Colored knot polynomials: HOMFLY in representation [2, 1]

Abstract

This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R = [2, 1]. The project involves several steps:(i) parametrization of big families of knots a la [1], (ii) evaluating Racah/mixing matrices for various numbers of strands in various representations a la [2], (iii) tabulating and collecting the results at [3]. In this paper we discuss only representation R = [2, 1] and construct all necessary ingredients that allow one to evaluate knot/links represented by three strand closed parallel braids with inserted double-fat fingers. In particular, it is used to evaluate knots from a 7-parametric family: this family contains over 80% of knots with up to 10 intersections, but does not include mutants.

Checks of integrality properties in topological strings

“…The progress is a cumulative effect of merging of the different research directions: i) reformulation of the RT formalism in the spaces of intertwining operators [73][74][75][76]- [80] with developments of the highest weight technique [31,33] and the eigenvalue conjecture [1,78] to evaluate the inclusive and exclusive Racah matrices [33]- [35], ii) representing the knot polynomials for all arborescent knots through the exclusive Racah matrices S andS [29], iii) developing the family technique [30,32,144,145] in order to adequately classify knots, at least, for calculational purposes, iv) applying Vogel's universality [154] to handle the adjoint representations and their descendants; important here is that deviations from the universality at the group theory level are not seen in knot polynomial calculus [1,141].…”

“…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.…”

On universal knot polynomials

“…We stress again that this answer should be used with a certain care, because the ambiguity issue is not fully resolved. The real resolution would come from lifting to the Vogel universal level of the full Rosso-Jones formula and, more generally, of the modern version of the Reshetikhin-Turaev formalism [56]- [78]. 2 Finally, some properties of universal polynomials are discussed in the last section.…”

“…It also has far going generalizations to arbitrary knot polynomials in braid realizations [65][66][67][68][69][70][71][72][73][74][75][76][77][78]. This formula treats differently the number m of strands in the braid and its length (evolution parameter) n: the m ←→ n symmetry of the answer, P…”

On rectangular HOMFLY for twist knots

“…The features of knot polynomials are still a set of mysteries, ranging from a hierarchical set of integrality properties to various RG-like evolutions in different parameters, especially in the space of representations R, while the standard methods of non-perturbative analysis, like Ward-identities, AMM/EO topological recursion, integrability techniques etc are not yet fully applicable. Development of the theory is still going through consideration of examples: particular knots K and particular representations R, for which a powerful technique is now developed [4]- [9]. At present stage these examples start being unified into the simplest families, either of knots or of representations.…”