On the homology of the space of knots

Details On the homology of the space of knots On the homology of the space of knots

2005-04-10

arxiv.org/abs/math/0504206v3

Geometry & Topology 13 (2009) 99--139

Mathematics

Geometric Topology

36 pages, 6 figures. v3: small revisions before publication

Scientific paper

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 of the authors, 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.

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