Knotted families from graspers

TL;DR

This work extends clasper/grope ideas to high‑dimensional embedding spaces by introducing grasper surgery, which yields explicit, decorated‑Lie‑tree indexed families of embeddings in Emb_ atural(C,M) for C ∈ {\mathbb{D}^1,\mathbb{S}^1} and d≥4. The authors construct multi‑families of embedded arcs inside a model d‑ball and couple them to graspers decorating by π_1(M), producing homotopy classes that map isomorphically onto decorated Lie trees under the relative Taylor tower evaluation, i.e. ev^{rel}_{n+1}∘\mathfrak{r}_n = Id on \mathsf{Lie}_{π_1M}(n). They connect these geometric constructions to Samelson/Whitehead products, Haefliger’s Borromean trefoil, and gropes/claspers, and show these classes realize the lower vanishing line of the Goodwillie–Weiss spectral sequence, with the n=2 case yielding the classical generator of π_{2(d-3)}(Emb_ atural(\mathbb{D}^1,\mathbb{D}^d)) ≅ \mathbb{Z}. The results illuminate the structure of embedding spaces in dimensions d≥4, unify geometric and algebraic formalisms, and provide a concrete, computable bridge between decorated trees and high‑dimensional knotted families, with further implications for embedding calculus and potential extensions to torsion phenomena.

Abstract

For any smooth manifold $M$ of dimension $d\geq4$ we construct explicit classes in homotopy groups of spaces of embeddings of either an arc or a circle into $M$, in every degree that is a multiple of $d-3$, and show that they are detected in the Taylor tower of Goodwillie and Weiss. The classes are obtained from families of string links constructed in the $d$-ball.

Figures (17)

• Figure 1: Two isotopic knots, obtained by degree $2$ clasper surgery on the unknot $\mathsf{u}$.
• Figure 2: A grasper in a $d$-manifold $M$ (here $d=3$) relative to $\mathsf{u}$ (the horizontal line) is a $d$-ball that "grasps" $\mathsf{u}$ in fixed subintervals $J_i$. The family $\mu_{\Gamma,d}$ has a particular linking pattern with the arcs $a_i$, that under $\mathbb{G}_n$ get identified with $\mathsf{u}(J_i)$.
• Figure 3: The $\mathsf{STU}^2$ relation is obtained by equating to zero the four terms on the right. They are in turn obtained from the graph on the left by resolving two vertices $v_0$ and $v_3$ as depicted.
• Figure 4: A degree $1$ grasper surgery on $\mathsf{u}$, giving $\mathfrak{r}_1(\Gamma^g)$.
• Figure 5: The arcs whose embedded commutators form a degree $2$ grasper surgery on $\mathsf{u}$, for $d=4$. The meridian 2-spheres $m_j(\mathbb{S}^{d-2})$ are drawn schematically, but should in fact not be contained in the depicted 3-dimensional slice (they are normal spheres to $\mathsf{u}$ and do not intersect it).

Theorems & Definitions (20)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 5.1
• proof : Proof of Theorem \ref{['thm:multi-family']}
• Proposition 6.1
• proof
• Proposition 6.2
• Theorem 6.3