Cubes, cacti, and framed long knots

TL;DR

This paper introduces an operad action on the Taylor tower of framed long knots, linking it with known structures and advancing the understanding of knot invariants and algebraic operations in knot theory.

Contribution

It defines a new operad action on the Taylor tower of framed long knots and establishes its compatibility with existing operad actions, enhancing the algebraic framework of knot invariants.

Findings

Proves space-level compatibility of operad actions on knot spaces.

Reproduces the nontriviality of a Browder bracket class.

Links operad actions with the Taylor tower and Vassiliev invariants.

Abstract

We define 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 space-level compatibility of our action with Budney's little 2-cubes action on the space of framed long knots itself. Our result improves upon previous joint work of the first author related to the conjecture that the Taylor tower for classical long knots is a universal Vassiliev invariant over the integers. As a corollary, we reprove the nontriviality of a certain Browder bracket class first detected by Sakai.