A Markov Chain Sampler
for Knot Diagrams

Harrison Chapman
Colorado State University
Andrew Rechnitzer
University of British Columbia


Special Session on Mathematical Methods for the Study
of the Three Dimensional Structure of Biopolymers

AMS 2018 Fall Western Sectional Meeting
SFSU, San Francisco CA
October 28th 2018

Knot Diagrams

Crossings are the primary components of knot diagram models

Knot Diagrams and Knot Shadows

4-valent Maps

A 4-valent map is a 4-valent graph embedded in the sphere

Knot Shadows

A knot shadow or plane curve is a 4-valent map with one link component, and vertices viewed as crossings

Knot Diagrams

A knot diagram is a knot shadow with crossing signs indicating depth

Knot Diagrams and Link Diagrams

Crossings as Self-contacts

Crossings may be viewed as self-contacts where enzymes may act to change DNA topology

Strand passage action, e.g. topo-IV

Coherent smoothing action, e.g. XerCD-dif-FtsK complex

Uniform Random Sampling

Direct sampling

No known algorithms for directly sampling knot diagrams uniformly


Related to lack of exact enumeration \(\{k_n\}\) for knot diagrams

Conjecture (Schaeffer-Zinn Justin '04)

\[ k_n \sim C\mu^n n^{\gamma-2}(1 + O(1/\log n)) \]

Uniform Random Sampling

Rejection sampling

Sample 4-valent maps uniformly, but only accept knot diagrams


  • Samples diagrams of specified size \(n\)
  • Distribution is exactly uniform across a size \(n\)
  • Knots are rare; most samples rejected
  • Knots of fixed type are exponentially rare among all knots

Uniform Random Sampling

Monte Carlo (MCMC) sampling

Sample states from Markov process that explores knot diagrams


  • Samples diagrams of all possible sizes
  • Stationary distribution approximately uniform across any given size
  • Only knot diagrams are sampled
  • Extends to sampling diagrams of any fixed type

Diagram Markov Chains

One step of a diagram Markov chain takes as input a knot diagram, performs with some probability a Reidemeister transition, and returns the resulting knot diagram

Shadow Markov Chain

Explore all knot shadows by ignoring crossing signs; get all knot diagrams by adding crossing information

Fixed Knot Type Markov Chain

Explore all diagrams of fixed knot type by respecting crossing signs (c.f. BFACF and lattice polygons)

Reidemeister Transitions


Aperiodic as there is always a chance a transition fails

Connected as all valid transitions have nonzero probability and;

Alexander-Briggs, Reidemeister '27

Any two knot diagrams of knot type \(K\) are related by a finite sequence of Reidemeister moves

We weight transition probabilities to enforce detailed balance, that \[ P(D \to N) \pi(D) = P(N \to D)\pi(N) \]

So the Markov chain is ergodic

Wang-Landau Transition Probabilities


MCMC sampling samples diagrams of arbitrary size, rather than some fixed \(n\)


Choose transition probabilities to enforce distribution on sizes sampled

Wang-Landau Transition Probabilities

Given a priori approximate enumeration data \(\{g_n\}\) so that \(g_n \approx k_n\):

Only perform transitions from \(n\)-crossing diagrams to \(m\) crossing diagrams with probability \(g_n/g_m\)

The approximate enumeration \(\{g_n\}\) can be calculated iteratively using the Markov process itself

Stationary Distribution

C–Rechnitzer '18

This Markov chain has stationary distribution where the probability that an \(n\)-crossing diagram \(D\) is sampled is, \[ \pi(D) \propto \frac{1}{g_n} \approx \frac{1}{k_n}. \]

Knot diagrams are sampled

  • uniformly for any given size, and
  • approximately each size is equally likely

Shadow Markov Chain

Shadow Markov chan is nicer than fixed knot type Markov chain;

Hass-Scott '94, de Graaf-Schrijver '97

There exist paths connecting two shadows \(C\) and \(D\) only involving states with \(\le \max(|C|,|D|)\) crossings

The shadow Markov chain is connected with maximum size limit

Explore statistics for knot shadows and convergence of distribution

A Random Walk Through Shadows

Gauss diagram animation for 10000 attempted transition steps

Sample Size Histogram

Statistics and Comparisons

Check distribution validity by comparing statistics:

Face degrees

Compare MCMC face degree statistics to rejection-sampled knot shadows and all 4-valent maps

Average \(v_2\) invariant

Compare MCMC average Casson invariant statistics to rejection-sampled knot shadows

Monogons in Shadows

In limit, \(\frac 13\) of faces in 4-valent maps are monogons

Find that both samples of knot shadows have similar counts of monogons, both different from all maps

Normalize statistics: Subtract off limit linear behavior for all maps

Monogons in Shadows

Other faces in Shadows

Similar to monogons, we get linear behavior for all other face degrees

Both knot shadow samplers yield similar counts

Normalize statistics: Subtract off limit \(y = n p_{k}\) behavior of all maps

\(n\)4-valent \(p_k\) MCMC \(p_k\)
1 \(\tfrac 13 = 0.\overline{3}\) \(0.35036 \pm 4(10^{-5})\)
2 \(\tfrac 16 = 0.1\overline{6}\) \(0.14056 \pm 3(10^{-5})\)
3 \(\tfrac {13}{108} = 0.12\overline{037}\) \(0.12257 \pm 3(10^{-5})\)
4 \(\tfrac {55}{648} \approx 0.08488\) \(0.08298 \pm 2(10^{-5})\)

2-gons and 4-gons are more rare in knot shadows; all other faces are more common

1–4-gons in Shadows

Average \(v_2\) Vassiliev Invariant

The average \(v_2\) invariant over all crossing assignments of a plane curve is a spherical curve invariant


The average \(v_2\) invariant grows linearly with the number of crossings \(n\)

In agreement with quadratic growth in the Petaluma model (Even Zohar et al. '16); \(m\) petals can correspond to \(O(m^2)\) crossings

Normalize statistics: Subtract off least-squares linear fit

Average \(v_2\) Vassiliev Invariant

Classical Questions

Efficiency and speed of convergence is related to classical questions


What is the diameter of any fixed knot type Reidemeister graph?

Lackenby '15

An unknot diagram \(D\) of \(n\) crossings can be reduced to the trivial diagram in at most \((236n)^{11}\) Reidemeister moves

(Unknotting) Experiments

Viewing Markov chain as "BFACF for diagrams" suggests experiments on diagrams, such as:

Experiment (Stolz et al. '17)

Find most probable unlinking pathways of E. coli DNA under XerCD-dif-FtsK site-specific recombination, using SAPs and local reconnection at edge-edge contacts


Repeat above, using knot diagrams and coherent smoothing at crossings

Thank you!

H. Chapman and A. Rechnitzer. A Markov chain sampler for plane curves. Submitted.