Planar decomposition of bipartite HOMFLY polynomials in symmetric representations

We generalize the recently discovered planar decomposition (Kauffman bracket) for the Hoste-Ocneanu-Millett-Freyd-Lickorish-Yetter (HOMFLY) polynomials of bipartite knot/link diagrams to (anti)symmetrically colored HOMFLY polynomials. Cabling destroys planarity, but it is restored after projection to (anti)symmetric representations. This allows one to go beyond arborescent-induced calculus, which so far produced the majority of results for colored polynomials. Technicalities include combinations of projectors, and these can be handled rigorously, without any guess work—what can be also useful for other considerations, where reliable quantization was so far unavailable. We explicitly provide simple examples of calculation of the HOMFLY polynomials in symmetric representations with the use of our planar technique. These examples reveal what we call the bipartite evolution and the bipartite decomposition of squares of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">R</a:mi></a:math> matrix eigenvalues in the antiparallel channel.