New York Journal of Mathematics
Volume 28 (2022), 367-401


Scott Baldridge, Louis H. Kauffman, and Ben McCarty

Unoriented Khovanov Homology

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Published: February 4, 2022.
Keywords: unoriented, Khovanov homology, virtual link, knot, Jones polynomial, Lee homology, parity, core, mantle, multicore decomposition.
Subject: 57K10, 57K12, 57K14, 57K18.

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.


Kauffman's work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (under contract number 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation). All three authors would like to thank William Rushworth for many helpful conversations and suggestions.

Author information

Scott Baldridge:
Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803, USA


Louis H. Kauffman:
Department of Mathematics, Statistics and Computer Science
University of Illinois at Chicago
Chicago, Illinois 60607-7045, USA, and
Department of Mechanics and Mathematics
Novosibirsk State University
Novosibirsk, Russia


Ben McCarty:
Department of Mathematical Sciences
University of Memphis
Memphis, TN 38152, USA