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 \(9_{35}\), \(9_{39}\), \(9_{43}\), \(9_{45}\), and \(9_{48}\) is exactly 9.
Corollary
Each of the knots \(9_{35}\), \(9_{39}\), \(9_{43}\), \(9_{45}\), and \(9_{48}\) has superbridge index equal to 4.
\(9_{35}\)
\(9_{39}\)
\(9_{43}\)
\(9_{45}\)
\(9_{48}\)
Proof.
\(\operatorname{b}(K) < \operatorname{sb}(K) \leq \frac{1}{2} \operatorname{stick}(K)\) 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 \(9_2\), \(9_3\), \(9_{11}\), \(9_{15}\), \(9_{21}\), \(9_{25}\), \(9_{27}\), \(10_8\), \(10_{16}\), \(10_{17}\), \(10_{56}\), \(10_{83}\), \(10_{85}\), \(10_{90}\), \(10_{91}\), \(10_{94}\), \(10_{103}\), \(10_{105}\), \(10_{106}\), \(10_{107}\), \(10_{110}\), \(10_{111}\), \(10_{112}\), \(10_{115}\), \(10_{117}\), \(10_{118}\), \(10_{119}\), \(10_{126}\), \(10_{131}\), \(10_{133}\), \(10_{137}\), \(10_{138}\), \(10_{142}\), \(10_{143}\), \(10_{147}\), \(10_{148}\), \(10_{149}\), \(10_{153}\), and \(10_{164}\) is less than or equal to 10.
The equilateral stick number of each of the knots \(10_3\), \(10_6\), \(10_7\), \(10_{10}\), \(10_{15}\), \(10_{18}\), \(10_{20}\), \(10_{21}\), \(10_{22}\), \(10_{23}\), \(10_{24}\), \(10_{26}\), \(10_{28}\), \(10_{30}\), \(10_{31}\), \(10_{34}\), \(10_{35}\), \(10_{38}\), \(10_{39}\), \(10_{43}\), \(10_{44}\), \(10_{46}\), \(10_{47}\), \(10_{50}\), \(10_{51}\), \(10_{53}\), \(10_{54}\), \(10_{55}\), \(10_{57}\), \(10_{62}\), \(10_{64}\), \(10_{65}\), \(10_{68}\), \(10_{70}\), \(10_{71}\), \(10_{72}\), \(10_{73}\), \(10_{74}\), \(10_{75}\), \(10_{77}\), \(10_{78}\), \(10_{82}\), \(10_{84}\), \(10_{95}\), \(10_{97}\), \(10_{100}\), and \(10_{101}\) is less than or equal to 11.
The equilateral stick number of each of the knots \(10_{76}\) and \(10_{80}\) is less than or equal to 12.
In particular, all knots up to 10 crossings have equilateral stick number \(\leq 12\).
\(10_{16}\)
\(10_{84}\)
Theorem [with Blair, Eddy, and Morrison]
The knots \(13n_{592}\) and \(15n_{41,127}\) have bridge index 4, superbridge index 5, and stick number 10.
\(13n_{592}\)
\(15n_{41,127}\)
Proof.
Surjective homomorphism \(\pi_1(S^3 \backslash 15n_{41,127}) \to S_5\), so
\(4 \leq \operatorname{b} < \operatorname{sb} \leq \frac{1}{2}\operatorname{stick} \leq 5\).
\(15n_{41,127}\)
\(\overline{13n_{592}}\)
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 \(\Rightarrow\) conserved quantity
The space of equilateral \(n\)-gons has lots of symmetries...
Rotations around \(n-3\) chords \(d_i\) by \(n-3\) angles \(\theta_i\) commute.
Theorem [with Cantarella]
The joint distribution of \(d_1,\ldots , d_{n-3}\) and \(\theta_1, \ldots , \theta_{n-3}\) are all uniform on their domains.
Therefore, sampling equilateral \(n\)-gons is equivalent to sampling random points in the convex polytope of \(d_i\)’s and random angles \(\theta_i\).
The \((n-3)\)-dimensional moment polytope \(\mathcal{P}_n \subset \mathbb{R}^{n-3}\) is defined by the triangle inequalities
Theorem [Smith, 1984]
For any convex polytope \(\mathcal{P}\), the hit-and-run Markov chain is uniformly ergodic with respect to Lebesgue measure on \(\mathcal{P}\).
The same algorithm works even better for tightly confined polygons: let \(d_i \leq R\) 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