These days I'm mainly working on the relationship of braids and quantization. Lots of people are interested in that these days, but lots more aren't, I bet, so let me briefly explain just a bit...
There's a knot invariant called the Conway polynomial that may be defined by essentially two rules. It's a polynomial in one variable, say ; let's call the polynomial assigned to the knot (or link) , . The knot has to be oriented; that is, one must draw little arrows tangent to it that say which way to go: . Okay:
Rule 1: If is the unknot (an unknotted circle), . This is sort of a normalization rule.
Rule 2: Suppose , , and are 3 knots (or links) differing at just one crossing (we're supposing them to be drawn as pictures in 2 dimensions).
At this crossing they look as follows:
looks like:
looks like:
looks like:
(All of them should have arrows pointing down. Any rotated version of this picture is fine too -- this is topology, after all!)
Then we have the ``skein relation''
It was Louis Kauffman, I believe, who first noted that this looks a lot
like the famous canonical commutation relations, or Heisenberg
relations:
Well, one could easily laugh this off as the ravings of someone who has been studying knot theory for too long, but it turned out that there was a deep connection. It was Turaev who first gave a precise formulation. He constructed an algebra from knots that involved a variable, , and such that as it converged (in some sense) to an algebra of loops on a two-dimensional surface. Projected onto a two-dimensional surface the knots and are the same, of course, so this makes some sense.
This however was only the tip of the iceberg...
© 1992 John Baez
baez@math.removethis.ucr.andthis.edu