HOMFLY polynomials in representation [3, 1] for 3-strand braids

Abstract

This paper is a new step in the project of systematic description of colored knot polynomials started in [1]. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R ⊗3 −→ Q with all possible Q, for R = [3, 1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3, 1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family (n, −1 | 1, −1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations.

Factorization of differential expansion for antiparallel double-braid knots

“…Using this matrix we can calculate H [3,3] for some arborescent knots, which can be made without the use of the second exclusive matrix S [33] . In examples these polynomials are consistent with available Vassiliev invariants and pass other checks from the list in [103]. Building of S fromS with the help of (2.1) and thus extension to arbitrary arborescent knots by the method of [36,37,94] is also straightforward.…”

“…Modern group theory is incapable to provide the answers beyond pure symmetric and antisymmetric representations [97][98][99][100] [103] and [104] for some inclusive Racah matrices for R = [3,1] and R = [2, 2] respectively). Further progress on these lines seems to be beyond the current computer capacities.…”

Checks of integrality properties in topological strings

“…In combination with the differential expansion method [142][143][144][145][146][147][148][149][150], this provides extensions to other rectangular representations. Further progress (for other nonrectangular representations) is expected after developing the ∆-technique briefly outlined in [33]. We are presently extending the work [77] investigating the highest weight method to determine polynomials of knots obtained from four or more strands carrying symmetric representation.…”

“…In order to get the Racah matrices, the simplest way is to look just at the highest weight vectors as elements in the abstract Verma modules. This formalism is successfully developed in [31] and [33] and has already allowed us to find the inclusive Racah matrices for R = [2,2] and even R = [3,1]. In combination with the differential expansion method [142][143][144][145][146][147][148][149][150], this provides extensions to other rectangular representations.…”

“…The HOMFLY polynomials for these representations and a list of the integers for more knots from the Rolfsen 20 is an arborescent and can also be obtained from 3-strand braid. The reason for our choice is that the exclusive Racah matrices necessary for evaluating the arborescent knots [32] are yet unavailable for the representation [3,1] [35], while the inclusive Racah matrices in this representation are known [33]. Hence, the integers in representations up to the fourth level can be constructed only for the knots that have 3-strand braid representations.…”