Archive | 2019
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. Spectacular success [1, 2] of the lasting program [3]-[9] to calculate colored knot polynomials [10] for antiparallel double braids (double twist knots) and Racah matrices [11] in all rectangular representations R from the evolution properties [12]-[16] of their differential expansions [13, 14, 17, 18], opens a way to attack the main problem of arborescent calculus [19,20]: understanding of non-rectangular representations. The main difference from rectangular case is that multiplicities occur in the product of representations, and this makes the notion of Racah matrices ambiguous. In the language of [20] this is described as the new gauge invariance and one of the problems is to define gauge-invariant arborescent vertices. However, before that there is a problem to calculate the Racah matrices S̄ and S, which enter the definition of ”fingers” and ”propagators”, connected by these vertices. These problems, are not fully unrelated, because S̄ and S in non-rectangular case are not gauge invariant – still one can ask what they are in a particular gauge. As suggested in [4], the key to evaluation of S̄ is differential expansion (DE) for twist knots [14] – which, once known, straightforwardly produces S̄ for rectangular R, because of spectacular (and still unexplained!) factorization property of the DE coefficients for double braids. S are then easily extractable as a diagonalizing matrix for S̄ – it is enough to solve a system of linear equations. However, for non-rectangular R the situation is worse: differential expansion for double braids includes not S̄ itself, but some gauge-invariant combination of its matrix elements, and also the linear system for S is degenerate and again provides only the information about gauge-invariant quantities. The problem therefore is to extract at this stages exactly the combinations, needed for arborescent calculus – and we do not yet know what they are. In other words, for non-rectangular R we face a whole complex of related problems, which is partly surveyed in [20], [7] and, especially, [8]. Whatever will be the resolution, the first step is going to be the differential expansion for twist knots – and it is still not fully known for non-rectangular R. It is the goal of the present paper to suggest a mixture of the results of [7] and [1, 2] to advance in this direction. We avoid repeating the whole story and refer to [2] for the latest summary and references. The crucial facts are the observation of [4] for the antiparallel double braid in Fig.1: H R = ∑ X⊂R⊗R̄ Z X R · F (m,n) X = ∑ X⊂R⊗R̄ Z X R · F (m) X F (n) X F (m) X = ∑ Y /X fXY · ΛY (1) and the second observation of [1, 2], that for rectangular R: