Introduction to Khovanov Homologies. II. Reduced Jones superpolynomials

Abstract

A second part of detailed elementary introduction into Khovanov homologies. This part is devoted to reduced Jones superpolynomials. The story is still about a hypercube of resolutions of a link diagram. Each resolution is a collection of non-intersecting cycles, and one associates a 2-dimensional vector space with each cycle. Reduced superpolynomial arises when for all cycles, containing a "marked" edge of the link diagram, the vector space is reduced to 1-dimensional. The rest remains the same. Edges of the hypercube are associated with cut-and-join operators, acting on the cycles. Superpartners of these operators can be combined into differentials of a complex, and superpolynomial is the Poincare polynomial of this complex. HOMFLY polynomials are practically the same in reduced and unreduced case, but superpolynomials are essentially different, already in the simplest examples of trefoil and figure-eight knot.

18 pages

References in corpus (2)

• The Refined Topological Vertex
• HOMFLY and superpolynomials for figure eight knot in all symmetric and antisymmetric representations

Cited by in corpus (16)

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• 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
• Cabling procedure for the colored HOMFLY polynomials
• Nimble evolution for pretzel Khovanov polynomials
• Checks of integrality properties in topological strings
• Planar decomposition of the HOMFLY polynomial for bipartite knots and links
• Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?
• Planar decomposition of bipartite HOMFLY polynomials in symmetric representations
• Bipartite expansion beyond biparticity
• Operator lift of Reshetikhin-Turaev formalism to Khovanov-Rozansky TQFTs
• On ambiguity in knot polynomials for virtual knots