A long knot is a smooth embedding \({\mathbb{R}}^1\hookrightarrow {\mathbb{R}}^d\) which coincides with the standard embedding in the complement of \([0,1]\). The authors study the Goodwillie-Weiss tower of the space of long knots \(\mathrm{emb}_{\mathrm c}({\mathbb{R}}^1, {\mathbb{R}}^d)\). There is a spectral sequence associated to the Goodwillie-Weiss tower, where the \(E^1\)-page consists of the homotopy groups of the layers of the tower. This spectral sequence is well understood rationally, by [G. Arone et al., Math. Res. Lett. 15, No. 1, 1–14 (2008; Zbl 1148.57033)]. In this paper, the authors consider the spectral sequence associated to the Goodwillie-Weiss tower, at a prime \(p\). They show that the differential \[ d^r_{-s,t}\colon E^r_{-s,t}\to E^r_{-s-r, t+r-1} \] vanishes \(p\)-locally, if \(r-1\) is not a multiple of \((p-1)(d-2)\) and if \[ t<2p-2+(s-1)(d-2). \] The proof of the result relies on the relation between the Goodwillie-Weiss tower and the little disks operad, and the existence of Galois symmetries on the little disks operad. The result yields a computation of some homotopy groups of the layers of the Goodwillie-Weiss tower, for \(d\geq 3\), where the range improves as the prime or the codimension increases.
The authors also prove a homological result: Let \(\overline{\mathrm{emb}}_{\mathrm c}({\mathbb{R}}^1, {\mathbb{R}}^d)\) denote the homotopy fiber of the map \[\mathrm{emb}_{\mathrm c}({\mathbb{R}}^1, {\mathbb{R}}^d) \hookrightarrow\mathrm{imm}_{\mathrm c}({\mathbb{R}}^1, {\mathbb{R}}^d). \] There is a homological Bousfield-Kan spectral sequence \(E^\ast(R)\) related to \(H_\ast(\overline{\mathrm {emb}}_{\mathrm c}({\mathbb{R}}^1, {\mathbb{R}}^d), R)\), where \(R\) is a ring of coefficients. The authors show that for a prime \(p\), the only possibly non-trivial differentials in \(E^\ast({\mathbb{Z}}_p)\) are \(d_{1+n(d-1)(p-1)}\) for \(n\geq 0\).