On possible existence of HOMFLY polynomials for virtual knots

Virtual knots are associated with knot diagrams, which are not obligatory planar. The recently suggested generalization from N=2 to arbitrary N of the Kauffman-Khovanov calculus of cycles in resolved diagrams can be straightforwardly applied to non-planar case. In simple examples we demonstrate that this construction preserves topological invariance -- thus implying the existence of HOMFLY extension of cabled Jones polynomials for virtual knots and links.Comment: 12 page

Evolution method and HOMFLY polynomials for virtual knots

Following the suggestion of arXiv:1407.6319 to lift the knot polynomials for virtual knots and links from Jones to HOMFLY, we apply the evolution method to calculate them for an infinite series of twist-like virtual knots and antiparallel 2-strand links. Within this family one can check topological invariance and understand how differential hierarchy is modified in virtual case. This opens a way towards a definition of colored (not only cabled) knot polynomials, though problems still persist beyond the first symmetric representation.Comment: 28 page

Free-Field Representation of Group Element for Simple Quantum Group

A representation of the group element (also known as ``universal ${\cal T}$-matrix'') which satisfies $\Delta(g) = g\otimes g$, is given in the form $$g = \left(\prod_{s=1}^{d_B}\phantom.^>\ {\cal E}_{1/q_{i(s)}}(\chi^{(s)}T_{-i(s)})\right) q^{2\vec\phi\vec H} \left(\prod_{s=1}^{d_B}\phantom.^<\ {\cal E}_{q_{i(s)}}(\psi^{(s)} T_{+i(s)})\right)$$ where $d_B = \frac{1}{2}(d_G - r_G)$, $q_i = q^{|| \vec\alpha_i||^2/2}$ and $H_i = 2\vec H\vec\alpha_i/||\vec\alpha_i||^2$ and $T_{\pm i}$ are the generators of quantum group associated respectively with Cartan algebra and the {\it simple} roots. The ``free fields'' $\chi,\ \vec\phi,\ \psi$form a Heisenberg-like algebra:$\psi^{(s)}\psi^{(s')} = q^{-\vec\alpha_{i(s)} \vec\alpha_{i(s')}} \psi^{(s')}\psi^{(s)}, & \chi^{(s)}\chi^{(s')} = q^{-\vec\alpha_{i(s)}\vec\alpha_{i(s')}} \chi^{(s')}\chi^{(s)}& {\rm for} \ s<s', \\ q^{\vec h\vec\phi}\psi^{(s)} = q^{\vec h\vec\alpha_{i(s)}} \psi^{(s)}q^{\vec h\vec\phi}, & q^{\vec h\vec\phi}\chi^{(s)} = q^{\vec h \vec\alpha_{i(s)}}\chi^{(s)}q^{\vec h\vec\phi}, & \\ &\psi^{(s)} \chi^{(s')} = \chi^{(s')}\psi^{(s)} & {\rm for\ any}\ s,s'.$We argue that the$d_G$-parametric ``manifold'' which$g$spans in the operator-valued universal envelopping algebra, can also be invariant under the group multiplication$g \rightarrow g'\cdot g''$. The universal${\cal R}$-matrix with the property that${\cal R} (g\otimes I)(I\otimes g) = (I\otimes g)(g\otimes I){\cal R}$is given by the usual formula$${\cal R} = q^{-\sum_{ij}^{r_G}||\vec\alpha_i||^2|| \vec\alpha_j||^2 (\vec\alpha\vec\alpha)^{-1}_{ij}H_i \otimes H_j}\prod_{ \vec\alpha > 0}^{d_B}{\cal E}_{q_{\vec\alpha}}\left(-(q_{\vec\alpha}- q_{\vec\alpha}^{-1})T_{\vec\alpha}\otimes T_{-\vec\alpha}\right).$$Comment: 68 page

Explicit examples of DIM constraints for network matrix models

Dotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W-constraints or loop equations or regularity condition for qq-characters) are also promoted to the DIM level, where they all become corollaries of a single identity.Comment: 46 page

Harer-Zagier formulas for knot matrix models

Knot matrix models are defined so that the averages of characters are equal to knot polynomials. From this definition one can extract single trace averages and generation functions for them in the group rank - which generalize the celebrated Harer-Zagier formulas for Hermitian matrix model. We describe the outcome of this program for HOMFLY-PT polynomials of various knots. In particular, we claim that the Harer-Zagier formulas for torus knots factorize nicely, but this does not happen for other knots. This fact is mysteriously parallel to existence of explicit beta = 1 eigenvalue model construction for torus knots only, and can be responsible for problems with construction of a similar model for other knots

Super-Schur Polynomials for Affine Super Yangian Y(gl^1∣1)\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})

We explicitly construct cut-and-join operators and their eigenfunctions -- the Super-Schur functions -- for the case of the affine super-Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This is the simplest non-trivial (semi-Fock) representation, where eigenfunctions are labeled by the superanalogue of 2d Young diagrams, and depend on the supertime variables $(p_k,\theta_k)$. The action of other generators on diagrams is described by the analogue of the Pieri rule. As well we present generalizations of the hook formula for the measure on super-Young diagrams and of the Cauchy formula. Also a discussion of string theory origins for these relations is provided.Comment: 27 pages, 3 figure