### Clayton Shonkwiler PRO

Mathematician and artist

/birs24

this talk!

Knot Theory Informed by Random Models and Experimental Data

April 4, 2024

- Symplectic tools lead to efficient algorithms for sampling random polygonal knots in confinement.
- Confinement boosts the likelihood of complicated knots:
*enriched sampling*.

**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 \(K\).

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

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 \(v\).

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

\(v_1\)

\(v_2\)

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

**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 \(\operatorname{sb}[4_1]=3\).

**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.

Examples provide upper bounds!

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

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.

**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 \(\theta_i\).

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

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 \(i\).

**Theorem **[Smith, 1984]

For any convex polytope \(\mathcal{P}\), the hit-and-run Markov chain is uniformly ergodic with respect to Lebesgue measure on \(\mathcal{P}\).

**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.

Generate trillions of random polygons in tight confinement and look for new examples.

Hyperbolic volume

check for uniqueness

(Knot ID, vertices)

DT code

KnotInfo’s

Knot ID

vertices

**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 \(\leq 12\).

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 \(\leq 12\).

\(10_{16}\)

\(10_{84}\)

**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

\(4 \leq \operatorname{b} < \operatorname{sb} \leq \frac{1}{2}\operatorname{stick} \leq 5\).

\(15n_{41,127}\)

\(\overline{13n_{592}}\)

**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.

**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.

**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.)

\(10_{76}\)

…or on Github, along with source code!

Results have also been added to KnotInfo

**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.

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 |

It is straightforward to generate large ensembles of small, complicated knots.

Funding: NSF & Simons Foundation

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

By Clayton Shonkwiler

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