Introduction to Khovanov Homologies. I. Unreduced Jones superpolynomial

TL;DR

This paper provides an elementary introduction to the Khovanov construction of superpolynomials, focusing on unreduced Jones superpolynomials for 2-strand braids, highlighting their complexity and foundational importance.

Contribution

It offers the first detailed exposition of unreduced Jones superpolynomials from first principles, emphasizing their role in developing and testing alternative approaches.

Findings

Unreduced superpolynomials contain more information than ordinary Jones polynomials.

The method applies to 2-strand braids, exemplified by the 5_1 knot.

Khovanov's construction remains the primary source for defining superpolynomials.

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.