My aim is like LeechLattice said, to have a canonical representation. So I could describe such a diagram with a word or a system of generators instead of pictures. So when I do computations involving such diagrams I can do it algebraically.

YCor: Perhaps not well-defined as I stated, but still answerable. Perhaps what I wanted is an algorithm that would obtain a local minimum of crossings numbers?

Thanks! Do you have any (readable) references for 1) The Jones obtained from the representation of $U_q(\mathfrak{sl}_2)$ and 2) The coloring/labeling of components?

Which conjecture(s) are you refering to? 1.5? Does my statement hold or is it still a conjecture?

Actually, I found an excellent Python implementation by David Eppstein in his PADS library. Works perfectly!