Introduction to Khovanov Homologies. II. Reduced Jones superpolynomials

TL;DR

This paper provides an elementary introduction to reduced Jones superpolynomials within Khovanov homologies, focusing on the hypercube of resolutions and the algebraic structures involved, highlighting differences from HOMFLY polynomials.

Contribution

It introduces the concept of reduced Jones superpolynomials, detailing their construction via hypercube resolutions and the associated algebraic operators, expanding understanding of Khovanov homologies.

Findings

Reduced superpolynomials differ significantly from unreduced ones in simple knots.

The hypercube of resolutions encodes the structure of the superpolynomial.

Superpolynomials capture richer information than HOMFLY polynomials.

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.