A Gordian Pair of Links

TL;DR

This paper constructs a pair of isotopic link configurations that are not thick isotopic, demonstrating a nuanced distinction in link isotopy types while maintaining total length.

Contribution

It introduces a novel pair of links that are isotopic but not thick isotopic, highlighting a subtle difference in link classification.

Findings

Existence of isotopic links not thick isotopic

Preservation of total length in the constructed pair

Advancement in understanding link isotopy distinctions

Abstract

We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.

Full text

A Gordian Pair of Links

Rob Kusner

Dept. of Mathematics & Statistics, University of Massachusetts, Amherst, MA 01003, USA

[profkusner@gmail.com

kusner@math.umass.edu](mailto:profkusner@gmail.com)

 and 

Wöden Kusner

Dept. of Mathematics, Vanderbilt University, Nashville, TN 37240, USA

[w.kusner@vanderbilt.edu

wkusner@gmail.com](mailto:w.kusner@vanderbilt.edu)

Work at the Aspen Center for Physics supported in part by National Science Foundation Award PHY-1607611; WK additionally supported by Austrian Science Fund (FWF) Project 5503 and NSF Award DMS-1516400.

ABSTRACT: We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.

Coward and Hass [5], using tools from [4], gave an example of physically distinct isotopic configurations for a $2$-component link: No isotopy can be performed while preserving the ropelength of each component; however length trading among components, which is more natural in the criticality theory [2, 3] for ropelength, is not allowed. Our configurations $\bf{R}$ and $\bf{W}$ are physically distinct in the broader length-trading sense appropriate for the Gordian unknot and unlink Problems [8]: Do nontrivial ropelength-critical configurations of unknots and unlinks exist? This problem arises in—and possibly obstructs—variational approaches [6] to the Smale Conjecture [7] via the space of unknots, and its generalization to spaces of unlinks [1].

Definition. A pair of link configurations is Gordian if the links are isotopic, but there is no isotopy between them with thickness at least $1$ which preserves total length.

In fact, we prove a stronger statement, in the context of link homotopy and Gehring [2] thickness:

Theorem. The configurations $\bf{R}$ and $\bf{W}$ minimize total ropelength in their common link homotopy class, but there is no link homotopy between them with Gehring thickness at least $1$ while preserving the total ropelength.

Because link homotopy is coarser than isotopy, and since the Gehring thickness constraint is more permissive than that for standard [4] thickness, a fortiori this is a Gordian pair.

Proof of Theorem. (i) For any minimizing configuration in this link homotopy class, each component must be a particular type of stadium curve [2, 4] surrounding 1, 2 or 4 disjoint unit disks; in the last case, there is an interval moduli space of such curves $C$, ranging between the square (depicted above) and equilateral-rhombic configurations of 4 unit disks. (ii) Define a map $\Pi$ from the space of minimizing link configurations in this link homotopy class to the space $\mathcal{C}_{4}(S^{1})$ of 4-point configurations on the circle, taking the given link configuration to the 4 intersection points of $C$ with the planar spanning disks for the 4 components linking $C$. (iii) The image of $\Pi$ lies in the closed subset of $\mathcal{C}_{4}(S^{1})$ where each intersection point lies in one of the 4 curved arcs of $C$, a deformation retract of $\mathcal{C}_{4}(S^{1})$. (iv) The 4-configurations $\Pi(\bf{R})$ and $\Pi(\bf{W})$ lie in distinct path components of $\mathcal{C}_{4}(S^{1})/\textrm{O}(2)$—corresponding to dihedral orders of 4 points on a circle—so there is no path between $\bf{R}$ and $\bf{W}$ in the moduli space of Gehring-ropelength minimizers. ∎

Remark. In forthcoming work (in part with Greg Buck), we develop tools giving a stronger result: The total Gehring ropelength must rise by at least $2$ in any isotopy (or link homotopy) between these minimizing link configurations.