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.
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 |
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].