AA GORDIAN PAIR OF LINKS
ROB KUSNER AND W ¨ODEN KUSNER
ABSTRACT: We construct a pair of isotopic link configurations that are not thick isotopic while preserving total length.
A Gordian Pair:
Configurations R (rotor) and W (wing) both minimize total ropelength in a common isotopy class; noisotopy between them preserves total ropelength. Coward and Hass [5], using tools from [4], gave an example of physically distinct isotopic config-urations for a 2-component link: No isotopy can be performed while preserving the ropelength ofeach component; however length trading among components, which is more natural in the criticalitytheory [2, 3] for ropelength, is not allowed. Our configurations R and W are physically distinctin 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 arisesin—and possibly obstructs—variational approaches [6] to the Smale Conjecture [7] via the space ofunknots, and its generalization to spaces of unlinks [1].
Definition.
A pair of link configurations is
Gordian if the links are isotopic, but there is noisotopy 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 R and W minimize total ropelength in their common link ho-motopy class, but there is no link homotopy between them with Gehring thickness at least whilepreserving the total ropelength. Because link homotopy is coarser than isotopy, and since the Gehring thickness constraint is morepermissive 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 com-ponent 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 Π fromthe space of minimizing link configurations in this link homotopy class to the space C ( S ) of 4-point configurations on the circle, taking the given link configuration to the 4 intersection pointsof C with the planar spanning disks for the 4 components linking C . (iii) The image of Π liesin the closed subset of C ( S ) where each intersection point lies in one of the 4 curved arcs of C ,a deformation retract of C ( S ). (iv) The 4-configurations Π( R ) and Π( W ) lie in distinct pathcomponents of C ( S ) / O(2)—corresponding to dihedral orders of 4 points on a circle—so there isno path between R and W in the moduli space of Gehring-ropelength minimizers. (cid:3) 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. a r X i v : . [ m a t h . G T ] A ug ROB KUSNER AND W ¨ODEN KUSNER
Remark.
In forthcoming work (in part with Greg Buck), we develop tools giving a strongerresult:
The total Gehring ropelength must rise by at least in any isotopy (or link homotopy)between these minimizing link configurations. References [1] Tara E Brendle and Allen E Hatcher. Configuration spaces of rings and wickets.
Commentarii MathematiciHelvetici , 88(1):131–162, 2013.[2] Jason Cantarella, Joseph HG Fu, Rob Kusner, John M Sullivan, and Nancy C Wrinkle. Criticality for the Gehringlink problem.
Geometry & Topology , 10(4):2055–2115, 2006.[3] Jason Cantarella, Joseph HG Fu, Robert B Kusner, and John M Sullivan. Ropelength criticality.
Geometry &Topology , 18(4):2595–2665, 2014.[4] Jason Cantarella, Rob Kusner, and John M Sullivan. On the minimum ropelength of knots and links.
Inventionesmathematicae , 150(2):257–286, 2002.[5] Alexander Coward and Joel Hass. Topological and physical link theory are distinct.
Pacific Journal of Mathemat-ics , 276(2):387–400, 2015.[6] Michael H Freedman, Zheng-Xu He, and Zhenghan Wang. M¨obius energy of knots and unknots.
Annals of Math-ematics , 139(1):1–50, 1994.[7] Allen E Hatcher. A proof of the Smale conjecture, Diff( S ) (cid:39) O(4).
Annals of Mathematics , 117(2):553–607, 1983.[8] Piotr Pieranski, Sylwester Przybyl, and Andrzej Stasiak. Gordian unknots. arXiv preprint physics/0103080 , 2001.
Dept. of Mathematics & Statistics, University of Massachusetts, Amherst, MA 01003, USA
E-mail address : [email protected], [email protected] Dept. of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
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