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

TL;DR

This paper introduces a simplified tensor-algebra method for constructing Khovanov-Rozansky invariants, making the process more accessible and applicable to all N without complex calculations.

Contribution

It presents a new, elementary tensor-algebra approach to Khovanov-Rozansky invariants that avoids matrix factorizations and extends naturally from SL(2) to SL(N).

Findings

Simplified tensor-algebra construction reproduces KR polynomials for all N.

Eliminates the need for matrix factorizations in computing invariants.

Provides a more natural and practical framework for knot polynomial calculations.

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.