  We introduce a Poincar\'{e} polynomial with two-variable $t$ and $x$ for
knots, derived from Khovanov homology, where the specialization $(t, x)$ $=$
$(1, -1)$ is a Vassiliev invariant of order $n$. Since for every $n$, there
exist non-trivial knots with the same value of the Vassiliev invariant of order
$n$ as that of the unknot, there has been no explicit formulation of a
perturbative knot invariant which is a coefficient of $y^n$ by the replacement
$q=e^y$ for the quantum parameter $q$ of a quantum knot invariant, and which
distinguishes the above knots together with the unknot. The first formulation
is our polynomial.
