Cubes, cacti, and framed long knots
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.
Keywords
Cite
@article{arxiv.2301.08858,
title = {Cubes, cacti, and framed long knots},
author = {Robin Koytcheff and Yongheng Zhang},
journal= {arXiv preprint arXiv:2301.08858},
year = {2025}
}
Comments
Main changes from v2 are in Section 7: added some background material; corrected, clarified, and expanded the proof of compatibility of the bracket on knots and the bracket on the spectral sequence; and added a new figure. Now 46 pages, 10 figures. Accepted for publication in AGT