This question is rather vague. Are there any natural situations which involve Laurent polynomials of the form
$$\sum q^{a\_i}\in\mathbb{Z}[q,q^{-1}]$$
where the $a\_i$'s are either Euler characteristics of some spaces (possibly all subspaces of one fixed space), or more generally, indices of some elliptic operators? I've stumbled across such a beast, but am unsure how to interpret it. I was thinking at first that it was an element of $K\_{S^1} (pt)$ or something, but in that case the exponents are telling us about which $S^1$ representations show up in the appropriate bundles, not the indices of the operators, right? Is there some relation with the index? (Please tell me if I'm talking nonsense! I don't really know this K-theory stuff). 


Maybe the answer I'm looking for doesn't involve K-theory, anyway. Does anyone have any ideas? I'd love to hear about any and everything!



Euler characteristics and operator indices as exponents for Laurent polynomials
1. Knot polynomials like the Jones polynomial
2. Perturbative expansions of Feynman Integrals
3. Heat kernel asymptotics, and other universal polynomials in characteristic classes.
4. Generating functions associated to combinatorial problems.
5. Poincare Polynomials of Topological Spaces.
6. Hilbert Polynomials.
7. Certain families of orthogonal polynomials, generally associated to representation
theory.
8. The A-polynomial


I am sure there are more.


