
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.
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A Gordian Pair of Links
Rob Kusner
Dept. of Mathematics & Statistics, University of Massachusetts, Amherst, MA 01003, USA
[email protected]](mailto:[email protected])
and
Wöden Kusner
Dept. of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
[email protected]](mailto:[email protected])
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 -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 and 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 which preserves total length.
In fact, we prove a stronger statement, in the context of link homotopy and Gehring [2] thickness:
Theorem. The configurations and minimize total ropelength in their common link homotopy class, but there is no link homotopy between them with Gehring thickness at least 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 , ranging between the square (depicted above) and equilateral-rhombic configurations of 4 unit disks. (ii) Define a map from the space of minimizing link configurations in this link homotopy class to the space of 4-point configurations on the circle, taking the given link configuration to the 4 intersection points of with the planar spanning disks for the 4 components linking . (iii) The image of lies in the closed subset of where each intersection point lies in one of the 4 curved arcs of , a deformation retract of . (iv) The 4-configurations and lie in distinct path components of —corresponding to dihedral orders of 4 points on a circle—so there is no path between and 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 in any isotopy (or link homotopy) between these minimizing link configurations.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] Jason Cantarella, Joseph HG Fu, Rob Kusner, John M Sullivan, and Nancy C Wrinkle. Criticality for the Gehring link problem. Geometry & Topology , 10(4):2055–2115, 2006.
- 3[3] Jason Cantarella, Joseph HG Fu, Robert B Kusner, and John M Sullivan. Ropelength criticality. Geometry & Topology , 18(4):2595–2665, 2014.
- 4[4] Jason Cantarella, Rob Kusner, and John M Sullivan. On the minimum ropelength of knots and links. Inventiones mathematicae , 150(2):257–286, 2002.
- 5[5] Alexander Coward and Joel Hass. Topological and physical link theory are distinct. Pacific Journal of Mathematics , 276(2):387–400, 2015.
- 6[6] Michael H Freedman, Zheng-Xu He, and Zhenghan Wang. Möbius energy of knots and unknots. Annals of Mathematics , 139(1):1–50, 1994.
- 7[7] Allen E Hatcher. A proof of the Smale conjecture, Diff ( S 3 ) ≃ similar-to-or-equals superscript 𝑆 3 absent ({S}^{3})\simeq O ( 4 ) 4 (4) . Annals of Mathematics , 117(2):553–607, 1983.
- 8[8] Piotr Pieranski, Sylwester Przybyl, and Andrzej Stasiak. Gordian unknots. ar Xiv preprint physics/0103080 , 2001.
