Hunt for 3-Schur polynomials

This paper describes our attempt to understand the recent success of Na Wang in constructing the 3-Schur polynomials, associated with the plane partitions. We provide a rather detailed review and try to figure out the new insights, which allowed to overcome the problems of the previous efforts. In result we provide a very simple definition of time-variables ${\bf P}_{i\geqslant j}$ and the cut-and-join operator $\hat W_2$, which generates the set of $3$-Schur functions. Some coefficients in $\hat W_2$ remain undefined and require more effort to be fixed

On an alternative stratification of knots

We introduce an alternative stratification of knots: by the size of lattice on which a knot can be first met. Using this classification, we find ratio of unknots and knots with more than 10 minimal crossings inside different lattices and answer the question which knots can be realized inside $3\times 3$ and $5\times 5$ lattices. In accordance with previous research, the ratio of unknots decreases exponentially with the growth of the lattice size. Our computational results are approved with theoretical estimates for amounts of knots with fixed crossing number lying inside lattices of given size.Comment: 12 page

Irreducible representations of simple Lie algebras by differential operators

We describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $\mathfrak{g}$. The Lie algebra generators are represented as first order differential operators in $\frac{1}{2} \left(\dim \mathfrak{g} - \text{rank} \, \mathfrak{g}\right)$ variables. All rising generators ${\bf e}$ are universal in the sense that they do not depend on representation, the weights enter (in a very simple way) only in the expressions for the lowering operators ${\bf f}$. We present explicit formulas of this kind for the simple root generators of all classical Lie algebras

3-Schurs from explicit representation of Yangian Y gl ̂ 1 Y(gl^1) \textrm{Y}\left({\hat{\mathfrak{gl}}}_1\right) . Levels 1–5

Abstract We suggest an ansatz for representation of affine Yangian Y gl ̂ 1 $$Y\left({\hat{\mathfrak{gl}}}_1\right)$$ by differential operators in the triangular set of time-variables P a,i with 1 ⩽ i ⩽ a, which saturates the MacMahon formula for the number of 3d Young diagrams/plane partitions. In this approach the 3-Schur polynomials are defined as the common eigenfunctions of an infinite set of commuting “cut-and-join” generators ψ n of the Yangian. We manage to push this tedious program through to the 3-Schur polynomials of level 5, and this provides a rather big sample set, which can be now investigated by other methods

Evolution properties of the knot's defect

The defect of differential (cyclotomic) expansion for colored HOMFLY-PT polynomials is conjectured to be invariant under any antiparallel evolution and change linearly with the evolution in any parallel direction. In other words, each ${\cal R}$-matrix can be substituted by an entire 2-strand braid in two different ways: the defect remains intact when the braid is antiparallel and changes by half of the added length when the braid is parallel

Hunt for 3-Schur polynomials

This paper describes our attempt to understand the recent success of Na Wang in constructing the 3-Schur polynomials, associated with the plane partitions. We provide a rather detailed review and try to figure out the new insights, which allowed to overcome the problems of the previous efforts. In result we provide a very simple definition of time-variables Pi⩾j and the cut-and-join operator Wˆ2, which generates the set of 3-Schur functions. Some coefficients in Wˆ2 remain undefined and require more effort to be fixed

Are Maxwell knots integrable?

We review properties of the null-field solutions of source-free Maxwell equations. We focus on the electric and magnetic field lines, especially on limit cycles, which actually can be knotted and/or linked at every given moment. We analyse the fact that the Poynting vector induces self-consistent time evolution of these lines and demonstrate that the Abelian link invariant is integral of motion. We also consider particular examples of the field lines for the particular family of finite energy source-free “knot” solutions, attempting to understand when the field lines are closed – and can be discussed in terms of knots and links. Based on computer simulations we conjecture that Ranada’s solution, where every pair of lines forms a Hopf link, is rather exceptional. In general, only particular lines (a set of measure zero) are limit cycles and represent closed lines forming knots/links, while all the rest are twisting around them and remain unclosed. Still, conservation laws of Poynting evolution and associated integrable structure should persist