Khovanov–Rozansky cycle calculus for bipartite links

Abstract Bipartite calculus is a direct generalization of Kauffman planar expansion from $$N=2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> 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 $$3^{n}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mn>3</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:math> -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.