Abstract
Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution -- that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots. In the case of rectangular representations $R=[r^s]$ we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots. This reduces the problem of rectangular exclusive Racah to constructing the answers for just a few twist knots. We develop a recent conjecture on the structure of differential expansion for the simplest members of this family (the trefoil and the figure-eight knot) and provide the exhaustive answer for the first unknown case of $R=[33]$. The answer includes HOMFLY of arbitrary twist and double-braid knots and Racah matrices $\bar S$ and $S$ -- what allows to calculate $[33]$-colored polynomials for arbitrary arborescent (double-fat) knots. For generic rectangular representations described in detail are only the contributions of the single- and two-floor pyramids, the way to proceed is explicitly illustrated by the examples of $R=[44]$ and $R=[55]$. This solves the difficult part of the problem, but the last tedious step towards explicit formulas for generic exclusive rectangular Racah matrices still remains to be made.
Highlights
Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution — that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, generalizing the one-parametric family of twist knots
In the case of rectangular representations R = [rs] we found an evidence that the corresponding differential expansion miraculously factorizes and can be obtained from that for the twist knots
Modular transformations are simpler than conformal blocks themselves, and one can proceed with knot polynomials even when conformal-block issues remain unsolved
Summary
In [102] exclusive Racah matrices S were extracted from the double evolution family of 3-strand knots: they diagonalize the double evolution matrix. The double evolution family of double-braid two-bridge knots defines Sdirectly: in the notation of [94]. X ,Y ∈R⊗Rand S is extracted from (2.1) as a diagonalizing matrix of TST. A knowledge of rectangular HOMFLY for the double-braid family can be used to obtain these Racah matrices in rectangular representations. This provides an alternative derivation for R = [22] — and coincidence with the result of [102] can serve as a check of our factorization hypotheses about the double-braid
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