Khovanov homology, its definitions and ramifications

Volume 184 / 2004

Oleg Viro Fundamenta Mathematicae 184 (2004), 317-342 MSC: 57M25, 57M27. DOI: 10.4064/fm184-0-18


Mikhail Khovanov defined, for a diagram of an oriented classical link, a collection of groups labelled by pairs of integers. These groups were constructed as the homology groups of certain chain complexes. The Euler characteristics of these complexes are the coefficients of the Jones polynomial of the link. The original construction is overloaded with algebraic details. Most of the specialists use adaptations of it stripped off the details. The goal of this paper is to overview these adaptations and show how to switch between them. We also discuss a version of Khovanov homology for framed links and suggest a new grading for it.


  • Oleg ViroDepartment of Mathematics
    Uppsala University
    Box 480
    S-751 06 Uppsala, Sweden

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