Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots

Clayton Shonkwiler

Colorado State University

shonkwiler.org

/davis21

this talk!

Key Points

  1. Symplectic tools lead to efficient algorithms for sampling random polygonal knots (in confinement).
  2. Confinement boosts the likelihood of complicated knots: enriched sampling.

Elementary Knot Invariants

Definition.

The stick number of a knot \(K\), denoted \(\operatorname{stick}[K]\), is the minimum number of segments needed to create a polygonal version of \(K\).

\(\operatorname{stick}[3_1]=6\)

Elementary Knot Invariants

If \(v \in S^2\) and \(\gamma\) a closed curve, let \(\operatorname{b}_v(\gamma)\) be the number of local maxima of the projection of \(\gamma\) to the line through \(v\).

\(\operatorname{b}_{v_1}(\gamma)=2\)

\(v_1\)

\(v_2\)

\(\operatorname{b}_{v_2}(\gamma)=3\)

Elementary Knot Invariants

Definition.

If \(\gamma\) is a closed curve, its bridge number is

and its superbridge number is

\operatorname{b}(\gamma) := \min_v \operatorname{b}_v(\gamma)
\operatorname{sb}(\gamma) := \max_v \operatorname{b}_v(\gamma).

The bridge index of a knot \(K\) is

and the superbridge index is

\operatorname{b}[K] := \min_{\gamma \sim K} \operatorname{b}(\gamma)
\operatorname{sb}[K] := \min_{\gamma \sim K} \operatorname{sb}(\gamma).

\(\operatorname{b}[4_1]=2\) and \(\operatorname{sb}[4_1]=3\).

(Most of) What is Known

Theorem [Kuiper, Jin]

For any nontrivial knot \(K\),

\(\operatorname{b}[K] < \operatorname{sb}[K] \leq \frac{1}{2} \operatorname{stick}[K].\)

Theorem [Jeon–Jin]

Every knot except \(3_1\) and \(4_1\) and possibly \(5_2\), \(6_1\), \(6_2\), \(6_3\), \(7_2\), \(7_3\), \(7_4\), \(8_4\), and \(8_9\) has superbridge index \(\geq 4\).

Theorem [Calvo]

Every knot except \(3_1\), \(4_1\), \(5_1\), \(5_2\), \(6_1\), \(6_2\), \(6_3\), \(8_{19}\), \(8_{20}\), \(3_1 \# 3_1\), and \(3_1\# -3_1\) has stick number \(\geq 9\).

Theorem [Kuiper, Jin, Adams et al., others]

The superbridge index is known for all torus knots, and the stick number is known for an infinite family of torus knots.

Mantra

Examples provide upper bounds!

Results

Theorem [with Eddy]

The 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.

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 \(9_2\), \(9_3\), \(9_{11}\), \(9_{15}\), \(9_{21}\), \(9_{25}\), \(9_{27}\), \(10_8\), \(10_{16}\), \(10_{17}\), \(10_{18}\), \(10_{56}\), \(10_{68}\), \(10_{82}\), \(10_{83}\), \(10_{85}\), \(10_{90}\), \(10_{91}\), \(10_{94}\), \(10_{100}\), \(10_{103}\), \(10_{105}\), \(10_{106}\), \(10_{107}\), \(10_{110}\), \(10_{111}\), \(10_{112}\), \(10_{115}\), \(10_{117}\), \(10_{118}\), \(10_{119}\), \(10_{126}\), \(10_{131}\), \(10_{133}\), \(10_{137}\), \(10_{138}\), \(10_{142}\), \(10_{143}\), \(10_{147}\), \(10_{148}\), \(10_{149}\), \(10_{153}\), and \(10_{164}\) is less than or equal to 10.

The equilateral stick number of each of the knots \(10_3\), \(10_6\), \(10_7\), \(10_{10}\), \(10_{15}\), \(\sout{10_{18}}\), \(10_{20}\), \(10_{21}\), \(10_{22}\), \(10_{23}\), \(10_{24}\), \(10_{26}\), \(10_{28}\), \(10_{30}\), \(10_{31}\), \(10_{34}\), \(10_{35}\), \(10_{38}\), \(10_{39}\), \(10_{43}\), \(10_{44}\), \(10_{46}\), \(10_{47}\), \(10_{50}\), \(10_{51}\), \(10_{53}\), \(10_{54}\), \(10_{55}\), \(10_{57}\), \(10_{62}\), \(10_{64}\), \(10_{65}\), \(\sout{10_{68}}\), \(10_{70}\), \(10_{71}\), \(10_{72}\), \(10_{73}\), \(10_{74}\), \(10_{75}\), \(10_{77}\), \(10_{78}\), \(\sout{10_{82}}\), \(10_{84}\), \(10_{95}\), \(10_{97}\), \(\sout{10_{100}}\), 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, \(4 \leq \mathrm{sb}[K]\leq \frac{1}{2}\mathrm{stick}[K] \leq 5\).

 

If \(\mathrm{sb} = 5\) for a 10-stick realization, then there exists \(v\in S^2\) so that

\(v^T [e_1 | -e_2 | \cdots | -e_{10} ] \)

has all positive entries. By Gordan’s theorem, this system of linear inequalities is feasible if and only if 

\([e_1 | -e_2 | \cdots | -e_{10}] u = 0\)

cannot be solved for a nonzero vector u with nonnegative entries.

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

  • 390+ billion polygons generated
  • Identified knot types of all but 59(!)
  • 93.3% unknots
  • 2484 distinct knot types, 2449 prime, including 11 different 16-crossing knots
  • \(\approx\)100,000 core-hours of CPU time (or 11.4 core-years)

Data for 10-gons

Random Walks

The freely jointed chain/random walk model for polymers dates back to Kuhn in the 1930s.

Modern polymer physics is based on the analogy between a polymer chain and a random walk.

Simulating random walks in \(\mathbb{R}^3\) is easy

Generate \(n\) independent uniform random points in \(\mathbb{R}^3\) according to your favorite probability distribution and treat them as an ordered list of edge vectors.

Viewpoint

Topologically constrained random walk \(\Leftrightarrow\) point in some (nice!) configuration space

A random polygonal knot is a random walk subject to a topological constraint.

Equilateral Polygons

The space of equilateral \(n\)-gons can be constructed as a symplectic reduction (see Kapovich–Millson and Hausmann–Knutson):

\operatorname{ePol}(n)=(S^2)^n/\!/\!_{\vec{0}}SO(3)

Continuous symmetry \(\Rightarrow\) conserved quantity

\(n-3\) commuting symmetries

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

More precisely, \(\operatorname{ePol}(n)\) is (almost) toric, and the \(d_i\) and \(\theta_i\) are action-angle coordinates.

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 \(\mathcal{P}\).

Simulations

Algorithm (w/ Cantarella)

A uniformly ergodic Markov chain for simulating random equilateral \(n\)-gons (implemented in plCurve).

More generally, a uniformly ergodic Markov chain for simulating random points from any toric symplectic manifold.

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 for New Bounds

KnotPlot

DT code

KnotInfo’s

KnotFinder

Knot ID

vertices

Questions

Our examples are all equilateral; are stick number and equilateral stick number distinct invariants?

\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

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

Other lower bounds on stick number or superbridge index?

Questions

Is there any connection to hyperbolic knot theory?

Most common 11-crossing knots among 10-gons:

DT name SnapPea Census Name
K11n38 K5_13
K11n92 [10 tetrahedra]
K11n12 K9_684
K11n19 K6_22

Also, K13n592 and K15n41127 from the theorem with Blair, Eddy, and Morrison are K8_301 and K6_37, respectively.

DT name SnapPea Census Name
K12n591 K8_154
K12n242 K3_1
K12n749 K8_121
K12n121 K6_10

Most common 12-crossing knots among 11-gons:

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

New computations of the superbridge index

Clayton Shonkwiler

Journal of Knot Theory and Its Ramifications doi:10.1142/S0218216520500960

The symplectic geometry of closed equilateral random walks in 3-space

Jason Cantarella and Clayton Shonkwiler

Annals of Applied Probability 26 (2016), no. 1, 549–596

Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots

By Clayton Shonkwiler

Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots

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