Factorization of differential expansion for non-rectangular representations

Abstract

Factorization of the differential expansion coefficients for HOMFLY-PT polynomials of double braids, discovered in arXiv:1606.06015 in the case of rectangular representations $R$, is extended to the first non-rectangular representations $R=[2,1]$ and $R=[3,1]$. This increases chances that such factorization will take place for generic $R$, thus fixing the shape of the DE. We illustrate the power of the method by conjecturing the DE-induced expression for double-braid polynomials for all $R=[r,1]$. In variance with rectangular case, the knowledge for double braids is not fully sufficient to deduce the exclusive Racah matrix $\bar{S}$ -- the entries in the sectors with non-trivial multiplicities sum up and remain unseparated. Still a considerable piece of the matrix is extracted directly and its other elements can be found by solving the unitarity constraints.

14 pages

References in corpus (4)

• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations
• On colored HOMFLY polynomials for twist knots
• Factorization of differential expansion for antiparallel double-braid knots
• On rectangular HOMFLY for twist knots

Cited by in corpus (11)

• Nimble evolution for pretzel Khovanov polynomials
• Tangle blocks in the theory of link invariants
• Eigenvalue hypothesis for multi-strand braids
• Distinguishing Mutant Knots
• Extension of KNTZ trick to non-rectangular representations
• On exclusive Racah matrices $\bar{S}$ for rectangular representations
• Planar decomposition of the HOMFLY polynomial for bipartite knots and links
• KNTZ trick from arborescent calculus and the structure of differential expansion
• Towards tangle calculus for Khovanov polynomials
• Pentad and triangular structures behind the Racah matrices
• Defect and degree of the Alexander polynomial