There are various (equivalent?) descriptions of a universal finite-type knot invariant, e.g. https://arxiv.org/abs/q-alg/9603010. They take the form of formal power series valued in Feynman diagrams (certain oriented decorated trivalent graphs). Each such Feynman diagram corresponds to a configuration space integral, and in particular is a function on the space of knots. In https://arxiv.org/abs/q-alg/9603010 these functions are denoted $I(\Gamma)$.

To evaluate the universal finite-type invariant on a knot we need to understand the values of $I(\Gamma)(K)$ on some particular knot $K$ for a Feynman diagram $\Gamma$. Is there a combinatorial way to do this? I am thinking of this process as ``expanding'' $K$ into a formal sum of Feynman diagrams: the coefficient $I(\Gamma)(K)$ tells me how many times $\Gamma$ appears. (Actually, we have to divide by an automorphism factor.)

This seems like it should be possible, because I have a different way of ``expanding'' $K$. (Well, I'm not quite sure this works, but I suppose that's part of the question.) By using the Vassiliev skein relation, we can expand $K$ as a sum of singular knot diagrams (with double points). These correspond to chord diagrams, which in turn correspond to Feynman diagrams via the STU relation. If we do this for long enough, we should be able to write $K$ as a sum of Feynman diagrams, at least up to some fixed order $n$. The coefficient of $\Gamma$ in this expansion should be something like the value of $I(\Gamma)(K)$.

Can you turn the approach of the third paragraph into a way to compute $I(\Gamma)(K)$? If not, is there a way to compute $I(\Gamma)(K)$ that avoids actually working with configuration space integrals?