Let's consider the space of long knots in $\mathbb R^n$, $n>3$. I know that there are many results (Vassiliev, Turchin, Sinha, Kontsevich) about different expressions of cohomology of this space. I think the last result is about convergency in $E^1$ term (Lambrechts, Turchin, and Volić - The rational homology of spaces of long knots in codimension $> 2$).

But my question is about what these cohomologies are precise. So, is it true that $H^1(\text{long knots in $\mathbb R^4$})=0$?

Does there exist some table with $H^i(\text{space of long knots in $\mathbb R^j$})$ at least for small $i$, $j$?