An. Morozov

Colored HOMFLY polynomials of knots presented as double fat diagrams

A bstractMany knots and links in S3 can be drawn as gluing of three manifolds with one or more four-punctured S2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the braiding matrices of four -strand braids. Incorporating the properties of four-point conformal blocks in WZNW models, we conjecture colored HOMFLY polynomials for these double fat graphs where the color can be rectangular or non-rectangular representation. With the recent work of Gu-Jockers, the fusion matrices for the non-rectangular [21] representation, the first which involves multiplicity is known. We verify our conjecture by comparing with the [21] colored HOMFLY of many knots, obtained as closure of three braids. The conjectured form is computationally very effective leading to writing [21]-colored HOMFLY polynomials for many pretzel type knots and non-pretzel type knots. In particular, we find class of pretzel mutants which are distinguished and another class of mutants which cannot be distinguished by [21] representation. The difference between the [21]-colored HOMFLY of two mutants seems to have a general form, with A-dependence completely defined by the old conjecture due to Morton and Cromwell. In particular, we check it for an entire multi-parametric family of mutant knots evaluated using evolution method.

Tabulating knot polynomials for arborescent knots

Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site. Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the family approach, and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY polynomials.

On the defect and stability of differential expansion

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 rth 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 in representation [3, 1] for 3-strand braids

A bstractThis paper is a new step in the project of systematic description of colored knot polynomials started in [1]. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R⊗3 −→ Q with all possible Q, for R = [3, 1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3, 1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family (n, −1 | 1, −1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.

Racah matrices and hidden integrability in evolution of knots

Abstract We construct a general procedure to extract the exclusive Racah matrices S and S ¯ from the inclusive 3-strand mixing matrices by the evolution method and apply it to the first simple representations R = [ 1 ] , [2], [3] and [ 2 , 2 ] . The matrices S and S ¯ relate respectively the maps ( R ⊗ R ) ⊗ R ¯ ⟶ R with R ⊗ ( R ⊗ R ¯ ) ⟶ R and ( R ⊗ R ¯ ) ⊗ R ⟶ R with R ⊗ ( R ¯ ⊗ R ) ⟶ R . They are building blocks for the colored HOMFLY polynomials of arbitrary arborescent (double fat) knots. Remarkably, the calculation realizes an unexpected integrability property underlying the evolution matrices.

Quantum Racah matrices and 3-strand braids in irreps R with | R | = 4

AbstractWe describe the inclusive Racah matrices for the first non-(anti)symmetric rectangular representation R = [2, 2] for quantum groups Uq(slN). Most of them have sizes 2, 3, and 4 and are fully described by the eigenvalue hypothesis. Of two 6 × 6 matrices, one is also described in this way, but the other one corresponds to the case of degenerate eigenvalues and is evaluated by the highest weight method. Together with the much harder calculation for R = [3, 1] and with the new method to extract exclusive matrices J and n

Eigenvalue hypothesis for multi-strand braids

overline J

Addendum to: Checks of integrality properties in topological strings

J¯ from the inclusive ones, this completes the story of Racah matrices for |R| ≤ 4 and allows one to calculate and investigate the corresponding colored HOMFLY polynomials for arbitrary 3-strand and arborescent knots.