During Winter 2019, we conducted a research on knots, especially about the invariants of knots (colourability & unknotting number).

Knots Theory is a branch of topology. There is an undergraduate/graduate course PMATH467/667 at University of Waterloo.

Key Words

low dimensional topology
mathematical knots
knot invariants
knot colouring
classification of knots
reduction of computational complexity
high performance computing


The poster reports on a colourability classification of mathematical knots with a crossing number up to 15.

After an introduction to mathematical knots and their colour invariants, a list of results and references, the poster shows figures of all 250 knots with a crossing number up to 10 coloured in one of their n-colourings [1].

Although the invariance of knot colouring is known for many years, a classification of knots with respect to colouring has so far not been available (see, for example, the KnotInfo database [2]).

The classification became possible through repeated algorithmic improvements of a computer program to speed it up multiple times and compute all n-colourings of knots very efficiently. For example, all colourings of all 250 knots with up to 10 crossings are computed in 3 sec on a desktop PC.

c nmax Knot B(c)
3 3 31 1.732
4 5 41 1.710
5 7 52 1.627
6 13 63 1.670
7 19 76 1.634
8 37 817 1.675
9 61 933 1.672
10 109 10115 1.684
11 199 11a301 1.698
12 353 12a1188 1.705
13 593 13a4620 1.703
14 1103 14a16476 1.714
15 1823 15a65606 1.710
Listing of \(B(c) = (n_{\max})^{1/(c-1)}\)

A first run of the computations up to crossing number 13 resulted in an approximate yet rather precise formula for the maximal value \(n_{\max}\) of n-colourability in dependence on the crossing number \(c: n_{\max} \approx 1.7^{(c-1)}\).

By using this result it was possible to speed up the computations again dramatically and complete them for the 59937 knots with crossing numbers up to 14 within 1 day and up to 15 in 2 weeks by running the computation in parallel on 10 CPU.

The colouring module used to perform the computations has been included in a freely available interactive workbench for knots [3].

A complete list of colourings is available under [4].


  1. R. H. Fox, Metacyclic invariants of knots and links, Canadian Journal of Mathematics 22 (1970) 193-201.
  2. J. C. Cha and C. Livingston, KnotInfo: Table of Knot Invariants, http://www.indiana.edu/~knotinfo
  3. T. Wolf, A Knot Workbench, Linux download: https://cariboutests.com/games/knots/AsciiKnots.tar.gz
  4. "Colour Classification of Knots with Crossing Number up to 15", https://cariboutests.com/qi/knots/colour3-15.txt


Colourings of the First 313230 Knots
Unknotting Numbers of the First 12965 Knots