Colored HOMFLY polynomials for the pretzel knots and links

Abstract

With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g + 1 two strand braids, parallel or antiparallel, and depend on g + 1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of g + 1 elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled SU q (N ) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many mutants, undistinguishable by symmetric HOMFLY polynomials, hence, the extension of our results to non-symmetric representations R is a challenging open problem. To this end, a non-trivial generalization of the suggested formula can be conjectured for entire family with arbitrary g and R.

Colored HOMFLY polynomials of knots presented as double fat diagrams

“…• The second is the conjectured expression [42][43][44][45][46][47] for knots in S 3 obtained from gluing three-manifolds with one or more four-punctured S 2 boundaries. We call such diagrams as double fat tree diagrams (see definitions below) .…”

“…There is lot of evidence for the eq. (1.2) from the conformal block method [42][43][44][45][46][47] and its advanced versions like the evolution [70], cabling [40] and differential expansion [39] methods for symmetric representations. In this paper, we will do similar verification for R = [21].…”

“…Expression (1.6) for each propagator (1.5) in (1.2) is usually interpreted as the ordinary time evolution in Chern-Simons theory, which is provided by monodromies of conformal block made from braiding matrices T ± , T 0 which are of three types (depending on the pair of adjacent strands that it acts on) and Racah matrices S, see [10] for the original idea and [43][44][45][46][47][50][51][52][53][54][55][56][57] for recent applications. In the case of arbitrary representation R, the form of the S matrix and the eq.…”

Checks of integrality properties in topological strings

“…Exclusive is, of course, much simpler than inclusive; however, involvement of the conjugate representation (inverted strand direction), is a considerable complication. The matrices S andS are known for all symmetric (and antisymmetric) representations R [26] and [152] and, by an outstanding effort, for R = [2, 1] [153].…”

Factorization of differential expansion for antiparallel double-braid knots

“…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.…”