Rasmussen invariant and the divisibility of Lee's class
We define a link invariant $s'_c(L)$ by a combination of some classical properties of its diagram $D$, and the $c$-divisibility of a specific class (generalized Lee's class) in the Khovanov-type link homology $H_c(D; R)$ determined by $D$ and an element $c$ in the base ring $R$. This invariant has many properties common to the Rasmussen's $s$-invariant: $s'_c$ is a link-concordance invariant, provides a lower bound for the slice genus, and the equality is sharp for positive knots. As a corollary an alternative proof for the Milnor conjecture follows. In particular if the base ring is the polynomial ring $\mathbb{Q}[h]$ and $c = h$, then $s'_h$ coincides with the $s$-invariant for knots.