Of course you can. In the Chern-Simons approach of Witten's you can get knot polynomials for any compact simple Lie group. (In fact, Chern-Simons theory for noncompact Lie groups is still not well-understood despite continuing progress, reviewed in a very recent paper of Witten's.) In fact, the knot polynomial associated to $\mathfrak{so}(3)$ is the Jones polynomial.

Edit: As pointed out in the comment below, the question is about knot polynomials constructed out of the R-matrix, whence my original answer is not relevant. I'll leave it here just in case someone else misunderstands the question.

Of course you can. In the Chern-Simons approach of Witten's you can get knot polynomials for any compact simple Lie group. (In fact, Chern-Simons theory for noncompact Lie groups is still not well-understood despite continuing progress, reviewed in a very recent paper of Witten's.) In fact, the knot polynomial associated to $\mathfrak{so}(3)$ is the Jones polynomial.