I am reading bar nathans paper about khovanov polynomials and am having a lot of trouble constructing the factor spaces.
So suppose V is a vector space with basis ${v_{+},v_{-}}$ and of degree $\pm 1$. We can construct the chain complex of a twisted unknot and get:
$0 \rightarrow V \otimes V \rightarrow V\{1\} \rightarrow 0$
So if we want to construct the factor spaces we need to compute $H(C) = \frac{\ker(C)}{im(C)}$.
if we look at the basis elements and map them with $m$ we get: $v_+ \otimes v_- \rightarrow v_-$, $v_- \otimes v_+ \rightarrow v_-$, $v_+ \otimes v_+ \rightarrow v_+$, $v_- \otimes v_- \rightarrow 0$
So $\ker(m) = \{v_- \otimes v_-, v_+ \otimes v_- - v_- \otimes v_+,v_+ \otimes v_- - v_- \otimes v_+\}$.
Now to get the two factor spaces we get:
$H_1 = \frac {\{v_- \otimes v_-, v_+ \otimes v_- - v_- \otimes v_+,v_+ \otimes v_- - v_- \otimes v_+\}.}{\{0\}}$.
$H_0 =\frac{V}{V} = \{0\}$. The q dimension of this space is $0$.
now my question is how does $H_1$ look like in terms of $V$ and how do we construct the q-dimension?
One can the find the paper here: http://arxiv.org/abs/math/0201043
Kees