Narrow Results

One of the spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group ${\mathrm{\text{SU}}}_q(2)$ through the Askey–Wilson polynomials, associated with the $q$-hypergeometric functions ${}_4{\varphi }_3$. Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary ${\mathrm{\text{SU}}}_q(N)$ — at least for exclusive Racah matrices $\bar{S}$. The natural question then is what substitutes the conventional $q$-hypergeometric polynomials when representations are more general? New advances in the theory of matrices $\bar{S}$, provided by the study of differential expansions of knot polynomials, suggest that these are multiple sums over Young sub-diagrams of the one which describes the original representation of ${\mathrm{\text{SU}}}_q(N)$. A less trivial fact is that the entries of the sum are not just the factorized combinations of quantum dimensions, as in the ordinary hypergeometric series, but involve non-factorized quantities, like the skew characters and their further generalizations — as well as associated additional summations with the Littlewood–Richardson weights.

Factorization of the differential expansion (DE) coefficients for colored HOMFLY-PT polynomials of antiparallel double braids, originally discovered for rectangular representations $R$, in the case of rectangular representations $R$, is extended to the first non-rectangular representations $R=[2,1]$ and $R=[3,1]$. This increases chances that such factorization will take place for generic $R$, thus fixing the shape of the DE. We illustrate the power of the method by conjecturing the DE-induced expression for double-braid polynomials for all $R=[r,1]$. In variance with the rectangular case, the knowledge for double braids is not fully sufficient to deduce the exclusive Racah matrix $\bar{S}$ — the entries in the sectors with nontrivial multiplicities sum up and remain unseparated. Still, a considerable piece of the matrix is extracted directly and its other elements can be found by solving the unitarity constraints.

This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation $R=[2,1]$. The project involves several steps:

• ⁽ⁱ⁾parametrization of big families of knots á la [A. Mironov and A. Morozov, arXiv:1506.00339],
• ⁽ⁱⁱ⁾evaluating Racah/mixing matrices for various numbers of strands in various representations á la [A. Mironov, A. Morozov and An. Morozov, J. High Energy Phys.03, 034 (2012), arXiv:1112.2654],
• ⁽ⁱⁱⁱ⁾tabulating and collecting the results at http://www.knotebook.org.

In this paper, we discuss only the representation $R=[2,1]$ and construct all necessary ingredients that allow one to evaluate knot/links represented by three-strand closed parallel braids with inserted double-fat fingers. In particular, it is used to evaluate knots from a 7-parametric family. This family contains over 80% of knots with up to 10 intersections, but does not include mutants.

Quantum ${ℛ}$-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 $S{U}_q(N)$ associated with each strand, one needs two matrices: ${{ℛ}}_1$ and ${{ℛ}}_2$. They are related by the Racah matrices ${{ℛ}}_2={𝒰}{{ℛ}}_1{{𝒰}}^{†}$. Since we can always choose the basis so that ${{ℛ}}_1$ is diagonal, the problem is reduced to evaluation of ${{ℛ}}_2$-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that ${{ℛ}}_2$-matrices could be transformed into a block-diagonal ones by the rotation in the sectors of coinciding eigenvalues. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of ${{ℛ}}_1$ matrix. In this case in order to get a block-diagonal matrix, one should rotate the ${{ℛ}}_2$ defined by the Racah matrix in the accidental sector by the angle exactly $\pm \frac{\pi }{4}$.