This morning, I’ve been looking through a very entertaining paper in which Roger Fenn distinguishes the left-hand trefoil from the right-hand trefoil in a way that could be explained to elementary school children.
R. Fenn, Tackling the trefoils.
One of the main questions in Knot Theory is the Tabulation Problem. Tabulate all knots up to crossings for the largest you can manage! Many modern tabulation techniques are quite high-tech algebraic/topological, using geometric structures on knot complements and the likes, but sometimes simple combinatorial techniques will do the job. A combinatorial invariant views a knot as a knot diagram, which it views in-turn as a network of crossings concatenated together in a plane, i.e. as a special sort of a tangle diagram. Knot diagrams are considered equivalent if they differ by a finite sequence of Reidemeister moves.
To classify knots, we identify knot invariants. A `good’ knot invariant should be both powerful and easy to compute.
Historically, the first knot invariants to strike this balance were the knot colourings, first considered by Tietze, and eventually explained in entirely elementary terms by Fox in 1956. The invariant is a yes-no answer to the following question:
Can you colour your knot by three colours such that all three colours are used, and at each crossing either one colour meets itself, or all three colours meet?
If you answer `yes’, then the knot is 3-colourable.
The trefoil is 3-colourable:
The unknot is not 3 colourable, because it has a diagram with only one arc, and therefore with only one colour. Three-colourability is easily seen to be a knot invariant, because it is conserved by Reidemeister moves:
In fact, with different language and in the fundamental group world, the above was the original proof that the trefoil is knotted!
Of course, there’s only so far a yes-no `boolean’ invariant can go- it sifts knots into two equivalence classes, but doesn’t do more than that. The Figure Eight knot, for example, is not 3-colourable, which distinguishes it from the trefoil but not from the unknot.
A large set of boolean invariants are much better than just one, though. By varying our pallete of colours, we can distinguish many more knots. We can colour knots with 5 colours instead of 3 (e.g. the Figure Eight knot is 5-colourable), play with the colouring rule so as to colour with general quandles, and our power to separate knots goes right up. As a matter of fact, a recent preprint of W. Edwin Clark, Mohamed Elhamdadi, Masahico Saito, Timothy Yeatman distinguishes all 2977 prime oriented knots, up to reversal and mirror image, with up to 12 crossings using just 26 quandles!
W.E. Clark, M. Elhamdadi, M. Saito, T. Yeatman, Quandle Colorings of Knots and Applications.
But what about distinguishing a knot from its mirror image, or a knot from its reverse, using colourings? Here, a quandle is no longer sufficient, because you can reflect and reverse any quandle colouring, so that any colouring of a knot uniquely induces a colouring of its reverse and of its mirror image.
So how can we distinguish a knot from its mirror image? Do we need heavy machinery? (Actually the Kauffman polynomial is not at all bad; but never mind).
It turns out that a slightly extended notion of a knot colouring does the job, at least for the left-hand trefoil and for the right-hand trefoil. Namely, the following colouring does the job:
I’ll explain what the colouring is, then how it distinguishes these two knots. First, make a parallel copy of each trefoil. Colour the outermost one in the usual way with three colours . As for the inner one, colour it also , but with the rule that it interacts only with the outer trefoil, changing colours whenever it passes over or under it on a different colour. Actually, we could just make the lower one always pass under the upper one, and make all of its `crossings’ virtual; but that’s a cosmetic quibble.
A quick standard verification shows that this notion of a doubled 3-colouring is invariant under Reidemeister moves. Note that the parallel trefoils are “connected”, and you can’t move one without moving the other.
So that’s the colouring… how do we now extract a knot invariant from it? Simple. Ignore all doubled crossings where only a single colour participates for the inner knot (there aren’t any in the above picture, but a Reidemeister 1 move for example would create one), all doubled crossings where the upper and lower colour agree for both undercrossing arcs (there is one such crossing in each knot). The set of coloured doubled crossings that left over is a knot invariant, up to global automorphism of the colours (renaming “blue” as “green” and “green” as “blue”, for example) and setting two copies of the same crossing equal to minus itself (itself with crossing sign reversed) and cancelling over-crossings with under-crossings with the same pattern.
As an invariant for coloured knots, this coincides with the `coloured untying invariant’ from my thesis:
D. Moskovich, Surgery untying of coloured knots.
It’s beautiful that Fenn can compute it combinatorially, generalize it hugely, and use it to distinguish the right-hand trefoil from the left-hand trefoil! Bravo!!!