Gordian split links in the Gehring ropelength problem

TL;DR

This work demonstrates that thickness constraints in the Gehring ropelength problem enforce nontrivial Gordian behavior across link classes. It constructs a 4-component thickly embedded Gordian split link that cannot be split by any thick homotopy, and proves that in every 2-component link-homotopy class there exist thick links that are local (but not global) minimizers, i.e., sinks, yielding Gordian pairs in all such classes. It also establishes a Gordian unlink in the 2-component case. The results collectively reveal significant energy-landscape features of ropelength under thickness constraints and provide constructive methods—via center-of-mass disks, Brouwer degree, mountain-climbing, and circle decompositions—to produce and certify Gordian phenomena.

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.

Figures (9)

• Figure 1: This plot shows the main curve $M$ in black, the two auxiliary curves $A^1$ and $A^2$ in green and the center curve $C$ in blue. The dotted line joins $M(a)$ and $M(b)$.
• Figure 2: A plot of the thickly embedded Gordian split link $L$
• Figure 3: Plot of the radius-$\tfrac{1}{2}$ tubes around the link $L$.
• Figure 4: A plot of the $XZ$-plane. The auxiliary curves $A^1$ and $A^2$ are plotted in green, the forbidden areas belonging to main curve and center curve are plotted in gray and blue respectively.
• Figure 5: A plot of the $XY$-plane. The main curve $M$ is plotted in black, the forbidden area belonging to the auxiliary curves $A^1$ and $A^2$ is plotted in green and the forbidden area belonging to center curve $C$ is plotted in blue.

Theorems & Definitions (74)

• Definition 4.1
• Definition 4.2
• Definition 4.3
• Definition 4.4
• Definition 4.5
• Definition 4.6
• Proposition 4.7
• proof
• Definition 4.8
• Definition 4.9