Conformally invariant functionals on the space of knots are introduced via extrinsic conformal geometry of knots and integral geometry on the space of spheres. Our functionals are expressed in terms of a complex-valued 2-form, which can be considered as the cross-ratio of a pair of infinitesimal segments of the knot. We show that our functionals detect the unknot as the total curvature does, and that their values explode if a knot degenerates to a singular knot with double points.

AMS 2000 Mathematics subject classification: Primary 57M25; 53A30. Secondary 57R17