Clayton Shonkwiler PRO
Mathematician and artist
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
/birs24
this talk!
Knot Theory Informed by Random Models and Experimental Data
April 4, 2024
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 [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!
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
Confined Polygons
If we want to sample polygons in rooted, spherical confinement of radius \(R\), then we simply add the constraints \(d_i \leq R\) for all \(i\).
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 in rooted spherical confinement (implemented in plCurve).
More generally, a uniformly ergodic Markov chain for simulating random points from any toric symplectic manifold.
Strategy
Generate trillions of random polygons in tight confinement and look for new examples.
Pipeline
Hyperbolic volume
check for uniqueness
(Knot ID, vertices)
Verification for New Bounds
DT code
KnotInfo’s
Knot ID
vertices
Exact stick numbers
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 \(\leq 12\).
Stick number bounds
The equilateral stick number of each of the knots \(9_2\), \(9_3\), \(9_{11}\), \(9_{15}\), \(9_{18}\), \(9_{21}\), \(9_{25}\), \(9_{27}\), \(10_8\), \(10_{16}\), \(10_{17}\), \(10_{18}\), \(10_{56}\), \(10_{68}\), \(10_{82}\), \(10_{83}\), \(10_{84}\), \(10_{85}\), \(10_{90}\), \(10_{91}\), \(10_{93}\), \(10_{94}\), \(10_{100}\), \(10_{103}\), \(10_{105}\), \(10_{106}\), \(10_{107}\), \(10_{110}\), \(10_{111}\), \(10_{112}\), \(10_{115}\), \(10_{117}\), \(10_{118}\), \(10_{119}\), \(10_{126}\), \(10_{131}\), \(10_{133}\), \(10_{137}\), \(10_{138}\), \(10_{142}\), \(10_{143}\), \(10_{147}\), \(10_{148}\), \(10_{149}\), \(10_{152}\), \(10_{153}\), and \(10_{164}\) is less than or equal to 10.
The equilateral stick number of each of the knots \(10_3\), \(10_6\), \(10_7\), \(10_{10}\), \(10_{15}\), \(\sout{10_{18}}\), \(10_{20}\), \(10_{21}\), \(10_{22}\), \(10_{23}\), \(10_{24}\), \(10_{26}\), \(10_{28}\), \(10_{30}\), \(10_{31}\), \(10_{34}\), \(10_{35}\), \(10_{38}\), \(10_{39}\), \(10_{43}\), \(10_{44}\), \(10_{46}\), \(10_{47}\), \(10_{50}\), \(10_{51}\), \(10_{53}\), \(10_{54}\), \(10_{55}\), \(10_{57}\), \(10_{58}\), \(10_{62}\), \(10_{64}\), \(10_{65}\), \(10_{66}\), \(\sout{10_{68}}\), \(10_{70}\), \(10_{71}\), \(10_{72}\), \(10_{73}\), \(10_{74}\), \(10_{75}\), \(10_{77}\), \(10_{78}\), \(10_{79}\), \(10_{80}\), \(\sout{10_{82}}\), \(\sout{10_{84}}\), \(10_{95}\), \(10_{97}\), \(\sout{10_{100}}\), and \(10_{101}\) is less than or equal to 11.
The equilateral stick number of each of the knots \(10_{76}\) and \(\sout{10_{80}}\) is less than or equal to 12.
In particular, all knots up to 10 crossings have equilateral stick number \(\leq 12\).
Stick number bounds
\(10_{16}\)
\(10_{84}\)
Higher-crossing knots
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
\(4 \leq \operatorname{b} < \operatorname{sb} \leq \frac{1}{2}\operatorname{stick} \leq 5\).
\(15n_{41,127}\)
\(\overline{13n_{592}}\)
Superbridge indices
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.
Theorem [—]
The knots \(9_3\), \(9_4\), \(9_6\), \(9_9\), \(9_{11}\), \(9_{13}\), \(9_{17}\), \(9_{18}\), \(9_{22}\), \(9_{23}\), \(9_{25}\), \(9_{27}\), \(9_{30}\), \(9_{31}\), and \(9_{36}\) all have superbridge index equal to 4.
Stick numbers for 11-crossing knots
Theorem [in progress]
All prime knots through 11 crossings have stick number \(\leq 13\).
Proof.
I found explicit equilateral examples of all but four 11-crossing knots.
The four missing were \(11a_{175}, 11a_{176}, 11a_{220}, 11a_{306}\), which are all 2-bridge knots, and hence have stick number \(\leq 13\) by a result of Hu–No–Oh.
There are 85 11-crossing knots for which the best bound on stick number is 13: all of these are alternating.
Superbridge index through 11 crossings
Corollary.
All prime 11-crossing knots have \(\operatorname{sb}[K] \leq 6\).
Theorem [—]
All prime knots through 10 crossings have \(\operatorname{sb}[K]\leq 5\).
Proof.
The only 10-crossing knots which could potentially have stick number 12 are \(10_{37}\) and \(10_{76}\).
\(10_{37}\) (cf. Adams et al.)
\(10_{76}\)
Tables and code
…or on Github, along with source code!
Results have also been added to KnotInfo
Conjectures
Conjecture.
For all knots \(K\) with \(c[K]\geq 6\), \(\operatorname{stick}[K] \leq c[K]+2\).
True for 2-bridge knots [Hu–No–Oh], and for \((p,q)\) torus knots with \(2\leq p < q < 3p\) [Adams et al., Jin, Bennett, Adams and Shayler, Johnson et al]. Best general bound is due to Hu–Oh: \(\operatorname{stick}[K] \leq \frac{3}{2}(c[K]+1)\).
Conjecture.
For all knots \(K\) with \(c[K] \geq 7\), \(\operatorname{sb}[K] \leq \lceil\frac{c[K]}{2}\rceil\).
True for torus knots [Kuiper]. Best general bound follows from Huh & Oh’s stick number bound: \(\operatorname{sb}[K] \leq \frac{3}{4}(c[K]+1)\).
Conjecture.
Stick number and equilateral stick number are distinct invariants.
\(\operatorname{stick}[9_{29}]=9\), \(\operatorname{eqstick}[9_{29}]\leq 10\); this is the only example I know where the bounds don’t agree.
Tiny but suggestive data on satellites
| Satellite | DT name | stick no. bound |
|---|---|---|
| Trefoil[–1/2] | K14n22180 | 12 |
| Trefoil[1/2] | K14n26039 | 12 |
| Trefoil[–3/2] | K15n59184 | 12 |
| Trefoil[3/2] | K15n115646 | 12 |
| Trefoil[–1] | K13n4587 | 13 |
| Trefoil[1] | K13n4639 | 13 |
| Trefoil[–3] | K15n40211 | 13 |
| Trefoil[3] | K15n124802 | 13 |
| Trefoil[–5] | K17ns12 | 13 |
| Trefoil[5] | K17ns1 | 13 |
Meta-takeaway
It is straightforward to generate large ensembles of small, complicated knots.
Thank you!
Funding: NSF & Simons Foundation
References
New stick number bounds from random sampling of confined polygons
Thomas D. Eddy and Clayton Shonkwiler
Experimental Mathematics 31 (2022), no. 4, 1373–1395
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 29 (2020), no. 14, 2050096
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
All prime knots through 10 crossings have superbridge index \(\leq 5\)
Clayton Shonkwiler
Journal of Knot Theory and Its Ramifications 31 (2022), no. 4, 2250023
Generating (and Computing with) Very Large Ensembles of Random Polygonal Knots
By Clayton Shonkwiler
