On a Poincar\'e polynomial from Khovanov homology and Vassiliev invariants

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.