Planar decomposition of bipartite HOMFLY polynomials in symmetric representations

TL;DR

The work expands planar decomposition methods from the fundamental HOMFLY case to symmetric colored HOMFLY polynomials, enabling planar, Kauffman-bracket–like calculations for a broad class of bipartite links built from lock tangles. It develops a rigorous projector calculus for representations [2], [3], and general [r], deriving recurrences and closed forms for planar components, chain evolutions, and knot closures (unknots, Hopf, twist, and double-braid knots). By expressing HOMFLY polynomials in symmetric representations as sums over a small number of planar diagrams with calculable coefficients, the approach bypasses heavy cabling-derived tensors and extends beyond arborescent methods. The results offer a practical framework for fast computation of colored HOMFLY invariants of bipartite links and hint at potential implications for colored Khovanov–Rozansky categorifications and related topological quantum field theory applications.

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.

Figures (24)

• Figure 1: The colored HOMFLY polynomial \ref{['WilsonLoopExpValue']} can be non-pertubatively calculated by contraction of matrices ${\cal R}^{ab}_{cd}$ and turning point operators ${\cal M}^{a\bar{b}}$ and their inverse while going along a link. Actually, matrices ${\cal R}^{ab}_{cd}$ are quantum $\cal R$-matrices of $U_q(\mathfrak{sl}_N)$ quantum group and $U$, $V$ are $U_q(\mathfrak{sl}_N)$-modules.
• Figure 2: The celebrated "Kauffman bracket" -- the planar decomposition of the ${\cal R}$-matrix vertex for the fundamental representation of $U_q(\mathfrak{sl}_2)$. In this case ($N=2$) the conjugate of the fundamental representation is isomorphic to it, thus, tangles in the picture has no orientation.
• Figure 3: Planar decomposition via the Kauffman bracket of a 2-cabled vertex. The smaller diagrams in the second and the third lines vanish when projecting to the first symmetric representation as all of them carry only the singlet representation on the closed end.
• Figure 4: Vertical AP lock from ALM. Also shown is the "opposite" of the vertical lock made from inverse vertices.
• Figure 5: A vertical iteration of the horizontal AP lock tangle called a vertical chain operator and its planar decomposition in the fundamental representation.