New Stick Number Bounds from Random Sampling of Confined Polygons

Clayton Shonkwiler

Colorado State University

http://shonkwiler.org

10.10.20

/utc20

This talk!

Collaborators

Ryan Blair

Cal State Long Beach

Thomas D. Eddy

Colorado State University

(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 \(\theta_i\).