# Extension of KNTZ trick to non-rectangular representations

## TL;DR

This paper extends the KNTZ trick to non-rectangular representations in knot theory, simplifying formulas for complex representations and addressing multiplicities and gauge invariance issues.

## Contribution

It introduces a reformulation of the universal-matrix precursor for non-rectangular representations, enabling simpler calculations in arborescent calculus.

## Key findings

- Reformulated formulas for [r,1] representations
- Demonstrated drastic simplification after reformulation
- Addressed multiplicities and gauge invariance issues

## 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.