Small Connected Quandles and Their Knot Colorings

This website contains computational results on small connected quandles and their knot colorings
obtained by W. Edwin Clark and Timothy Yeatman.

Background

Go to Quandle Cocycle Knot Invariants for backgound material on quandles and their knot colorings.

Applications of these computational results can be found at the website Quandle Colorings and Invariants of Knots in the Table.

The paper Quandle Colorings and of Knots and Applications is based on computations posted in this and the above mentioned websites.

References

• The GAP Group, GAP -- Groups, Algorithms, and Programming, Version 4.4.12; 2008. (http://www.gap-system.org).
• Maple 15- Magma package-copywrite by Maplesoft, a division of Waterloo Maple, Inc, 1981--2011.
• McCune, W., Prover9 and Mace4, http://www.cs.unm.edu/~mccune/Prover9, 2005--2010.

• Cha, J. C.; Livingston, C., KnotInfo: Table of Knot Invariants, http://www.indiana.edu/~knotinfo, May 26, 2011.

  The list of 12965 knots contains the sequence Knot[i], i = 1,...,12965, that are (representatives of) all prime oriented knots with at most 13 crossings. The first entry of the list Knot[i] is the name of the knot, the second is the braid index, the third is the length of the braid, the fourth is the braid description of the knot. These are from J. C. Cha and C. Livingston, http://www.indiana.edu/~knotinfo/, (up to 12 crossings) and the remaining 13 crossing knots from Alexander Stoimenow, http://stoimenov.net/stoimeno/homepage/ptab/index.html.

• Vendramin, L., RIG - a GAP package for racks and quandles, May 22, 2011, http://code.google.com/p/rig/.

  The quandles in the RIG package are left distributive. To conform with our convention that quandles are right distributive, we use the transpose of the quandle matrices in RIG and we call them RIG quandles. The quandle in RIG denoted by SmallQuandle(n,j) we denote by C[n,j]. We also denote the i-th quandle in lexicographical order on the pairs (n,j) by RIG[i]. A file containing these quandle can be found below.

Small connected quandles and their properties

• 431 RIG quandles
  RIG[i], i=1,...,431, are the 431 connected quandles with orders < 36. The file also includes a list RIGindices[i] = [n,j] where RIG[i] is the j-th connected quandle of order n in the RIG package of connected quandles. These are transposes of the 431 connected quandles in the GAP package RIG.
• 790 RIG quandles
  RIG[i],i=1,...,790, are the 790 connected quandles with orders < 48. The list expands the above list, and was obtained by Leandro Vendramin in 2014.
• Properties of rig quandles
  (Characterizations such as whether they are Alexander, kei, latin, faithful, are listed.)
• Further properties of rig quandles
  (Further properties such as subquandles, products, duals, abelian extensions are listed.)
• Non-faithful connected quandles of order at most 47 that are conjugation quandles

Other families of connected quandles

• A list of 26 quandles
  Q[i], i=1,...,26, are quandles which suffice to distinguish (up to orientation and mirror image) the 2977 oriented knots with at most 12 crossings. At the bottom of the file is a list dual such that dual[i]=j iff the dual quandle of Q[i] is Q[j].
• A list of 23 quandles
  T[i], i=1,...,23, are quandles which distinguish the 1058 (out of 1366) chiral or negative amphicheiral knots with at most 12 crossings from their mirror images.
  A list of the 1058 knots and T[i] that distinguish them from their mirror images.
• Some families of quandles
  Non-abelian version of Alexander quandles, and families of quandle operations parametrized by group elements are listed.
• A list of 60 connected quandles
  Matrices of quandles used to distinguish knots from their mirrors, reserses, and reversed mirrors.
  A list of quandles, knots and the ("Psi") tangle invariant values.

Knot coloring matrices

• A 431 x 2977 coloring matrix M for the 2977 knots by the RIG quandles
  The (i,j)-entry of the matrix M is the number of non-trivial colorings by i-th rig quandle RIG[i] of the j-th knot in the table at KnotInfo.
• A 790 x 2977 coloring matrix M790 for the 2977 knots by the RIG quandles
  The (i,j)-entry of the matrix M790 is the number of non-trivial colorings by i-th rig quandle RIG[i] (of order up to 47) of the j-th knot in the table at KnotInfo. This matrix expands the above matrix and was obtained in 2014.
• A 26 x 2977� coloring matrix M for the 2977 knots by the quandles Q[i], i = 1,...,26
  The (i,j)-th entry of the matrix M is the number of non-trivial colorings of Knot[j] by quandle Q[i] from the file of 26 quandles.
• A 12 x 12965� coloring matrix NC for the 12965 knots with at most 13 crossing by the first 12 RIG quandles
  The quandles used are C[3,1] through C[7, 5].

Computer programs

• C program
  This file is a compressed file which contains our fastest C program for finding the number of colorings of a knot by a quandle.
  
• GAP functions
  This file is a file of GAP functions for finding the number of colorings of a knot by a quandle, and other miscellaneous functions for creating and manipulating quandles with GAP. These are independent of the RIG package for GAP.
  
• Maple procedures
  This file is a file of Maple procedures for finding the number of colorings of a knot by a quandle, and other miscellaneous procedures for creating and manipulating quandles with Maple. If used with the Maple package Magma, the file QuandleProcedures.txt should be read in before loading the Magma package (since some procedures coincide with procedures in Magma.

Acknowledgement