Volume 13, issue 1 (2009)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 n, 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 n for n > 3, as well as consequences for the space of smooth embeddings of S1 in S3 and embeddings of S1 in 3.

Keywords
knots, embeddings, spaces, cubes, homology
Mathematical Subject Classification 2000
Primary: 58D10, 57T25
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
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