Related papers: Factorization of differential expansion for non-re…
Recent results of J.Gu and H.Jockers provide the lacking initial conditions for the evolution method in the case of the first non-trivially colored HOMFLY polynomials H_{[21]} for the family of twist knots. We describe this application of…
We generalize the recently discovered planar decomposition (Kauffman bracket) for the HOMFLY polynomials of bipartite knot/link diagrams to (anti)symmetrically colored HOMFLY polynomials. Cabling destroys planarity, but it is restored after…
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…
Construction of (colored) knot polynomials for double-fat graphs is further generalized to the case when "fingers" and "propagators" are substituting R-matrices in arbitrary closed braids with m-strands. Original version of arXiv:1504.00371…
The recently suggested bipartite analysis extends the Kauffman planar decomposition to arbitrary $N$, i.e. extends it from the Jones polynomial to the HOMFLY polynomial. This provides a generic and straightforward non-perturbative calculus…
Basing on evaluation of the Racah coefficients for SU_q(3) (which supported the earlier conjecture of their universal form) we derive explicit formulas for all the 5-, 6- and 7-strand Wilson averages in the fundamental representation of…
If a knot is represented by an m-strand braid, then HOMFLY polynomial in representation R is a sum over characters in all representations Q\in R^{\otimes m}. Coefficients in this sum are traces of products of quantum R-matrices along the…
Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the…
Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid carrying arbitrary $SU(N)$ representation is still tedious. For a class of rank $r$ symmetric representations, $[r]$-colored HOMFLY-PT $H_{[r]}$ evaluation becomes…
As a new step in the study of rectangularly-colored knot polynomials, we reformulate the prescription of arXiv:1606.06015 for twist knots in the double-column representations $R=[rr]$ in terms of skew Schur polynomials. These, however, are…
The HOMFLY-PT polynomial is a two-parameter knot polynomial that admits a character expansion, expressed as a sum of Schur functions over Young diagrams. The Harer-Zagier (HZ) transform, which converts the HOMFLY--PT polynomial into a…
Racah matrices and higher $j$-symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced…
Explicit answer is given for the HOMFLY polynomial of the figure eight knot $4_1$ in arbitrary symmetric representation R=[p]. It generalizes the old answers for p=1 and 2 and the recently derived results for p=3,4, which are fully…
We elaborate on the simple alternative from arXiv:1308.5759 to the matrix-factorization construction of Khovanov-Rozansky (KR) polynomials for arbitrary knots and links in the fundamental representation of arbitrary SL(N). Construction…
Quantum $\mathcal{R}$-matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation $T$ of $SU_q(N)$ associated with each strand one…
We study factorizations of HOMFLY polynomials of certain knots and oriented links. We begin with a computer analysis of knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for…
Recently it was shown that the (Ooguri-Vafa) generating function of HOMFLY polynomials is the Hurwitz partition function, i.e. that the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group…
We obtain a novel formula for characteristic polynomials of deformations of the Braid arrangement using the notion of levels of regions. As an application, we recover and strengthen results of Chen et al. on the characteristic polynomial of…
We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related…
This paper studies Symmetric Determinantal Representations (SDR) in characteristic 2, that is the representation of a multivariate polynomial P by a symmetric matrix M such that P=det(M), and where each entry of M is either a constant or a…