On genus expansion of superpolynomials

Abstract

Recently it was shown that the (Ooguri-Vafa) generating function of HOMFLY polynomials is the Hurwitz partition function, i.e. that the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and expansion through Vassiliev invariants explicitly demonstrate this phenomenon. In the present letter we claim that the superpolynomials are not functions of such a type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are β-deformed to Hamiltonians of the Calogero-Moser-Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev and genus expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials. However, even for the thin knots the beta-deformation is non-innocent: already in the simplest examples it seems inconsistent with the postivity of colored superpolynomials in non-(anti)symmetric representations, which also happens in I.Cherednik's (DAHA-based) approach to the torus knots.

17 pages

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• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• A slow review of the AGT correspondence
• Towards matrix model representation of HOMFLY polynomials
• On the defect and stability of differential expansion
• Racah matrices and hidden integrability in evolution of knots
• Cut-and-join structure and integrability for spin Hurwitz numbers
• Differential expansion and rectangular HOMFLY for the figure eight knot
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Rectangular superpolynomials for the figure-eight knot
• Distinguishing Mutant Knots
• Checks of integrality properties in topological strings
• Knot polynomials for twist satellites
• On possible existence of HOMFLY polynomials for virtual knots
• Quantum Racah matrices and 3-strand braids in representation [3,3]
• Colored Alexander polynomials and KP hierarchy
• Gaussian distribution of LMOV numbers
• Algebra of quantum $\mathcal{C}$-polynomials
• Matrix model and dimensions at hypercube vertices