Dataset DOI: 10.5061/dryad.zpc866tnv

Description of the data and file structure

Files are stored in plain text formats.

Files and variables

mean_number_of_summands_and_tv_distance_to_poisson.tsv

This tab-separated file contains computed values of λ_K(n), our estimate of the expected number of factors of knot type K in a random self-avoiding n-gon, together with standard errors and total variation (TV) distance from the empirical distribution to the Poisson distribution with the same mean.

_observed_summands.tsv

These tab-separated files contain observation data for the number of N-gons with m summands of knot type K. In each file, the first entry of each row is N, and the remaining entries list the various observed m's and the number of polygons with that many factors of K.

For example, the file 5_2_observed_summands.tsv contains observation data of the 5₂ knot. The sixth row is:

32768 0 268317594 1 79454 2 8

This indicates that, among the 32,768-gons, we saw 268,317,594 polygons with no 5₂ factors, 79,454 with one 5₂ factor, and 8 with two 5₂ factors.

pfruns-large.zip

This file contains the raw data underlying the summary and computed data in the above files. This file expands to a directory pfruns-large containing a number of directories with names of the form 2-to-the-n-gons for n an integer ranging from 10 to 27. Each such directory contains the raw data for 2ⁿ-gons.

At the next level down in the directory hierarchy is a collection of 64 (for n ≤ 26) or 128 (for n = 27) directories with names of the form xxxx, corresponding to different parallel runs of the polygonal fold Markov chain.

Finally inside each pfruns-large/2-to-the-n-gons/xxxx directory are 3 files:

• Info.m is a Wolfram Language package file containing an Association object with summary information about the run. For reproducibility, perhaps the most important information can be found in "Initialization". In particular, "Git Hash" gives the Git hash of the version of Knoodle that produced this run.
• MacLeod.txt gives the prime factors of recorded polygon in the run. The format of this file is explained more below.
• Tools_Log.txt records any warnings generated during the run.

MacLeod.txt

MacLeod.txt gives the knot type of each recorded polygon in a highly compressed format. Each line of this file is in one of the following forms:

• u <n> This indicates a run of n consecutive unknots.
• k This is the start of a new nontrivial knot. The lines between this k and the next u <n> or k line give the prime factors of this knot.
• T+ <n> There are n positive trefoils in the knot.
• T- <n> There are n negative trefoils in the knot.
• F8 <n> There are n figure-eight factors in the knot.
• s <0/1> | <c>: This is a factor which is not recognizably a trefoil or figure-eight. The <0/1> is 1 if this factor is provably minimal, and 0 otherwise. The <c> is a short MacLeod code for the knot which can be converted by Knoodle to a planar diagram object for further analysis. Because MacLeod codes are not guaranteed to be backwards-compatible, please be sure to analyze these codes with the version of Knoodle indicated in the "Git Hash" in the associated Info.m file.

For example:

• Lines 1–10 of pfruns-large/2-to-the-16-gons/afsb/Macleod.txt are
  u 5
k
T+ 1
u 2
k
T+ 2
T- 1
k
T+ 1
u 15
  This indicates that this run started with 5 unknots, followed by a positive trefoil, another 2 unknots, a composite knot with 2 positive trefoil factors and 1 negative trefoil factor, then a positive trefoil, and then 15 consecutive unknots.
• Lines 17–21 of pfruns-large/2-to-the-25-gons/bfsb/MacLeod.txt are:
  k
T+ 40
T- 34
F8 2
s 1 | 31 21 15 21 23
  Since this is the fifth instance of k in the file, this indicates that the fifth recorded 2²⁵-gon in this run was a knot with 40 positive trefoil factors, 34 negative trefoils, 2 figure-eights, and one additional factor with a diagram with short MacLeod code 31 21 15 21 23. This is a 5-crossing diagram which is provably minimal (hence the 1 after the s) because it is reduced and alternating. Knoodle easily identifies this factor as a 5₂ knot.

Code/software

All data was generated by and analyzed with Knoodle. The Git hash of the version of Knoodle which produced and analyzed a particular run is recorded in "Git Hash" in the associated Info.m file.