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 $> 2$](http://palmer.wellesley.edu/%7Eivolic/pdf/Papers/VassilievCollapseFinal-G%26T.pdf)).


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 $i$, $j$?



First cohomology of the space of long knots in ℝ⁴
Long knots in $\mathbb R^4$ form a simply-connected space. I pointed this out in my survey paper [A Family of Embedding Spaces](https://arxiv.org/abs/math/0605069). The primary tool used to prove it is what's called the *embedding calculus* due to Goodwillie, Klein and Weiss.


Let $\mathcal K\_{n,j}$ denote the space of long embeddings of $\mathbb R^j$ into $\mathbb R^n$. The same survey paper above shows that $\mathcal K\_{n,j}$ is $(2n-3j-4)$-connected, and the $(2n-3j-3)$-rd homotopy group is computed. It turns out to be either $\mathbb Z$ or $\mathbb Z\_2$ depending on a parity issue, assuming $2n-3j-3 \geq 0$.


When $2n-3j-3<0$ there are a few cases where some of these homotopy groups are computed, due to Haefliger and Kervaire. When $2n-3j-3>0$ many of the rational groups have been computed by Victor Turchin in [On the other side of the bialgebra of chord diagrams](https://arxiv.org/abs/math/0411436). But very few integral homotopy or homology groups have been computed as of yet.


