On the homology of the space of knots

Abstract

Consider the space of long knots in ${ℝ}^n$, $K_{n,1}$. 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 $K_{3,1}$ 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 $K_{3,1}$.

This leads to some natural questions about the homotopy type of the space of long knots in ${ℝ}^n$ for $n>3$, as well as consequences for the space of smooth embeddings of $S^1$ in $S^3$ and embeddings of $S^1$ in ${ℝ}^3$.