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

Structure and classification of intangible assets in industrial enterprises

http://vsed.oneu.edu.uaIn this article, the author attempts to identify and analyze the main prerequisites of agro-industrial integration in Ukraine at the institutional level: the institutional environment and transaction costs; forms and mechanisms of interaction between the participants, the measures of state support. The implicit constraints of integration processes in the investigated area were examined. A combined approach to a reasonable assessment of the benefits of integration was proposed. Finding a balance between the obvious economic interests and implicit constraints will actively involve farmers in the integration processes and thus contribute to the removal of the domestic agricultural sector out of the crisis

Implicit attitudes towards food and the self in sub-clinical eating disorder pathology

This thesis focuses on the assessment of implicit cognition in disordered eating behaviour, and specifically on the role of implicit attitudes towards the self and food in sub-clinical levels of eating disorder (ED) pathology. Chapter I reviews key theoretical approaches to implicit cognition and an assessment of the properties of implicit attitudes. Chapter II discusses methodological approaches to implicit attitude assessment, with a focus on the Implicit Attitude Test (IAT) which was used in the empirical studies. Chapter III (Study 1) outlines research in implicit cognition in ED pathology and assesses differences in implicit self-esteem (i.e., implicit attitude towards the self) between high-pathology and low-pathology participants. While the results of the study supported the presence of a self-esteem discrepancy in the high-pathology group, the findings were insufficient to pursue further research in this area. The next three studies were focused on the role implicit food attitudes. Chapter IV (Study 2) introduces the hypothesis that implicit food attitudes and ego depletion may play a role in elevated ED pathology. However, the ego depletion procedure used was not successful and the results were therefore insufficient to either support or reject the hypothesis. Chapter V (Study 3) assessed similar questions to Study 2, but from the angle of automatic approach and avoidance towards food stimuli and dietary restraint failure. Contrary to expectations, the results suggested that it is unlikely that either ego depletion or a positive implicit attitude towards high-fat foods contributes to dietary restraint failure. Chapter VI (Study 4) was designed to clarify the findings of Study 3 and found that implicit food attitudes may change following novel goal activation or changes in behaviour. Chapter VII is a general discussion which summarises the findings, discusses the outcomes within the broader context of implicit cognition theories, and proposes directions for further study