Question on linear maps defined in Khovanov homology There are two linear maps $m:V \otimes V \rightarrow V$ and $\Delta:V \rightarrow V\otimes V$ in the definition of the differential of Khovanov homology. So my question is why do they map elements as below?
$$m:v_+ \otimes v_- \rightarrow v_-$$
$$m:v_+ \otimes v_+ \rightarrow v_+$$
$$m:v_- \otimes v_+ \rightarrow v_-$$
$$m:v_- \otimes v_- \rightarrow 0$$
and
$$\Delta:v_+ \rightarrow v_+ \otimes v_-+v_-\otimes v_+$$
$$\Delta:v_- \rightarrow v_- \otimes v_-$$
Thank you!
 A: First note that there are other definitions you could choose.  Eun Soo Lee uses an altered differential (introduced in section 4 of this paper), and the resulting theory was used to great effect by Jake Rasmussen in his paper on slice genus.
For discussion of Khovanov's differential, I recommend Bar-Natan's explanation in his colorful introduction to Khovanov homology.  Paraphrasing section 3.2, the differential ought to be of degree 0 and be invariant under any reordering of the cycles.  This requires that the multiplication and comultiplication maps be of degree $-1$ and be commutative and co-commutative, respectively.  Working in $\mathbb{Z}/2\mathbb{Z}$, this determines $m$ and $\Delta$.
A: The maps are chosen so that the resulting $KC(L)$ (which is freely generated by the Khovanov generators) is a chain complex, i.e. $d^2=0$. In $\mathbb{Z}/2$-coefficients (or other coefficients if we have already introduced the signs in the differential), this equation can be translated into the combinatorial statement: 
"Given Khovanov generators $x\ge_2 y$ (i.e. $y$ can be reached from $x$ after performing 2 moves, such as fusing two circles or splitting a circle), let $v,w$ be the corresponding binary vectors to $x,y$, respectively, and let $u_1, u_2$ be the other 2 vectors in the 2-face containing $v$ and $w$. Consider the set of Khovanov generators $\{ z | x\ge_1 z\ge_1 y\}$. Then the number of such generators corresponding to $u_1$ equals the number of such generators corresponding to $u_2$."
Which happens to be satisfied by Khovanov's differential. Notice that, as the above answer points out, there are other differentials which also lead to valid link homology theories, such as Bar-Natan's one in his Khovanov's homology for tangles and cobordisms paper.
