Embedding Tower of String Links

• Embedding tower of string links is a systematic framework that approximates the homotopy type of smooth embeddings using polynomial stages derived from Goodwillie–Weiss calculus.
• It constructs n-excisive approximations as homotopy limits over localized data, effectively capturing key invariants including Milnor’s μ-invariants.
• The framework connects algebraic models, graph complexes, and operadic structures to provide concrete computations of finite-type invariants and link complement properties.

An embedding tower of string links refers to the systematic approximation of the homotopy type and invariants of the space of smooth embeddings of multiple disjoint intervals (string links) into a fixed manifold (typically $D^3$), using the methods of Goodwillie–Weiss embedding calculus. This tower yields a sequence of “polynomial” approximations (or stages) whose increasing complexity captures progressively finer geometric, homotopical, and link-theoretic information—including all invariants of finite type, such as Milnor’s $\mu$-invariants. The embedding tower is central for understanding the algebraic and homotopical structures underlying spaces of links and their invariants, providing both finite-stage truncations and deep connections to graph complexes, operads, homotopy-theoretic integration, and configuration space models.

1. Definition of the Embedding Tower for String Links

Let $P = \bigsqcup_{i=1}^k D^1$ (the disjoint union of $k$ intervals) and consider smooth embeddings of $P \times I$ into $D^d$, with endpoints fixed at prescribed positions on $\partial D^d$. The embedding tower approximates the space $\operatorname{Emb}_\partial(P \times I, D^d)$ by a sequence of “polynomial” approximations $T_n\,\operatorname{Emb}(P \times I, D^d)$, constructed as follows:

• One defines the full subcategory $\mathrm{Disc}_{\leq n,\,\partial,\,d}$ of at most $n$ standard disks (with boundary structure) in $d$-manifolds.
• The $n$-th stage is the right Kan extension (or equivalently, a derived mapping space of presheaves):

$$T_n\,\operatorname{Emb}(M, N) = \mathrm{Map}_{\mathrm{Psh}(\mathrm{Disc}_{\leq n, \partial, d})}\left(\iota_n^* E_M, \iota_n^* E_N\right)$$

where $E_X(-) = \operatorname{Emb}(-, X)$ and $\iota_n$ is the inclusion of $\mathrm{Disc}_{\leq n,\,\partial,\,d}$ into all $d$-manifolds with boundary.

Applied to $M = k I = \bigsqcup_{i=1}^k (D^1 \times I)$ and $N = D^3$, the sequence

$$\cdots \to T_n \operatorname{Emb}(kI, D^3) \to T_{n-1} \operatorname{Emb}(kI, D^3) \to \cdots \to T_1 \operatorname{Emb}(kI, D^3)$$

is the embedding tower for $k$-component string links (Jin, 6 Nov 2025).

Each stage $T_n$ is an $n$-excisive (or degree $\leq n$) approximation, in the sense of Goodwillie calculus.

2. Construction of the Embedding Tower and Polynomial Approximations

The tower is fundamentally constructed from localized data: for open subsets $U \subset P \times I$, one considers the functor $$U \mapsto \overline{\operatorname{Emb}}_c(U, \mathbb{R}^d)$$ (embeddings fixing the ends). The $k$-th Taylor approximation $T_k$ is defined as a homotopy limit over all covers of $U$ by at most $k$ subintervals or disks: $$T_k\,\overline{\operatorname{Emb}}_c \simeq \operatorname{holim}_{V \in \mathcal{O}_k} \overline{\operatorname{Emb}}_c(V, \mathbb{R}^d)$$ where $\mathcal{O}_k$ is the poset of opens covered by $\leq k$ coordinate disks (Tsopméné et al., 2015).

In the “punctured links” model (Koytcheff, 2011), each stage $T_n$ can alternatively be realized as the homotopy limit of a diagram of spaces of string links with up to $n$ “punctures” (removed subintervals) in each component: $$T_n(L_k) = \operatorname{holim} \left\{ E_n(k) \to E_{n-1}(k) \to \cdots \to E_0(k) \right\}$$ with chain-level and function-theoretic models also available.

The tower converges (in the sense that $$\operatorname{Emb}_\partial(P \times I, D^d) \simeq \operatorname{holim}_n T_n \operatorname{Emb}(P \times I, D^d)$$) when the ambient dimension satisfies $d \geq 2 \max_i m_i + 2$ (Tsopméné et al., 2015).

3. Algebraic Models and Graph Complexes for Homology and Homotopy

Each stage of the embedding tower admits an explicit algebraic model in terms of chain complexes built from “hairy graphs.” The main results of (Tsopméné et al., 2015) show:

• Rational homology is computed by the homology of a direct sum of complexes $\Gamma_k$ of finite “hairy” graphs with $k$ labelled external (univalent) vertices, edges, and colored “hairs” for each link component.
• Generator types:
  • External vertices: correspond to points on a link component.
  • Internal vertices (valence $\geq 3$): correspond to multipoint interactions.
  • Edges (possibly with loops/multiple edges), with differential given by vertex expansions (internal) or edge contractions (connected).

Theorems: $$C_*\left(\overline{\operatorname{Emb}}_c(\bigsqcup_i \mathbb{R}^{m_i}, \mathbb{R}^d); \mathbb{Q}\right) \simeq \bigoplus_{k=0}^\infty \Gamma_k$$ and

$$(Q \otimes \pi_*)\left(\overline{\operatorname{Emb}}_c(\bigsqcup_i \mathbb{R}^{m_i}, \mathbb{R}^d)\right) \cong H_*\left(\bigoplus_k \Gamma_k^\pi\right)$$

where $\Gamma_k^\pi$ is the connected part (primitive) (Tsopméné et al., 2015).

This framework generalizes the classical computation of spaces of long knots and connects directly to the Goodwillie–Weiss tower and functor calculus (Tsopméné et al., 2015, Jin, 6 Nov 2025).

4. Functorial n-Excisive Complements and the Stallings-Type Theorem

A structural innovation is the passage to “complement data” in the $n$-excisive world. To any $T_n$-embedding, one can functorially associate an $n$-excisive complement $C_n$; this complement is uniquely determined by a universal property in presheaves (Jin, 6 Nov 2025): $$\overline{C}_n \colon \mathrm{Psh}(\mathrm{Mfld}_{\partial, d})^{op} \longrightarrow P^{\partial/}$$ with $\overline{C}_n$ compatible with $P_n$, the $n$-excisive Goodwillie approximation functor.

The Stallings-type theorem in this context asserts:

For all $\eta \in T_n \mathrm{Emb}(P \times I, D^d)$, the induced boundary map $$> D^{d-1} \setminus i(P) \xrightarrow{\simeq} C_n(\eta) >$$ is an equivalence. Thus, every $T_n$-complement of a string link embedding is $n$-excisively equivalent to the fixed model complement $D^{d-1} \setminus i(P)$ (Jin, 6 Nov 2025).

Alexander duality underlies the identification of homology of complements, and the proof utilizes covers by at most $n$ disks and their Thom-space models.

5. Detection of Milnor Invariants and the Role of the Tower

A principal geometric application of the embedding tower concerns finite-type invariants:

• The $n$-th stage $T_n \operatorname{Emb}(kI, D^3)$ detects all length-$\leq n+1$ Milnor $\mu$-invariants of $k$-component string links (Jin, 6 Nov 2025).
• The map

$$\pi_0 \operatorname{Emb}(kI, D^3) \rightarrow \pi_0 T_n \operatorname{Emb}(kI, D^3) \xrightarrow{\pi_0 \mathscr{A}_n} \operatorname{Aut}\left(F(k)/F(k)_{n+1}\right)$$

recovers the classical Artin representation, where $F(k)$ is the free group on $k$ generators and $F(k)_{n+1}$ its $(n+1)$th lower central series subgroup (Jin, 6 Nov 2025).

Using the Magnus expansion or Milnor–Fox calculus, the non-commutative coefficients in the truncated automorphisms correspond precisely to Milnor invariants of length $\leq n+1$.

For the triple linking number ($\mu_{123}$), the embedding calculus tower provides the precise stage $n \geq 2$ at which $\mu_{123}$ is detected, via a Pontrjagin–Thom (Bott–Taubes style) construction realized on $T_n(L_3)$ (Koytcheff, 2011).

6. Homotopy Theory, Factorization of Invariants, and Operadic Structures

The embedding tower for string links canonically relates to mapping spaces, configuration spaces, and operad actions:

• At the first nontrivial stage, $T_{(1,1,\dots,1)}$ of the multivariable Taylor tower, the fundamental $\check\kappa$-invariant of Koschorke for string links is fully detected as the map to path components of a mapping space into configuration space $\pi_0 \operatorname{map}_a(I^n, (C, n))$ (Cohen et al., 2015).
• The $\check\kappa$–invariant completely separates homotopy string link classes and provides a rigid factorization of Koschorke's closed-link $\kappa$-invariant (Cohen et al., 2015).
• All finite-type invariants (“Vassiliev-type”) factor through the Taylor tower, with type-$n$ invariants supported by $T_{2n}$ or $T_{(2n,2n,\dots,2n)}$ depending on the number of punctures per strand (Koytcheff, 2011).

The operad of little intervals ($\mathcal{L}_1$) acts naturally on both the space of string links and the mapping spaces within the tower, reflecting composition by stacking (Cohen et al., 2015).

7. Implications, Examples, and Further Directions

The embedding tower unifies the calculation of rational homology/homotopy for spaces of high-dimensional string links via explicit graph complexes, whose generators correspond to Milnor invariants and higher-order linking data (Tsopméné et al., 2015).

In the stable range ($d \geq 2 \max_i m_i + 2$), the entire rational theory is controlled by finite graph-complexes. In codimension $> 2$, the conjectural extension of these models (supported by partial results) would provide a full computation of rational homotopy types via graphs (Tsopméné et al., 2015).

Through the functorial Stallings-type rigidity, the embedding tower explains the behavior of nilpotent quotients and automorphism groups associated to string link complements (Jin, 6 Nov 2025).

The embedding calculus bridge to knot and link invariants via configuration space models and the Taylor tower is used to geometrize and compute all finite-type (Vassiliev) invariants—including, through explicit Pontrjagin–Thom constructions, classical invariants such as Milnor's triple linking number (Koytcheff, 2011).

In summary, the embedding tower of string links provides a functorial, algebraic, and geometric framework within which all classical and quantum invariants of link spaces are organized and related via the deep structure of the Goodwillie–Weiss calculus and homotopical algebra.