Introduction to Khovanov homologies. III. A new and simple tensor-algebra construction of Khovanov–Rozansky invariants

Abstract

We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N−1=1 was based on the cut-and-join calculus of the planar cycles, which are involved rather artificially. We substitute them by alternative and natural set of cycles, not obligatory planar. Then the whole construction is straightforwardly lifted from SL(2) to SL(N) and reproduces Khovanov–Rozansky (KR) polynomials, simultaneously for all values of N. No matrix factorization and related tedious calculations are needed in such approach, which can therefore become not only conceptually, but also practically useful.