Homotopy Approximations to the Space of Knots, Feynman Diagrams, and a Conjecture of Scannell and Sinha

Scannell and Sinha considered a spectral sequence to calculate the rational homotopy groups of spaces of long knots in Rn, for n ≥ 4. At the end of the paper they conjecture that when n is odd, the terms on the antidiagonal at the E2 stage precisely give the space of primitive Feynman diagrams related to the theory of Vassiliev invariants. In this paper we prove that conjecture. This has the application that the path components of the terms of the Taylor tower for the space of long knots in R3 are in one-toone correspondence with quotients of the module of Feynman diagrams, even though the Taylor tower does not actually converge. This provides strong evidence that the stages of the Taylor tower give rise to universal Vassiliev knot invariants in each degree.