FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2005, VOLUME 11, NUMBER 4, PAGES 127-152

**The Khovanov complex for virtual links**

V. O. Manturov

Abstract

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One of the most outstanding achievements of the modern knot theory is
Khovanov's categorification of Jones polynomials.
In the present paper, we construct the homology theory for virtual
knots.
An important obstruction to this theory (unlike the case of classical
knots) is the nonorientability of "atoms"; an atom is
a two-dimensional combinatorial object closely related with
virtual link diagrams.
The problem is solved directly for the field $$**Z**_{2}, and
also by using some geometrical constructions applied to atoms.
We discuss a generalization proposed by Khovanov; he modifies the
initial homology theory by using the Frobenius extension.
We construct analogues of these theories for virtual knots, both
algebraically and geometrically (by using atoms).

Location: http://mech.math.msu.su/~fpm/eng/k05/k054/k05411h.htm

Last modified: November 25, 2005