TITLE: Why are knot invariants best organized as polynomials?
QUESTION [13 upvotes]: 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.

REPLY [4 votes]: 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.