#### Volume 13, issue 1 (2009)

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On the homology of the space of knots

### Ryan Budney and Fred Cohen

Geometry & Topology 13 (2009) 99–139
##### 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 ${ℝ}^{3}$.

##### Keywords
knots, embeddings, spaces, cubes, homology
##### Mathematical Subject Classification 2000
Primary: 58D10, 57T25
Secondary: 57M25, 57Q45
##### Publication
Received: 2 July 2008
Revised: 14 September 2008
Accepted: 4 September 2008
Preview posted: 22 October 2008
Published: 1 January 2009
Proposed: John Morgan
Seconded: Ralph Cohen, Steve Ferry
##### Authors
 Ryan Budney Department of Mathematics and Statistics University of Victoria Victoria BC Canada V8W 3P4 Fred Cohen Department of Mathematics University of Rochester Rochester NY 14627 USA