Bipartite expansion beyond biparticity

TL;DR

This paper extends bipartite analysis to all knots, showing positive decompositions of HOMFLY polynomials exist beyond bipartite knots and exploring implications for knot theory and polynomial invariants.

Contribution

It demonstrates that positive decompositions of HOMFLY polynomials are possible for all knots, not just bipartite ones, and proposes explanations and criteria for bipartite realizations.

Findings

Positive decomposition exists for arbitrary knots.

Bipartite expansion applies beyond bipartite knots.

Criteria based on Jones polynomials for bipartite realization.

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 $\phi$, $\bar\phi$ 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.