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:

(Here is momentum, is position, and is Planck's constant). Of course it looks more like it if you call the variable ``'', but the real thing is to note that the two kinds of crossings in and are analogous (somehow) to the different orderings in and .

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