Geometry & Topology 13 (2009) 99–139

DOI: 10.2140/gt.2009.13.99

Abstract

Consider the space of long knots in Rⁿ, 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 Rⁿ for n > 3, as well as consequences for the space of smooth embeddings of S¹ in S³ and embeddings of S¹ in R³.

Keywords

knots, embeddings, spaces, cubes, homology

Mathematical Subject Classification

Primary: 57T25, 58D10

Secondary: 57M25, 57Q45

References

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