### New Stick Number Bounds from Random Sampling of Confined Polygons

Clayton Shonkwiler

http://shonkwiler.org

10.10.20

/utc20

This talk!

### Collaborators

Ryan Blair

Cal State Long Beach

Thomas D. Eddy

(now Fountain)

Nathaniel Morrison

Cal State Long Beach

(now Lancaster University)

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

Theorem [with Eddy]

The equilateral stick number of each of the knots $$9_2$$, $$9_3$$, $$9_{11}$$, $$9_{15}$$, $$9_{21}$$, $$9_{25}$$, $$9_{27}$$, $$10_8$$, $$10_{16}$$, $$10_{17}$$, $$10_{18}$$, $$10_{56}$$, $$10_{82}$$, $$10_{83}$$, $$10_{85}$$, $$10_{90}$$, $$10_{91}$$, $$10_{94}$$, $$10_{100}$$, $$10_{103}$$, $$10_{105}$$, $$10_{106}$$, $$10_{107}$$, $$10_{110}$$, $$10_{111}$$, $$10_{112}$$, $$10_{115}$$, $$10_{117}$$, $$10_{118}$$, $$10_{119}$$, $$10_{126}$$, $$10_{131}$$, $$10_{133}$$, $$10_{137}$$, $$10_{138}$$, $$10_{142}$$, $$10_{143}$$, $$10_{147}$$, $$10_{148}$$, $$10_{149}$$, $$10_{153}$$, and $$10_{164}$$ is less than or equal to 10.

The equilateral stick number of each of the knots $$10_3$$, $$10_6$$, $$10_7$$, $$10_{10}$$, $$10_{15}$$, $$\sout{10_{18}}$$, $$10_{20}$$, $$10_{21}$$, $$10_{22}$$, $$10_{23}$$, $$10_{24}$$, $$10_{26}$$, $$10_{28}$$, $$10_{30}$$, $$10_{31}$$, $$10_{34}$$, $$10_{35}$$, $$10_{38}$$, $$10_{39}$$, $$10_{43}$$, $$10_{44}$$, $$10_{46}$$, $$10_{47}$$, $$10_{50}$$, $$10_{51}$$, $$10_{53}$$, $$10_{54}$$, $$10_{55}$$, $$10_{57}$$, $$10_{62}$$, $$10_{64}$$, $$10_{65}$$, $$10_{68}$$, $$10_{70}$$, $$10_{71}$$, $$10_{72}$$, $$10_{73}$$, $$10_{74}$$, $$10_{75}$$, $$10_{77}$$, $$10_{78}$$, $$\sout{10_{82}}$$, $$10_{84}$$, $$10_{95}$$, $$10_{97}$$, $$\sout{10_{100}}$$, 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

Theorem []

The knots $$8_1$$, $$8_2$$, $$8_3$$, $$8_5$$, $$8_6$$, $$8_7$$, $$8_8$$, $$8_{10}$$, $$8_{11}$$, $$8_{12}$$, $$8_{13}$$, $$8_{14}$$, $$8_{15}$$, $$9_7$$, $$9_{16}$$, $$9_{20}$$, $$9_{26}$$, $$9_{28}$$, $$9_{32}$$, and $$9_{33}$$ all have superbridge index equal to 4.

$$8_{10}$$

$$9_7$$

Proof.

For each knot, $$\mathrm{sb}(K)\leq \frac{1}{2}\mathrm{stick}(K) \leq 5$$ and, using a result of Jeon and Jin, $$4 \leq \mathrm{sb}(K)$$.

Can show that $$\mathrm{sb} = 5$$ is equivalent to a linear system of inequalities with no solution.

### Results

Tables of best current stick number and superbridge index bounds in our papers​

…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

### 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)

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?

Other lower bounds on stick number or superbridge index?

# Thank you!

### References

New stick number bounds from random sampling of confined polygons

Thomas D. Eddy and Clayton Shonkwiler

Experimental Mathematics, to appear, arXiv:1909.00917

Knots with exactly 10 sticks

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

Journal of Knot Theory and Its Ramifications 29 (2020), no. 3, 2050011