Planar decomposition of bipartite HOMFLY polynomials in symmetric representations

TL;DR

This paper extends the planar decomposition method for HOMFLY polynomials to symmetric representations, enabling calculations beyond arborescent calculus and providing explicit examples of bipartite decomposition in knot invariants.

Contribution

It introduces a generalized planar decomposition technique for (anti)symmetric HOMFLY polynomials, overcoming limitations of previous methods and enabling rigorous calculations without guesswork.

Findings

Successfully computes HOMFLY polynomials in symmetric representations using planar techniques.

Reveals bipartite evolution and decomposition of R-matrix eigenvalues in knot theory.

Provides explicit examples demonstrating the effectiveness of the new method.

Abstract

We generalize the recently discovered planar decomposition (Kauffman bracket) for the 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 to go beyond arborescent 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 $\mathcal{R}$-matrices eigenvalues in the antiparallel channel.