This year I managed to attend the annual Joint Mathematics Meetings, this year held in New Orleans, LA. It was greatly enjoyable, but one talk in particular irritated me. It manages to tie into a large number of useful subjects, so it will provide for many, many posts to explain what I’m talking about to the lay audience. Unfamiliar terms will eventually be defined in later posts and this one will be updated with links to the explanations.
Prinarily, the talk claimed to be motivated by a problem in DNA splicing. A strand is cut and one end is joined to a nearby free end of another DNA strand. In the middle of the splice, we have three strands of DNA all meeting in one central area. You’ll remember that the basic structure of DNA is basically a twisted ladder. Here we ignore the rungs of the ladder and just consider the rails, which normall twist around each other. Ath the splice, though, the six rails — two for each of the three strands — tangle up in the center somehow. The problem, as I understand it, is to determine how they are tangled.
The talk completely did not address how the knot theory shown was supposed to determine the biology, nor how the inputs to the mathematics were to be read off from the biology. Despite the fact that this connection is a — if not the — major question in any work of applied mathematics, I will not deal with it now. I am not, nor have I ever been, a mathematical biologist.
The approach presented was to attempt to color the tangle in a similar way to more classical knot colorings. Specifically, we label each arc in the diagram with an element of the group so that when arcs labelled and meet at an overcrossing arc labelled , the labels satisfy . Rather than finding solutions for various values of , the authors wrote down the relations — one for each crossing — that any coloring would have to satisfy. These (linear!) equations are grouped in a matrix and the authors searched for various matrices that could arise from various tangles.
In particular they noted that for knots there are as many crossings as there are arcs (prove it!), so there are as many relations as there are variables, and the presentation matrix is square. With free ends, however, there are more variables and so the matrix is rectangular. The authors broke the matrix into four blocks and argued that when the whole matrix was put into a certain form one block was zero and two of the others were invariants of the tangle. The last they seemed to disregard.
Really what they were doing is determining a presentation matrix for the “fundamental involutory quandle” of the tangle. I was able to determine with a few questions that the speaker did recognize this phrase, at least in the context of knots (no free ends). The problem I have here is twofold.
Firstly, matrices are horrible, horrible things until the very end when you absolutely must do some calculations. Matrices depend on a lot of essentially arbitrary choices, and different choices give rise to different — though in some sense “equivalent” — matrices. If you want to study the quandle, study the quandle already! Don’t put so much stock in some array of numbers that introduces all sorts of artifacts to cloud the discussion.
Secondly, extending the notion of the fundamental involutory quandle to tangles with free ends has already been done. It’s a special case of part of my dissertation. The cospan construction handles the fundamental involutory quandle beautifully, and it was depressing to see it shackled into the ugly dungeon of matrices.
I’ll go into it more another time, but the idea of the cospan construction here is that we can talk about the fundamental involutory quandle of the edge of the tangle where no tangling goes on and consider how that edge includes into the whole tangle. Since we’re always considering the case of six points on the edge the quandle there is uniquely determined by that fact. In fact, if you insist on drawing out matrices to talk about the fundamental involutory quandle-cospan you get three matrices to consider: one for the edge, one for the whole tangle, and one for the inclusion mapping. These are exactly the three non-zero blocks the authors (should have) considered, but each one of them now has a meaning. One is not just an invariant, though, it’s completely determined by the fact that there are six strands coming in.
It’s one thing to recognize in a talk a bad reconstruction of a result you know exists. It’s another level to recognize a bad reconstruction of something you did.