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.
So I’m up until 4:30 scrawling something out. What’s the only thing that wouldn’t annoy me to no end waking me up at 9:00?