Empirical analysis of many colored knot polynomials, made possible by recent computational advances in Chern-Simons theory, reveals their stability: for any given negative N and any given knot the set of coefficients of the polynomial in r-th symmetric representation does not change with r, if it is large enough. This fact reflects the non-trivial and previously unknown properties of the differential expansion, and it turns out that from this point of view there are universality classes of knots, characterized by a single integer, which we call defect, and which is in fact related to the power of Alexander polynomial.HOMFLY polynomials are Wilson-loop averages in 3d Chern-Simons theory [1], which in this simplest model depend only on the topology of the Wilson line (knot). Therefore one can separate and study the group-theory properties of observables -and this is a non-trivial and very interesting problem, for a brief summary of results see [2]. From the quantum field theory perspective knot polynomials are direct generalization of conformal blocks, and this relation [3] , which was temporarily postponed because of the insufficient "experimental" material.In this note we describe empirically obtained properties of these expansions for symmetric representations [r] (where r is the length of the single-line Young diagram). It looks like there are different universality classes of such expansions, characterized by a single integer, which we call "defect" δ K . Moreover, these newly observed properties allow to identify 2(δ K + 1) with the power of Alexander polynomial and lead to a peculiar stability property of symmetrically colored HOMFLY for large enough r: what stabilizes is not the polynomial itself, but the set of its coefficients -i.e. something like the "coordinates" g r,j , introduced in [24]. Theoretical analysis of these observations, proofs and extension to non-(anti)symmetric representations are beyond the scope of the present text.

K-theoretic Donaldson–Thomas counts of curves in toric and many related threefolds can be computed in terms of a certain canonical 3-valent tensor, the K-theoretic equivariant vertex. In this paper we derive a formula for the vertex in the case when two out of three entries are nontrivial. We also discuss some applications of this result.

As a new step in the study of rectangularly-colored knot polynomials, we reformulate the prescription of [1] for twist knots in the double-column representations R = [rr] in terms of skew Schur polynomials. These, however, are mysteriously shifted from the standard topological locus, what makes further generalization to arbitrary R = [r s ] not quite straightforward.Knot theory is an old and respected branch of mathematics, but recently it also became one of the rapidly developing branches of theoretical physics. This is because the knot polynomials [2] appeared to provide exact non-perturbative answers to Wilson-line averagesin 3d Chern-Simons theory [3] -one of the simplest members of the family of physically relevant Yang-Mills theories.In (1) q is made from the coupling constant k, q = exp 2πi k+N , and A = q N -from the parameter N of the gauge group Sl(N ). Remarkably, in these variables the average is a Laurent polynomial -provided the space-times is simply-connected. Despite Chern-Simons is topological theory, i.e. has nearly trivial dynamics in space-time, dependencies of physical quantities on the other parameters (coupling constants etc) are quite non-trivial -and provide a good model and polygon for the study of renormalization-group and boundary-condition properties. Moreover, from this point of view Chern-Simons seems less trivial than, say, the comprehensible sectors of N = 4 SYM theory (in particular, its integrability properties are far more sophisticated) -still it is exactly solvable,but not yet solved. Added to this are deep connections of Chern-Simons theories to conformal field theory and various string models, especially to toric Calabi-Yau compactifications. The features of knot polynomials are still a set of mysteries, ranging from a hierarchical set of integrality properties to various RG-like evolutions in different parameters, especially in the space of representations R, while the standard methods of non-perturbative analysis, like Ward-identities, AMM/EO topological recursion, integrability techniques etc are not yet fully applicable. Development of the theory is still going through consideration of examples: particular knots K and particular representations R, for which a powerful technique is now developed [4]- [9]. At present stage these examples start being unified into the simplest families, either of knots or of representations. This paper is about a mixture: we provide an exact answer for a one parametric family of twist knots Tw m in a one-parametric family of two-column rectangular representations R = [rr]. It is a new small step along the line, originated in [10,11] In the present paper we address one of the important claims of [17], which in reformulation of [18] states that the rectangular HOMFLY polynomials for defect-zero knots (those where Alexander polynomial has degree one), in particular for the twist family Tw m , can be represented as 1