Factorization of differential expansion for antiparallel double-braid knots

TL;DR

This paper introduces a new approach to compute Racah matrices for colored knot polynomials using double-evolution of antiparallel double-braids, simplifying calculations for rectangular representations.

Contribution

It demonstrates that differential expansion factorizes for antiparallel double-braids, reducing Racah matrix computation to a few twist knots, and provides explicit results for certain rectangular representations.

Findings

Differential expansion factorizes for antiparallel double-braids.

Explicit Racah matrices are obtained for specific rectangular representations.

The approach simplifies the calculation of colored knot polynomials for arborescent knots.

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.