Data Abstract A few extra words
Nevertheless, we give two different extensions of the Khovanov homology for tangles in the virtual case. The main problem
is that a straightforward extension is not possible in a nice way, since an open saddle can close of as three different
surfaces, i.e. a multiplication, a comultiplication or even a Möbius cobordism.
Note that, if we assume that the characteristic
of the underlying ring is not two, the latter one is zero. The problem that arises is, in order to maintain that
the complex is a well-defined chain complex, one has to define some open saddles to be zero. But, after “tensoring” local pieces
together, the corresponding saddles could be non-zero anymore. Hence, to solve this problem, we introduce another piece of data which we call
indicator. This indicator is a number that is either or .
In order to use the local construction in a global picture, we introduce a specific way of calculation that we call dot-calculus. See in the picture below.
As an application of the whole progress we show that Lee's variant of a virtual knot is trivial which is the first input one needs
if one want to define a virtual Rasmussen invariant.
A main tool to prove that the virtual Khovanov homology of a virtual knot is trivial, we prove that the two generators of the homology are precisely given by
so-called non-alternating resolutions (as an example see the upper right resolution in the picture below) and that a virtual knot has exactly two of these resolutions. This is a completely combinatorial
observation that tells us something about the combinatorial nature of a virtual knot.