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Does anyone have a good explanation for why Knot invariants tend to be well organized as polynomials? What exactly is going on and why don't we often see polynomial invariants for classifying other geometric objects? For instance, why don't we just say that the Alexander invariants are a finite set of numbers? Presumably, the organizing of them into polynomials shows off some inherent geometric properties of knot description and combination which is easily encapsulated in the multiplication and addition of polynomials. I would be happy (and not surprised) if this also helps me understand why Lie algebra representations also often arise, especially in more recently discovered invariants.

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We do see polynomial invariants for classifying other geometric objects: to any reasonably nice space you can associate its Poincare polynomial whose coefficients are the corresponding Betti numbers. The Poincare polynomial is multiplicative under product by the Kunneth formula, and for particularly nice spaces (complex flag manifolds, etc.) these polynomials are particularly nice (q-analogues of binomial coefficients, etc.). The reason people don't talk about Poincare polynomials is that they want the extra data corresponding to the (co)homology groups themselves, and this in turn is a... –  Qiaochu Yuan Dec 19 '10 at 21:18
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...natural motivation for the whole program of categorifying knot invariants. Also, I think this would be a fine question for MO; all the knot theorists who would be able to answer your question here are also there. –  Qiaochu Yuan Dec 19 '10 at 21:21
    
Although perhaps you are asking a simpler question than I thought, in which case one answer is that the HOMFLY polynomial is multiplicative under connected sum, and I imagine similar statements are true for other knot polynomials. But there are also reasons to be interested in the evaluation of knot polynomials at particular values coming from quantum groups and statistical mechanics. –  Qiaochu Yuan Dec 19 '10 at 21:37
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I think one reason the Poincare polynomial does not get as much discussion is that, for someone interested in spaces, it's a reflection of a fairly "plain" commutative monoid structure on spaces: the operations being disjoint union and cartesian product of spaces. –  Ryan Budney Dec 19 '10 at 22:56

2 Answers 2

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I disagree with the hypothesis of the question -- knot invariants do not tend to be organized as polynomials. The Alexander module is not a polynomial. The fundamental group of the complement is not a polynomial. The complement of a knot is not a polynomial. These are some of the most fundamental invariants of a knot, and they're not polynomials.

That said, polynomial invariants do occur in knot theory and there are a few good reasons. One is that knots up to isotopy have the structure of a commutative monoid under the connect-sum operation. Invariably, when you start talking about functions on an object $O$, you like to talk about the vector space/module structure of such a function space, so you like to talk about formal linear combinations of objects from $O$. Formal addition and connect-sum turns knots into a commutative ring. Polynomial rings are free commutative rings, so they're fundamental in discussions about the structure of commutative rings. In the special case of knots in $\mathbb R^3$, it's a theorem of Schubert's, that knots under these two operations are a free commutative ring (the free generators being the "prime" knots). So in some sense knots can be thought of as a polynomial ring.

So the upshot of this is the study of "knot invariants" together with the connect-sum operation on knots prejudices the discussion towards one where you are concerned with commutative rings, and so commutative-ring valued invariants. So that's why polynomial invariants occur so often.

If you were to study different internal structures on knots, you perhaps prejudice the types of invariants you use to study knots. Tangle categories and categorical invariants would be another example. The knot complement and geometrization would be another.

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The only polynomial invariants of knots that I can think of are the Alexander polynomial on one hand, and the Jones polynomial and its generalizations (HOMFLYPT et.al.) on the other (ignoring things like the 2-loop polynomial). It's clear why the Alexander polynomial should be a polynomial- it's the generating function for torsion numbers of the knot, as explained in Rolfsen, and the Alexander module is finitely presented.
Why the Jones polynomial should be a polynomial is a much more interesting question, addressed in the paper Is the Jones polynomial of a knot really a polynomial? by Garoufalidis and Le. From the TQFT perspective, we would have expected quantum invariants to be power series, but actually for knots the Jones polynomial is a polynomial, which is quite interesting.
So I suppose that I don't know the ultimate reason that knots have interesting polynomial invariants. I feel that there has got to be a wonderful secret hiding here! I would guess that it ultimately has to do with the rich local algebraic structure of an algebra over a modular operad generated by crossings, and maybe trivalent vertices and other stuff.

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