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We elaborate on the simple alternative from arXiv:1308.5759 to thematrix-factorization construction of Khovanov-Rozansky (KR) polynomials forarbitrary knots and links in the fundamental representation of arbitrary SL(N).Construction consists of 2 steps: first, with every link diagram with mvertices one associates an m-dimensional hypercube with certain q-graded vectorspaces, associated to its 2^m vertices. A generating function for q-dimensionsof these spaces is what we suggest to call the primary T-deformation of HOMFLYpolynomial -- because, as we demonstrate, it can be explicitly reduced tocalculations of ordinary HOMFLY polynomials, i.e. to manipulations with quantumR-matrices. The second step is a certain minimization of residues of this newpolynomial with respect to T+1. Minimization is ambiguous and is actuallyspecified by the choice of commuting cut-and-join morphisms, acting along theedges of the hypercube -- this promotes it to Abelian quiver, and KR polynomialis a Poincare polynomial of associated complex, just in the original Khovanov'sconstruction at N=2. This second step is still somewhat sophisticated -- thoughincomparably simpler than its conventional matrix-factorization counterpart. Inthis paper we concentrate on the first step, and provide just a mnemonictreatment of the second step. Still, this is enough to demonstrate that all thecurrently known examples of KR polynomials in the fundamental representationcan be easily reproduced in this new approach. As additional bonus we get asimple description of the DGR relation between KR polynomials andsuperpolynomials and demonstrate that the difference between reduced andunreduced cases, which looks essential at KR level, practically disappearsafter transition to superpolynomials. However, a careful derivation of allthese results from cohomologies of cut-and-join morphisms remains for furtherstudies.

Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY