Extension of KNTZ trick to non-rectangular representations

Abstract: We claim that the recently discovered universal-matrix precursor for the $F$ functions, which define the differential expansion of colored polynomials for twist and double braid knots, can be extended from rectangular to non-rectangular representations. This case is far more interesting, because it involves multiplicities and associated mysterious gauge invariance of arborescent calculus. In this paper we make the very first step -- reformulate in this form the previously known formulas for the simplest non-rectangular representations [r,1] and demonstrate their drastic simplification after this reformulation.