HOMFLY for twist knots and exclusive Racah matrices in representation [333]

TL;DR

This paper advances the calculation of HOMFLY polynomials for twist knots by extending Racah matrices extraction to more complex representations, revealing new phenomena in the coefficients that could lead to broader generalizations.

Contribution

It introduces a novel approach to derive Racah matrices for triple-column and triple-hook representations, expanding the understanding of HOMFLY polynomials for twist knots.

Findings

Deviation of coefficient $f_{[332]}^{[21]}$ from skew dimension observed

Extension of Racah matrices extraction to more complex representations

Potential for further generalizations in knot polynomial calculations

Abstract

Next step is reported in the program of Racah matrices extraction from the differential expansion of HOMFLY polynomials for twist knots: from the double-column rectangular representations R=[rr] to a triple-column and triple-hook R=[333]. The main new phenomenon is the deviation of the particular coefficient $f_{[332]}^{[21]}$ from the corresponding skew dimension, what opens a way to further generalizations.