Knot invariants from Virasoro related representation and pretzel knots

We remind the method to calculate colored Jones polynomials for the plat representations of knot diagrams from the knowledge of modular transformation (monodromies) of Virasoro conformal blocks with insertions of degenerate fields. As an illustration we use a rich family of pretzel knots, lying on a surface of arbitrary genus g, which was recently analyzed by the evolution method. Further generalizations can be to generic Virasoro modular transformations, provided by integral kernels, which can lead to the Hikami invariants.Comment: 29 page

S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality

A recent claim that the S-duality between 4d SUSY gauge theories, which is AGT related to the modular transformations of 2d conformal blocks, is no more than an ordinary Fourier transform at the perturbative level, is further traced down to the commutation relation [P,Q]=-i\hbar between the check-operator monodromies of the exponential resolvent operator in the underlying Dotsenko-Fateev matrix models and beta-ensembles. To this end, we treat the conformal blocks as eigenfunctions of the monodromy check operators, what is especially simple in the case of one-point toric block. The kernel of the modular transformation is then defined as the intertwiner of the two monodromies, and can be obtained straightforwardly, even when the eigenfunction interpretation of the blocks themselves is technically tedious. In this way, we provide an elementary derivation of the old expression for the modular kernel for the one-point toric conformal block.Comment: 15 page

Colored knot polynomials for Pretzel knots and links of arbitrary genus

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformation of toric conformal block.Comment: 5 page

Electronic Structure of Transition-Metal Dicyanamides Me[N(CN)2_2]2_2 (Me = Mn, Fe, Co, Ni, Cu)

The electronic structure of Me[N(CN)$_2$]$_2$ (Me=Mn, Fe, Co, Ni, Cu) molecular magnets has been investigated using x-ray emission spectroscopy (XES) and x-ray photoelectron spectroscopy (XPS) as well as theoretical density-functional-based methods. Both theory and experiments show that the top of the valence band is dominated by Me 3d bands, while a strong hybridization between C 2p and N 2p states determines the valence band electronic structure away from the top. The 2p contributions from non-equivalent nitrogen sites have been identified using resonant inelastic x-ray scattering spectroscopy with the excitation energy tuned near the N 1s threshold. The binding energy of the Me 3d bands and the hybridization between N 2p and Me 3d states both increase in going across the row from Me = Mn to Me = Cu. Localization of the Cu 3d states also leads to weak screening of Cu 2p and 3s states, which accounts for shifts in the core 2p and 3s spectra of the transition metal atoms. Calculations indicate that the ground-state magnetic ordering, which varies across the series is largely dependent on the occupation of the metal 3d shell and that structural differences in the superexchange pathways for different compounds play a secondary role.Comment: 20 pages, 11 figures, 2 table

Spin States of Cobalt Ions in the Bulk and on the Surface of LaCoO3 Probed by X-ray Absorption, Emission, and Photoelectron Spectra

We present X-ray photoelectron, Co L2,3 and O K X-ray absorption, as well as Co Kβ1,3 X-ray emission spectroscopy results of studies of the spin states of trivalent cobalt ions in single-crystal cobaltite LaCoO3. We show that at room temperature, in the bulk of a LaCoO3 single crystal, Co3+ ions are in the low-spin state, while high-spin Co2+, high-spin Co3+, low-spin Co3+ , and probably also intermediate-spin Co3+ ions are localated on the surface