Gordian split links in the Gehring ropelength problem

TL;DR

This paper investigates the properties of thick and thickly embedded links in three-dimensional space, introducing new concepts and explicit examples related to the Gehring ropelength problem, including non-split links and local minima.

Contribution

It defines thickly embedded links, constructs explicit examples of non-split thick links that cannot be split via thick homotopy, and identifies local minima for ropelength in various link classes.

Findings

Explicit example of a thickly embedded 4-component link that cannot be split by thick homotopy

Construction of thick links with 2 components that are local minima for ropelength

First explicit construction of local minima for ropelength in this context

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, we define a thickly embedded link to be a thick link whose open radius-$\tfrac{1}{2}$ normal disk bundles of all components are embedded. The Gehring ropelength problem asks how large the sum of the lengths of the components of a thick (respectively thickly embedded) link must be, given the link homotopy (respectively isotopy) class of the link. A thick homotopy (isotopy) is a link homotopy (isotopy) of a thick (thickly embedded) link that preserves thickness throughout, and such that during the homotopy the total length of the link never exceeds the initial total length. These notions of thick homotopy and isotopy are more permissive than other notions of physical link isotopies in which the length of each individual component must remain constant. We construct an explicit example of a thickly embedded 4-component link which is topologically split but cannot be split by a thick homotopy, and thick links in every homotopy class with 2 components that are non-global local minima for ropelength. This is the first time such local minima for ropelength have been explicitly constructed. In particular, we construct a thick 2-component link in the link homotopy class of the unlink which cannot be split through a thick homotopy.