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.

However, I think you are interested in how to shift the dimensions of the homology groups of the trace free representations of a knot group into SU(2) in order to get a homology theory that categorifies the Jones polynomial.  

Conjecturally, you are working out the Instanton homology of Kronheimer and Mrowka.
A good model of what you are trying to do, is the finite dimensional Floer homology interpretation of Floer's original theory, as conjectured by Atiyah.  I think this was worked out by Dietmar Saloman and Katrina Wehrheim.

Since you need to work out what is going on at the reducible representations to get a full answer, you need to be working in a stratified space.  If you want to see that dealt with properly I recommend Kevin Walker's book in the Princeton University Press series on his extension of Casson's invariant to Rational Homology spheres.

The place you need to work is in the space of bundles with parabolic structures over a $2n$-punctured sphere.  The idea is in Jacobsson and Rubinsztein's paper. All that work to
get the symplectic structure obscures what is going on. The paper of Jeffrey, Weinstein, Guruprassad,
and Huebschmann is about finding a finite dimensional model for doing symplectic reduction to get the moduli space.  I think there are cleaner developments of the symplectic structure.

The space is actually a stratified symplectic manifold.  A point that people miss, is this.
The cycles you are intersecting (in Xiao Song's setting, which is also Rubinsztein and Jacobsson's setting) have logarithmic perversity.  However, it turns out that in a space of the form X\times X,
logarithmic perversity homology is carried by middle perversity homology. That's why Goresky and MacPhereson could do the Lefschetz fixed point formula for intersection homology. You can see that idea articulated in a paper of Andy Nicas and me,  on an intersection homology invariant of knot complements.  I don't know if that is the right way to go though.

Trying to follow this all through conceptually is probably not humanly possible. Here is my real advice:

Work a whole bunch of examples till you see a pattern. :)