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

We elaborate on the simple alternative from arXiv:1308.5759 to the matrix-factorization construction of Khovanov-Rozansky (KR) polynomials for arbitrary knots and links in the fundamental representation of arbitrary SL(N). Construction consists of 2 steps: first, with every link diagram with m vertices one associates an m-dimensional hypercube with certain q-graded vector spaces, associated to its 2^m vertices. A generating function for q-dimensions of these spaces is what we suggest to call the primary T-deformation of HOMFLY polynomial -- because, as we demonstrate, it can be explicitly reduced to calculations of ordinary HOMFLY polynomials, i.e. to manipulations with quantum R-matrices. The second step is a certain minimization of residues of this new polynomial with respect to T+1. Minimization is ambiguous and is actually specified by the choice of commuting cut-and-join morphisms, acting along the edges of the hypercube -- this promotes it to Abelian quiver, and KR polynomial is a Poincare polynomial of associated complex, just in the original Khovanov's construction at N=2. This second step is still somewhat sophisticated -- though incomparably simpler than its conventional matrix-factorization counterpart. In this paper we concentrate on the first step, and provide just a mnemonic treatment of the second step. Still, this is enough to demonstrate that all the currently known examples of KR polynomials in the fundamental representation can be easily reproduced in this new approach. As additional bonus we get a simple description of the DGR relation between KR polynomials and superpolynomials and demonstrate that the difference between reduced and unreduced cases, which looks essential at KR level, practically disappears after transition to superpolynomials. However, a careful derivation of all these results from cohomologies of cut-and-join morphisms remains for further studies.Comment: 146 pages; some points clarified, some typos correcte

Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?

$R$-coloured knot polynomials for $m$-strand torus knots $Torus_{[m,n]}$ are described by the Rosso-Jones formula, which is an example of evolution in $n$ with Lyapunov exponents, labelled by Young diagrams from $R^{\otimes m}$. This means that they satisfy a finite-difference equation (recursion) of finite degree. For the gauge group $SL(N)$ only diagrams with no more than $N$ lines can contribute and the recursion degree is reduced. We claim that these properties (evolution/recursion and reduction) persist for Khovanov-Rozansky (KR) polynomials, obtained by additional factorization modulo $1+{\bf t}$, which is not yet adequately described in quantum field theory. Also preserved is some weakened version of differential expansion, which is responsible at least for a simple relation between {\it reduced} and {\it unreduced} Khovanov polynomials. However, in the KR case evolution is incompatible with the mirror symmetry under the change $n\longrightarrow -n$, what can signal about an ambiguity in the KR factorization even for torus knots. }Comment: 23 p

New and Old Results in Resultant Theory

Resultants are getting increasingly important in modern theoretical physics: they appear whenever one deals with non-linear (polynomial) equations, with non-quadratic forms or with non-Gaussian integrals. Being a subject of more than three-hundred-year research, resultants are of course rather well studied: a lot of explicit formulas, beautiful properties and intriguing relationships are known in this field. We present a brief overview of these results, including both recent and already classical. Emphasis is made on explicit formulas for resultants, which could be practically useful in a future physics research.Comment: 50 pages, 15 figure