Assume you have a knot "colored" with the irrep of a quantum Lie algebra with its parameter at $q=1+\epsilon$ where $\epsilon$ is small. When it is zero, all crossings turn virtual. Does it make any sense to do a Taylor series of a crossing (in form of a skein relation), writing something like $overcross=virtual+\epsilon*quartic casimir+...,undercross=virtual-\epsilon*...$ (quarticcasimir stands for "I have no idea :-)
I experimented a bit; at least $(overcross-undercross)/\epsilon$ seems to be "compatible" with the formalism. E.g. take the R matrix $R$ of the defining irrep $7$ of $G_2$. Its eigenvalues are $1/q^2(27),-q^6(7),q^{12}(1),-1(14)$, where the multiplicities are in brackets (and are the dimensions in $7\bigotimes 7=27\bigoplus 7\bigoplus 1\bigoplus 14$; the exponents of $q$ are the quadratic casimirs). Then $A=lim_{\epsilon\rightarrow 0} (R-R^{-1})/\epsilon$ has eigenvalues $-4(27),-12(7),24(1),0(14)$, which "looks good" (read: the multiplicities are OK, and you surely noted the eigenvalues are the quadratic casimirs times 2).