Unoriented Khovanov Homology

Document Type

Article

Publication Date

2022

Abstract

The Jones polynomial and Khovanov homology of a classical link are invariants that depend upon an initial choice of orientation for the link. In this paper, we give a Khovanov homology theory for unoriented virtual links. The graded Euler characteristic of this homology is proportional to a similarly-defined unoriented Jones polynomial for virtual links, which is a new invariant in the category of non-classical virtual links. The unoriented Jones polynomial continues to satisfy an important property of the usual one: for classical or even virtual links, the unoriented Jones polynomial evaluated at one is two to the power of the number of components of the link. As part of extending the main results of this paper to non-classical virtual links, a new framework for computing integral Khovanov homology based upon arc labeled diagrams is described. This framework can be efficiently and effectively implemented on a computer. We define an unoriented Lee homology theory for virtual links based upon the unoriented version of Khovanov homology.

Publication Source (Journal or Book title)

NEW YORK JOURNAL OF MATHEMATICS

First Page

367

Last Page

401

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