Khovanov-Rozansky cycle calculus for bipartite links

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.

40 pages

References in corpus (30)

• Khovanov's homology for tangles and cobordisms
• The Refined Topological Vertex
• Knot Invariants and Topological Strings
• On Khovanov's categorification of the Jones polynomial
• Khovanov-Rozansky Homology and Topological Strings
• Knots, links and branes at large N
• Matrix Models, Geometric Engineering and Elliptic Genera
• Polynomial invariants for torus knots and topological strings
• sl(3) link homology
• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Chern-Simons Theory in the Temporal Gauge and Knot Invariants through the Universal Quantum R-Matrix
• Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid
• Colored HOMFLY polynomials of knots presented as double fat diagrams
• New Quantum Obstructions to Sliceness
• On universal knot polynomials
• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Computing Khovanov-Rozansky homology and defect fusion
• Nimble evolution for pretzel Khovanov polynomials
• Tangle blocks in the theory of link invariants
• Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?
• Knot polynomials for twist satellites
• Checks of integrality properties in topological strings
• Gaussian distribution of LMOV numbers
• Planar decomposition of the HOMFLY polynomial for bipartite knots and links
• Towards tangle calculus for Khovanov polynomials
• The action of full twist on the superpolynomial for torus knots
• Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
• Bipartite expansion beyond biparticity
• Khovanov-Rozansky homology for infinite multi-colored braids

Cited by in corpus (2)

• Khovanov--Rozansky matrix factorization reduction for bipartite links
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs