Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial

Abstract

An elementary introduction to Khovanov construction of superpolynomials. Despite its technical complexity, this method remains the only source of a definition of superpolynomials from the first principles and therefore is important for development and testing of alternative approaches. In this first part of the review series we concentrate on the most transparent and unambiguous part of the story: the unreduced Jones superpolynomials in the fundamental representation and consider the 2-strand braids as the main example. Already for the 5_1 knot the unreduced superpolynomial contains more items than the ordinary Jones.

33 pages

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• Introduction to Khovanov Homologies. III. A new and simple tensor-algebra construction of Khovanov-Rozansky invariants
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• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
• On genus expansion of superpolynomials
• HOMFLY polynomials in representation [3,1] for 3-strand braids
• Factorization of differential expansion for antiparallel double-braid knots
• On rectangular HOMFLY for twist knots
• Nimble evolution for pretzel Khovanov polynomials
• Are Khovanov-Rozansky polynomials consistent with evolution in the space of knots?
• On possible existence of HOMFLY polynomials for virtual knots
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• Evolution method and HOMFLY polynomials for virtual knots
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• Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?
• Bipartite expansion beyond biparticity
• Algebra of quantum $\mathcal{C}$-polynomials
• Khovanov-Rozansky Homologies and Cabling
• Towards formalization of the soliton counting technique for the Khovanov-Rozansky invariants in the deformed $\mathcal{R}$-matrix approach
• On geometric bases for quantum A-polynomials of knots
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• Matrix model and dimensions at hypercube vertices
• Khovanov--Rozansky matrix factorization reduction for bipartite links
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs