/davis21

this talk!

- 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!

**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 \(\leq 12\).

**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 \(\leq 12\).

\(10_{16}\)

\(10_{84}\)

**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.

…or on Github, along with source code!

Results have also been added to KnotInfo

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

- 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

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.*

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.

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

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

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

**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 (implemented in plCurve).

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

The same algorithm works *even better* for tightly confined polygons: let \(d_i \leq R\) for all \(i\).

Hyperbolic volume

check for uniqueness

(Knot ID, vertices)

DT code

KnotInfo’s

Knot ID

vertices

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?

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:

New stick number bounds from random sampling of confined polygons

Thomas D. Eddy and Clayton Shonkwiler

*Experimental Mathematics*, to appear, arXiv:1909.00917

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