Clayton Shonkwiler PRO
Mathematician and artist
/cville20
This talk!
Ryan Blair
Cal State Long Beach
Thomas D. Eddy
Colorado State University
Nathaniel Morrison
Cal State Long Beach
Funding: Simons Foundation
Theorem [with Eddy]
The stick number and equilateral stick number of each of the knots 935, 939, 943, 945, and 948 is exactly 9.
Corollary
Each of the knots 935, 939, 943, 945, and 948 has superbridge index equal to 4.
935
939
943
945
948
Proof.
b(K)<sb(K)≤21stick(K) and each of these knots has bridge index 3.
Proof.
By work of Calvo, the stick number is bounded below by 9, and we found 9-stick examples.
Theorem [with Eddy]
The equilateral stick number of each of the knots 92, 93, 911, 915, 921, 925, 927, 108, 1016, 1017, 1056, 1083, 1085, 1090, 1091, 1094, 10103, 10105, 10106, 10107, 10110, 10111, 10112, 10115, 10117, 10118, 10119, 10126, 10131, 10133, 10137, 10138, 10142, 10143, 10147, 10148, 10149, 10153, and 10164 is less than or equal to 10.
The equilateral stick number of each of the knots 103, 106, 107, 1010, 1015, 1018, 1020, 1021, 1022, 1023, 1024, 1026, 1028, 1030, 1031, 1034, 1035, 1038, 1039, 1043, 1044, 1046, 1047, 1050, 1051, 1053, 1054, 1055, 1057, 1062, 1064, 1065, 1068, 1070, 1071, 1072, 1073, 1074, 1075, 1077, 1078, 1082, 1084, 1095, 1097, 10100, and 10101 is less than or equal to 11.
The equilateral stick number of each of the knots 1076 and 1080 is less than or equal to 12.
In particular, all knots up to 10 crossings have equilateral stick number ≤12.
1016
1084
Theorem [with Blair, Eddy, and Morrison]
The knots 13n592 and 15n41,127 have bridge index 4, superbridge index 5, and stick number 10.
13n592
15n41,127
Proof.
Surjective homomorphism π1(S3\15n41,127)→S5, so
4≤b<sb≤21stick≤5.
15n41,127
13n592
Table of best current stick number bounds in our paper…
…or on Github, along with source code!
Results have also been added to KnotInfo
Generate hundreds of billions of random polygons in tight confinement and look for new examples.
Data for 10-gons
Continuous symmetry ⇒ conserved quantity
The space of equilateral n-gons has lots of symmetries...
Rotations around n−3 chords di by n−3 angles θi commute.
Theorem [with Cantarella]
The joint distribution of d1,…,dn−3 and θ1,…,θn−3 are all uniform on their domains.
Therefore, sampling equilateral n-gons is equivalent to sampling random points in the convex polytope of di’s and random angles θi.
The (n−3)-dimensional moment polytope Pn⊂Rn−3 is defined by the triangle inequalities
Theorem [Smith, 1984]
For any convex polytope P, the hit-and-run Markov chain is uniformly ergodic with respect to Lebesgue measure on P.
The same algorithm works even better for tightly confined polygons: let di≤R for all i.
Hyperbolic volume
check for uniqueness
(Knot ID, vertices)
DT code
KnotInfo’s
Knot ID
vertices
Only two examples where the best bound on stick number is different from the best bound on equilateral stick number:
Are there more (low-crossing) examples? Are these distinct invariants?
Other strategies for generating large ensembles of random polygons in tight confinement?
New stick number bounds from random sampling of confined polygons
Thomas D. Eddy and Clayton Shonkwiler
Ryan Blair, Thomas D. Eddy, Nathaniel Morrison, and Clayton Shonkwiler
Journal of Knot Theory and Its Ramifications, doi:10.1142/S021821652050011X
By Clayton Shonkwiler