We show that the space of long knots in an euclidean space of dimension
larger than three is a double loop space, proving a conjecture by Sinha. We
construct also a double loop space structure on framed long knots, and show
that the map forgetting the framing is not a double loop map in odd dimension.
However there is always such a map in the reverse direction expressing the
double loop space of framed long knots as a semidirect product. A similar
compatible decomposition holds for the homotopy fiber of the inclusion of long
knots into immersions. We show also via string topology that the space of
closed knots in a sphere, suitably desuspended, admits an action of the little
2-discs operad in the category of spectra. A fundamental tool is the
McClure-Smith cosimplicial machinery, that produces double loop spaces out of
topological operads with multiplication.Comment: 16 page
