On the homology of the space of knots

Abstract

Consider the space of long knots in ℝn, Kn,1. This is the space of knots as studied by V Vassiliev. Based on previous work [Budney: Topology 46 (2007) 1–27], [Cohen, Lada and May: Springer Lecture Notes 533 (1976)] it follows that the rational homology of K3,1 is free Gerstenhaber–Poisson algebra. A partial description of a basis is given here. In addition, the mod–p homology of this space is a free, restricted Gerstenhaber–Poisson algebra. Recursive application of this theorem allows us to deduce that there is p–torsion of all orders in the integral homology of K3,1. This leads to some natural questions about the homotopy type of the space of long knots in ℝn for n>3, as well as consequences for the space of smooth embeddings of S1 in S3 and embeddings of S1 in ℝ3.