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.
Highlights
Interesting are results obtained for entire families of knots or links
In this paper we reported the results about the HOMFLY polynomials for the pretzel knots, which are a natural generalization of the torus knots from g = 1 to arbitrary genus g, for which an exhaustive answer like the Rosso-Jones formula can presumably be found
We found a well-structured exhaustively explicit answer for arbitrary g in allsymmetric representations, and the Rosso-Jones formula arises as its very special case
Summary
We begin with this simplest example, which is the simplest possible case of the Rosso-Jones formula (1.1). Since the HOMFLY polynomials in symmetric representations do not distinguish the mutant knots [58], with the help of mutation one can permute nk ↔ nk+1 This enhanced symmetry reduces the number of necessary initial conditions and, more formulas can be obtained and more are the chances to observe regularities, leading to discovery of generic expressions. Our formulas (3.22) and (3.24) provide the explicit answer for arbitrary pretzel link in arbitrary symmetric representation These formulas (3.1) are perfectly consistent with (and, partly inspired by) the arbitrary genus results of [61] for the Jones polynomials. The pretzel links and knots provide us with an ample set of examples of the HOMFLY polynomials in all (anti)symmetric representations. Our results (3.22) and (3.24) open a road for obtaining many more A-polynomials, though these expressions literally are not suitable and still have to be reshuffled: they have no form of a qhypergeometric polynomial and the existing software implementing Zeilberger’s algorithm for the hypergeometric sums [74, 75] can not be immediately used
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