Embedding spaces of split links

TL;DR

This work determines the motion group of a split link by expressing the fundamental group $\pi_1(\mathcal{E}(L))$ as a triple semidirect product involving the Fouxe–Rabinovitch automorphisms, the piecewise motion groups $\pi_1(\mathcal{E}(L_i))$, and the permutation group $P_L$ of isotopic pieces. The authors introduce a semi-simplicial space of separating systems, Sep_•, to linearize the study of $\pi_1(\mathcal{E}(L))$ and relate it to the homotopy types of the individual piece spaces $\mathcal{E}(L_i)$, as well as to the configuration spaces of points in the complements. They extend the results to embeddings in $S^3$ via Inner automorphism quotients and prove a contractibility result for the separating-systems fibre, enabling a concrete description of the motion group and its finite presentability under mild hypotheses. The approach unifies and extends previous knot- and link-space analyses (Brendle–Hatcher, Wattenberg, Goldsmith) into a cohesive framework that also informs potential higher homotopy and homology studies, with applications to 4-manifold diffeomorphisms. Overall, the paper provides a computable, structurally transparent description of the motion groups of split links and a robust toolkit for studying their higher homotopy structure.

Abstract

We study the homotopy type of the space $E(L)$ of unparametrised embeddings of a split link $L=L_1\sqcup \ldots \sqcup L_n$ in $\mathbb{R}^3$. Our main result is a simple description of the fundamental group, or motion group, of $E(L)$, and we extend this to a description of the motion group of embeddings in $S^3$. The main tool we build is a semi-simplicial space of separating systems, which we show is homotopy equivalent to $E(L)$. This combinatorial object provides a gateway to studying the homotopy type of $E(L)$ via the homotopy type of the spaces $E(L_i)$.

Figures (5)

• Figure 1: Separating systems for a link $\rho \in \mathop{\mathrm{\mathcal{E}}}\nolimits(L)$ with three pieces. Only the bottom right separating system is not essential, because the unbounded component is homeomorphic to $\mathop{\mathrm{Int}}\nolimits(S^2\times I)$.
• Figure 2: An essential separating system $\Sigma$ for $\rho=\rho_1\sqcup \rho_2\sqcup \rho_3 \in \mathop{\mathrm{\mathcal{PE}}}\nolimits(L)$ and its dual rooted $L$-tree $T_\Sigma$.
• Figure 3: Three moves on $L$-trees.
• Figure 4: Replacing $(F,f)$ with $(F_1,f_1)$ when $k=1$. The image of $F$ is shown in tan, and the image of $F_1$ in tan and blue. The link of $\Sigma_p$ in $S^1$ is $\Lambda_1$ and $\Lambda_2$. The homotopy from $f$ to $f_1$ replaces the star of $\Sigma_p$ in the boundary of $\mathop{\mathrm{im}}\nolimits(F)$ with the star of $\Sigma_p'$ in the boundary of $\mathop{\mathrm{im}}\nolimits(F_1)$. It linearly shifts the weight from $\Sigma_p$ to $\Sigma_p'$, which pushes the arc $\Lambda_1-\Sigma_p-\Lambda_2$ across the two rightmost blue triangles to $\Lambda_1-\Sigma_p'-\Lambda_2$.
• Figure 5: A schematic picture of surgery on $S_q$ with respect to $D$. $S_p,~S_q$ are represented by the black circles on the left and right respectively, and $D$ is the arc in red on $S_p$. The $\delta$-neighbourhood of $S_q$ is light red while the $\epsilon$-neighbourhood of $S_p$ is shown in light blue. The surgered sphere $S_q^+$ is the union of the green and orange arc in the interior of $S_q$, while $S_q^-$ is the union of the orange and green arc in the interior of $S_p$.

Theorems & Definitions (78)

• Theorem A
• Corollary B
• Theorem C
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Example 2.4
• Definition 2.5
• Remark 2.6
• Remark 2.7