Colored Kauffman Homology and Super-A-polynomials

Abstract

We study the structural properties of colored Kauffman homologies of knots. Quadruple-gradings play an essential role in revealing the differential structure of colored Kauffman homology. Using the differential structure, the Kauffman homologies carrying the symmetric tensor products of the vector representation for the trefoil and the figure-eight are determined. In addition, making use of relations from representation theory, we also obtain the HOMFLY homologies colored by rectangular Young tableaux with two rows for these knots. Furthermore, the notion of super-A-polynomials is extended in order to encompass two-parameter deformations of PSL(2,C) character varieties.

47+21 pages, 17 figures, 5 tables; Ancillary file attached on the right; v2,v3,v4 minor corrections and references added

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Cited by in corpus (25)

• Colored HOMFLY polynomials of knots presented as double fat diagrams
• Colored HOMFLY polynomials for the pretzel knots and links
• Universal Racah matrices and adjoint knot polynomials. I. Arborescent knots
• On universal knot polynomials
• On colored HOMFLY polynomials for twist knots
• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• Sequencing BPS Spectra
• A slow review of the AGT correspondence
• Racah matrices and hidden integrability in evolution of knots
• Differential expansion and rectangular HOMFLY for the figure eight knot
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Factorization of differential expansion for antiparallel double-braid knots
• Rectangular superpolynomials for the figure-eight knot
• Perspectives of differential expansion
• Checks of integrality properties in topological strings
• On Factorization of Generalized Macdonald Polynomials
• On possible existence of HOMFLY polynomials for virtual knots
• Quantum Racah matrices up to level 3 and multicolored link invariants
• Evolution method and HOMFLY polynomials for virtual knots
• Khovanov polynomials for satellites and asymptotic adjoint polynomials
• Towards formalization of the soliton counting technique for the Khovanov-Rozansky invariants in the deformed $\mathcal{R}$-matrix approach
• Topological entanglement and hyperbolic volume
• On knots, complements, and 6j-symbols
• On ambiguity in knot polynomials for virtual knots
• On an alternative stratification of knots