Gordian split links in the Gehring ropelength problem

Abstract

A thick link is a link in $\mathbb{R}^3$ such that each component of the link lies at distance at least $1$ from every other component. Strengthening the notion of thickness, a thickly embedded link is a thick link whose open radius-$\tfrac{1}{2}$ normal disk bundles of all components are embedded. A thick homotopy is a link homotopy of a thick link that preserves thickness and total length throughout. A thick isotopy is a link isotopy of a thickly embedded link that preserves thick-embeddedness and total length throughout. We construct an isotopically Gordian split link, that is, a thickly embedded 4-component link which is topologically split but which cannot be split by a thick isotopy. This is the first time a Gordian split link is shown to exist in this most permissive setting where length trading between components is allowed. We then prove for the first time that local, non-global minima for Gehring ropelength exist. In particular, we construct a 2-component homotopically Gordian unlink, that is, a link in the link homotopy class of the unlink which cannot be split by any thick homotopy.