Cubes, cacti, and framed long knots

Abstract

ROBIN KOYTCHEFF AND YONGHENG ZHANG A. We de ne an action of the operad of projective spineless cacti on each stage of the Taylor tower for the space of framed 1-dimensional long knots in any Euclidean space. By mapping a subspace of the overlapping intervals operad to the subspace of normalized cacti, we prove a compatibility of our action with Budney's little 2-cubes action on the space of framed long knots itself. This compatibility is at the level of the multiplication and the Browder and Dyer-Lashof operations in homology. Our result improves upon previous joint work of the rst author related to the conjecture that the Taylor tower for classical long knots is a universal Vassiliev invariant over the integers.