As I discussed last time, coloring a knot with any abelian group is secretly using the dihedral quandle associated to that group. This is an involutory quandle with action . The reason knot coloring works out so nicely is that the axioms of (involutory) quandles line up with the Reidemeister moves.
But for the moment we’re stuck with picking this or that involutory quandle and counting how many colorings it gives for a given knot. Different quandles give different coloring numbers, and we’d like to find a better way of thinking of them all at once. We’re going to construct a new involutory quandle from a knot that captures all of them.
Take any diagram of the knot we’re interested in. Remember the knot table if you want to pick one out. Now each arc in the diagram has to get some color, no matter what quandle we’re using to color it. Instead of picking a color from a specific quandle, let’s just slap a label like , , or on each arc. Be sure to use a different label for each different arc.
Now those labels will generate an involutory quandle. We can throw them together with the two quandle compositions to get “words” like . These words, of course, are subject to the normal quandle equivalences, but we need more relations for our purposes. At each crossing the values in a coloring have to satisfy a certain relation, so we’re going to build that right into our quandle. If the arcs labeled and meet under the crossing arc labeled , then we must have .
This seems to depend on the choice of a diagram, though. Well, it sort of does, but any Reidemeister move gives an isomorphism of quandles relating the two sides. For example, performing the first one splits an arc into two pieces. Say label becomes and . Then the relations we introduce say that . But the axioms of quandles say that , so and we can just drop one of these generators and the relation we’ve now “used up”. Try to find the isomorphisms for the other two moves. This justifies calling the quandle we’ve constructed (up to isomorphism) “the” fundamental involutory quandle of the knot .
So what’s a coloring? A coloring assigns an element of some quandle to each arc of the knot diagram. But arcs in the diagram are just generators of the fundamental quandle. That is, a coloring is a function that takes generators of the fundamental quandle to a selected target quandle. If it plays nicely with the relations between the generators, it will be a quandle homomorphism. In fact it does, precisely because we picked the relations between the generators to be exactly those required by colorings. A given relation comes from a crossing, and every coloring of a knot obeys the same restrictions at crosings.
In the end we’ve found that the set of all colorings of by an involutory quandle is the set of quandle homomorphisms , so the number of -colorings is the cardinality of this set. If we have a good understanding of quandles and their homomorphisms, we can read off coloring numbers by involutory quandles from the fundamental involutory quandle.
At long last I come to quandles. I know there are some readers who have been waiting for this, but I wanted to at least get through a bunch of group theory before I introduced them, because they tend to feel a bit weirder so it’s good to warm up before jumping into them.
The story of quandles really begins back in the late ’50s and early ’60s when John Conway and Gavin Wraith considered the wrack and ruin that’s left when you violently rip away the composition from a group and just leave behind its conjugation action. This is a set with an operation , and it’s already a quandle. The part of the structure they considered, though, has lost its ‘w’ and become known as a “rack”.
In 1982, David Joyce independently discovered these structures while a student under Peter Freyd working on knot theory with a very categorical flavor (hmm.. sounds familiar). He called them “quandles” because he wanted a word that didn’t mean anything else already, and when the term popped into his head he just liked the sound of it. There are other names for similar structures, but “quandle” is the one that really took hold, partially because there are a lot of unusual algebraic structures that aren’t good for much but their own interest, but “quandle” was the term the knot theorists picked up and ran with.
Actually, after hearing one of my talks Dr. Freyd mentioned that Joyce had come up with a lot of good things while a student, but he (Freyd) never thought much would come of quandles. In the end quandles have become the biggest thing to come out of his (Joyce’s) thesis.
Okay, so let’s get down to work. There are three axioms for the structure of a quandle, and I’ll go through them in the reverse of the usual order for reasons that will become apparent. We start with a set with two operations, written and .
The third and most important axiom is that distributes over itself: . A set with one operation satisfying this property is called a “shelf”, leading to Alissa Crans’ calling the property “shelf-distributivity”. No, I’m not going to let her live down making such an awful pun, mostly because she beat me to it. We can verify that conjugation in a group satisfies this property:
The second axiom is that the two operations undo each other: . Some authors just focus on the one operation and insist that for every and the equation have a unique solution. Our second operation is just what gives you back that solution. A shelf satisfying (either form of) this axiom is called a rack. We again verify this for conjugation, using conjugation by the inverse as our second operation:
Finally, the first axiom is that . Indeed, for a group we have . This axiom makes a rack into a quandle.
One more specialization comes in handy: we call a quandle “involutory” if . Equivalently (by the second axiom), . That is, acts on by some sort of reflection, and acting twice restores the original.
As a bit of practice, check that in a rack the second operation is also self-distributive. That is, . Also verify that if we start with an abelian group (writing group composition as addition), the operation makes the set of elements of into an involutory quandle.