blueollie

Progress: math and running

Workout notes: I did my “Cornstalk 8” plus “out and back” for roughly 10.1-10.2 miles in 1:46:05; the 8 portion was 1:23:40 (average). But it was…roughly 60 F (15.5 C) and wet; highly unusual for Illinois in January. I kept the pace conservative because of my pitiful 14 last Saturday; no problems. But it was about 2:20 slower than a week ago.

I’ve also been tackling a tough mathematics problem. Part of the problem involved a very tedious calculation; one which is highly prone to “transcription” errors.

(for the curious: there is a way of finding an invariant for fundamental groups. If one has two groups presented as generators and relations; e. g. ${x, y| xyx=yxy }$ it is, in general, difficult to see if one group is isomorphic (equivalent) to another. So one thing one can do is to find a map from the groups to something that is easer to manipulate; in this case one can get a map from groups to polynomials (called Alexander polynomials). Isomorphic groups map to “equivalent” polynomials (up to multiplication by a unit in the ring). One way to get the map is to use Fox Calculus. In doing so, you take a “partial derivative” of the relations in the group; the “product rule” for this calculus is slightly different than the product rule that one usually encounters in elementary calculus.

The problem is that this process is every error prone, especially if the calculations are long.

However, in my case, the object of interest is, in fact, equivalent to a known object that has been tabulated.

I was studying this object:

which I found could be deformed into this object:

Yes, I confirmed this by playing with colored yarn!

How did I know to do this? I used software to compute the hyperbolic volume of (the complement of) the first object and then matched that volume to known links:

And that gave me the data that I needed.

My point: having all of this data available really helped save me a couple of weeks of work; perhaps that is one reason why it is easier to make progress in some areas of mathematics, and perhaps why it might be a good time to review old and still unsolved problems and clean some of them up.

Anyway that is my goal.