Khovanov-Rozansky cycle calculus for bipartite links

TL;DR

This paper develops a comprehensive cycle-calculus framework for Khovanov–Rozansky polynomials restricted to bipartite links composed of antiparallel lock tangles. By lifting the Jones 2-hypercube to a 3-hypercube and introducing N-dimensional cycle spaces with cut-and-join–type morphisms plus a new gluing operation, the authors provide a tractable, matrix-factorization–free approach to KR polynomials in this locus. They delineate explicit algorithms, derive KR invariants for a broad class of bipartite diagrams, and demonstrate the method through multiple examples including Hopf, trefoil, and figure-eight knots, as well as precursors and dualities with precursor Jones polynomials. The work shows that superpolynomials can be computed in a more accessible, planar framework for bipartite knots, with potential implications for understanding KR invariants, their continuity in N, and connections to topological string theory and refined Chern–Simons theory.

Abstract

Bipartite calculus is a direct generalization of Kauffman planar expansion from $N=2$ to arbitrary $N$, applicable to the restricted class of knots which are entirely made of antiparallel lock tangles. Whenever applicable, it allows a straightforward generalization of the Khovanov calculus without a need of the technically complicated matrix factorization used for arbitrary $N$ in the Khovanov-Rozansky (KR) approach. The main object here is the $3^{n}$-dimensional hypercube with $n$ being the number of bipartite vertices. Maps, differentials, complex and Poincaré polynomials are straightforward and indeed reproduce the Khovanov-Rozansky polynomials in the known cases. This provides a great simplification of the Khovanov-Rozansky calculus on the bipartite locus, what can make it an accessible tool for the study of superpolynomials.

Figures (47)

• Figure 1: 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 have no orientation. The resolutions are of two types which we denote 0 and 1. These numbers are values of $\alpha_i$, see the text below.
• Figure 2: The Hopf link and its hypercube of resolutions.
• Figure 3: The trefoil knot $3_1$ and its hypercube of resolutions.
• Figure 4: Denotations for a crossing and its mirror.
• Figure 5: The Hopf link and its Khovanov complex.