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

Clayton Shonkwiler

Colorado State University

shonkwiler.org

/davis21

this talk!

Key Points

Symplectic tools lead to efficient algorithms for sampling random polygonal knots (in confinement).
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.

– Alexander Grosberg

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 \(\theta_i\).