I’ll explain how the absolute Galois group of the rationals acts on 
a space which is closely related to the space of all knots. The path components
of this space form a finitely generated abelian group which is, conjecturally, a
universal receptacle for (integral) finite-type knot invariants. The added Galois 
symmetry allows us to extract new information about its homotopy and 
homology. This is joint work with Geoffroy Horel.