New Stick Number Bounds from Random Sampling of Confined Polygons

Clayton Shonkwiler

Colorado State University

http://shonkwiler.org

03.14.20

/cville20

This talk!

Collaborators

Ryan Blair

Cal State Long Beach

Thomas D. Eddy

Colorado State University

Nathaniel Morrison

Cal State Long Beach

Funding: Simons Foundation

Results

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 \(\leq 12\).

Results

\(10_{16}\)

\(10_{84}\)

Results

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

\(4 \leq \operatorname{b} < \operatorname{sb} \leq \frac{1}{2}\operatorname{stick} \leq 5\).

\(15n_{41,127}\)

\(\overline{13n_{592}}\)

Results

Table of best current stick number bounds in our paper​

…or on Github, along with source code!

Results have also been added to KnotInfo

Strategy

Generate hundreds of billions of random polygons in tight confinement and look for new examples.

Some Numbers

  • 220 billion polygons generated
  • Identified knot types of all but 59(!)
  • 93.3% unknots
  • 2455 distinct knot types, 2420 prime, including 11 different 16-crossing knots
  • 50,000 core-hours of CPU time (or 5.7 core-years)

Data for 10-gons

Equilateral Polygons

Continuous symmetry \(\Rightarrow\) conserved quantity

The space of equilateral \(n\)-gons has lots of symmetries...

\(n-3\) commuting symmetries

Rotations around \(n-3\) chords \(d_i\) by \(n-3\) angles \(\theta_i\) commute.

Chord distributions

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 \(\theta_i\).

A polytope

The \((n-3)\)-dimensional moment polytope \(\mathcal{P}_n \subset \mathbb{R}^{n-3}\) is defined by the triangle inequalities

0 \leq d_i \leq 2
1 \leq d_i + d_{i-1}
|d_i - d_{i-1}| \leq 1
0 \leq d_{n-3} \leq 2

Reconstruction

Sampling

Theorem [Smith, 1984]

For any convex polytope \(\mathcal{P}\), the hit-and-run Markov chain is uniformly ergodic with respect to Lebesgue measure on \(\mathcal{P}\).

Confined Polygons

The same algorithm works even better for tightly confined polygons: let \(d_i \leq R\) for all \(i\).

Pipeline

plCurve

vertices

PD code

HOMFLY

SnapPy

Hyperbolic volume

pyknotid

check for uniqueness

(Knot ID, vertices)

Verification

KnotPlot

DT code

KnotInfo’s

KnotFinder

Knot ID

vertices

Questions

Only two examples where the best bound on stick number is different from the best bound on equilateral stick number:

\operatorname{stick}(9_{29}) = 9 \qquad \operatorname{eqstick}(9_{29})\leq 10
\operatorname{stick}(10_{79}) \leq 11 \qquad \operatorname{eqstick}(10_{79})\leq 12

Are there more (low-crossing) examples? Are these distinct invariants?

Other strategies for generating large ensembles of random polygons in tight confinement?

Thank you!

References

Knots with exactly 10 sticks

Ryan Blair, Thomas D. Eddy, Nathaniel Morrison, and Clayton Shonkwiler

Journal of Knot Theory and Its Ramifications, doi:10.1142/S021821652050011X

New Stick Number Bounds from Random Sampling of Confined Polygons

By Clayton Shonkwiler

New Stick Number Bounds from Random Sampling of Confined Polygons

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