On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant

Abstract

We compute in many classes of examples the first potentially interesting homotopy group of the space of embeddings of either an arc or a circle into a manifold M M of dimension d ≥ 4 d\geq 4 . In particular, if M M is a simply connected 4-manifold the fundamental group of both of these embedding spaces is isomorphic to the second homology group of M M , answering a question posed by Arone and Szymik. The case d = 3 d=3 gives isotopy invariants of knots in a 3-manifold that are universal of Vassiliev type ≤ 1 \leq 1 , and reduce to Schneiderman’s concordance invariant.