Can one use the real lie algebra so(3) to get knot polynomials? If so, do they have a skein relation (I presume they would, if they come from R-matrices in some standard way. If so, is the R-matrix written down anywhere?) I've only ever seen complex Lie algebras used to get knot polynomials, but I'm not sure if this is for some fundamental reason, or just that no one has bothered to investigate the real ones. 



SO(3) knot polynomials
I don't see how taking a real form would *change* your knot invariants at all. Whatever it meant would be a number which would then not change when you base extend to the complex numbers.


From the quantum group perspective U\_q(so\_3) and U\_q(su\_2) mean the same thing (as far as I understand it), but for so\_3 the natural representation is 3-dimensional, whereas for su\_2 the natural representation is 2-dimensional.


So the R-matrix you want is just the R-matrix for the *three dimensional* representation of U\_q(su\_2). I'm not sure if that's been written down somewhere that's easy to find, but it's a good exercise to work out yourself. From the diagrammatic perspective there's something called the "Yamada polynomial" which gives the so\_3 knot polynomial (just as the Jones polynomial gives the su\_2 knot polynomial).


The other thing people mean when they say SO(3) instead of SU(2) is that you only look at the category of representations of U\_q(su\_2) which come from deformations of representations of su\_2 which lift to the lie group SO(3), that is only the representations whose highest weights are in the root lattice. The best place I know to learn about this perspective (and in particular what it means in terms of 3-manifold invariants which is what Jose mentioned) 
are the papers of Steve Sawin, for example [this one](http://arxiv.org/abs/math/9905010).


