A one-dimensional embedding complex

We give the first explicit computations of rational homotopy groups of spaces of “long knots” in Euclidean spaces. We define a spectral sequence which converges to these rational homotopy groups whose E1 term is defined in terms of familiar Lie algebras. For odd k we establish a vanishing line for this spectral sequence, show the Euler characteristic of the rows of this E1 term is zero, and make calculations of E2 in a finite range.