Bipartite expansion beyond biparticity

TL;DR

This work extends the planar decomposition framework from bipartite knots to arbitrary knots by introducing positive decomposition (PD) of the fundamental HOMFLY polynomial, with coefficients that are non-negative integers in the variables $\phi$, $\bar{\phi}$ and $D$, modulo the relation $G=\phi+\bar{\phi}+\phi\bar{\phi} D=0$. It clarifies how PD specializes to bipartite expansion (BE) for diagrams built from antiparallel locks and demonstrates the equivalence with Kauffman calculus at $N=2$; importantly, PD exists for all knots, not only bipartite ones, and there are two distinct regimes: chiral PD (unique and positive) and non-chiral PD (ambiguous, with multiple positive representatives differing by multiples of $G$). The paper provides explicit examples of chiral PD up to 10 crossings, discusses non-chiral PD including “fake BE” phenomena, and introduces a precursor Jones polynomial criterion to test for potential underlying bipartite realizations. It also extends the discussion to non-bipartite diagrams via a bipartite-calculus-like expansion and explores the more complex symmetric representation $[2]$, highlighting additional variables and potential selection rules. The results illuminate a largely new and rich structure in knot polynomials, with implications for non-perturbative Chern–Simons theory and possible connections to Khovanov-type frameworks, while leaving open questions about canonical PD definitions, minimality, and deep links to BE. The work thus positions PD as a universal, albeit nuanced, feature of HOMFLY polynomials with potential broad applications in knot theory and quantum topology.

Abstract

The recently suggested bipartite analysis extends the Kauffman planar decomposition to arbitrary $N$, i.e. extends it from the Jones polynomial to the HOMFLY polynomial. This provides a generic and straightforward non-perturbative calculus in an arbitrary Chern--Simons theory. Technically, this approach is restricted to knots and links which possess bipartite realizations, i.e. can be entirely glued from antiparallel lock (two-vertex) tangles rather than single-vertex $R$-matrices. However, we demonstrate that the resulting positive decomposition (PD), i.e. the representation of the fundamental HOMFLY polynomials as positive integer polynomials of the three parameters $φ$, $\barφ$ and $D$, exists for arbitrary knots, not only bipartite ones. This poses new questions about the true significance of bipartite expansion, which appears to make sense far beyond its original scope, and its generalizations to higher representations. We have provided two explanations for the existence of the PD for non-bipartite knots. An interesting option is to resolve a particular bipartite vertex in a not-fully-bipartite diagram and reduce the HOMFLY polynomial to a linear combination of those for smaller diagrams. If the resulting diagrams correspond to bipartite links, this option provides a PD even to an initially non-bipartite knot. Another possibility for a non-bipartite knot is to have a bipartite clone with the same HOMFLY polynomial providing this PD. We also suggest a promising criterium for the existence of a bipartite realization behind a given PD, which is based on the study of the precursor Jones polynomials.

Figures (13)

• Figure 1: The planar decomposition of the positive (in the first line) and negative (in the second line) lock vertices in the topological framing. In bipfund, we used the vertical framing, but the topological one is more convenient for our considerations, thus, we use it throughout the present paper.
• Figure 2: The celebrated Kauffman bracket -- the planar decomposition of the ${\cal R}$-matrix vertex for the fundamental representation of $\mathfrak{sl}_q(2)$. 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: Analogues of the first and the second Reidemeister moves for bipartite diagrams.
• Figure 4: I. The diagram that reproduces one of 8 bipartite knots depending on signs of the bipartite vertices. Table on the left give knots up to 12 crossings with the corresponding HOMFLY polynomial (a diagram with any other combination of signs is equivalent to one of these 8 ones). One of these knots is $10_{140}$. II. The knot $10_{140}$ as the Montensinos knot $K(\frac{2}{3},-\frac{2}{3},\frac{1}{4})$. The two diagrams may correspond to the two "local minima" of PD.
• Figure 5: Example of an equivalence transformation of a bipartite diagram that changes PD to a positive polynomial that is a multiple of $\phi+\bar{\phi}+D\phi\bar{\phi}$. An initial bipartite diagram (living inside a big circle) is arbitrarily split into two bipartite 4-tangles ${\rm BT}_1$ and ${\rm BT}_2$. The new answer for the HOMFLY polynomial (coming from the last line) is not a "local minimum".