classifications of handlebody knotting levels based on spine properties and develops effective obstructions using quandle invariants and homology tools.

Findings

Defined multiple knotting levels based on spine properties

Established obstructions using quandle-coloring invariants

Connected intrinsic and extrinsic knotting properties

Abstract

Given a (genus 2) cube-with-holes M, i.e. the complement in S^3 of a handlebody H, 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.