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# The Conway Polynomial

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...

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