### 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.

- 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

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

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

**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:

\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?

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

- 1,850