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See the previous post in this series for context. See the bottom of the first post in the series for a table of contents.

In this post, we construct a combinatorial invariant of unit weaves and prove that it detects the unweave. The major mathematical content of this post is a citation to the paper Strange Actions of Groups on Spheres by Freedman & Skora; in particular, they claim the essence of the main proposition of this post by synthetic geometry in Lemma 3.2.

This note concenrs both the finite and infinite case; for the rest of the post, \(1 \leq n \leq \infty\).

The linking graph

We first warm up with some unit weave constructions; recall that weaves have underlying links, and pairs of components in links have linking numbers.

Construction. Let \(\tilde w \in \cW_n\) be a unit weave realization, and let \(\pi:\cW_n \rightarrow \RR^{3n} = \operatorname{Map}(n,\RR^3)\) be projection to the centers of circles. The linking grpah of \(\tilde w\) is the embedded graph \(L(\tilde w) \subset \RR^3\) whose vertices embed as \(\pi(\tilde w)\), with edges drawn linearly between vertices if the corresponding circles have nonzero linking number.

The fact that this is invariant is obvious:

Proposition 1. Let \(\tilde w, \tilde u\) be unit weave realizations; there is a continuous map from equivalences \(\tilde w \sim \tilde u\) to ambient isotopies \(L(\tilde w) \sim L(\tilde u)\); in particular, the underlying graph is a weave invariant. Hence, for \(w \in W_n\), we define \(L(w)\) to be the underlying graph of \(L(\tilde w)\) for any realization \(\tilde w\) of \(w\).

Note that \(L\) is naturally functorial under isometries; that is, \(O(3)\) acts on \(\cW_n\), and this is naturally mapped to an action of the stabilizer \(G\) of \(\tilde w\) on \(L(\tilde w)\) as an embedded graph. This further maps to an action of \(G\) on \(L(w)\) as a graph. In particular, when \(w\) is linear, \(L(w)\) has a natural \(\ZZ\) action.

We will explore this in the next post.

The ring dilation trick for units

Construction. The \(n\)th part of the space of scaled chainmail weaves* is defined by*

$$\begin{align*} \cW'_n :&= \cbr{\prn{(r,\cbr{x_i,P_i}_{i \in [n]}} \mid \not \exists i\neq j \text{ with } y \in P_i \cap P_j \text{ s.t. } d(x_i,y) = d(x_j,y) = r)}\\ & \subset \RR \times \prn{\RR^3 \times \RRP^2}^n_{\Sigma_n}; \end{align*}$$

explicitly, this is the space of \(n\) nonintersecting circles of radius \(r\) in \(\RR^3\).

The following proposition is obvious.

Proposition 2. The “divide by \(r\)” map \(\cW'_n \rightarrow \cW_n\) is a trivial \(\RR\)-fiber bundle; in particular, it is a homotopy equivalence, so it induces a bijection on \(\pi_0\).

We’ll freely use \(\cW'_n\) when useful from now on, which will allow us to avoid the hassle of explicit “ring resizing” tricks.

For the following argument, let \(\rho:\cW'_n \rightarrow \RR\) be the projection to the \(\RR\) coordinate, i.e. the radius, and for \(c,c' \in \abs{w}\), let \(d(c,c')\) denote the distance between their centers. The following lemma is clear, but useful.

Lemma 3. Let \(w \in \cW'_n\) be a unit weave such that for all pairs \(c,c' \in \abs{w}\), we have \(d(c,c') > \rho(w)/2\). Then, \(w\) is the unweave.

Linking detects the unit unweave

Proposition 4. Suppose \(w\) is a unit weave of (possibly infinite) size \(n\) such that \(L(w)\) is trivial, i.e. no pairs of components are linked. Then, \(w\) is the unweave.

To prove this, we use Freedman & Skora Lemma 3.2; under the embedding \(\RR^3 \subset S^3 \subset \delta B^4\), there exist unique hemispheres bounded by each component of \(\tilde w\) intersecting \(S^3\) perpendicularly. Doubling about \(S^3\), these turn into standard metric 2-spheres; by an explicit euclidean geometry argument, any pair of such spheres must have intersection either empty, a circle which intersects \(S^3\), or a single point contained in \(S^3\). In particular, since the intersection of such circles with \(S^3\) is \(w\), any pair’s intersection doesn’t intersect \(S^3\), and hence they are pairwise non-intersecting.

Passing through the standard “thickened stereographic projection” map \(B^4 - \cbr{0} \cup \cbr{(1,0,0) \times I} \simeq \RR^3 \times \RR\), this yields a collection of pairwise nonintersecting hemispheres in \(\RR^3 \times I\) who intersection with \(\RR^3 \times \cbr{0}\) is \(w\); this implies that they have the same radius \(r\), so their sintersection with \(\RR^3 \times \cbr{t}\) is in \(\cW'_n\) for \(t < r\), yielding a path \(h_t\) in \(\cW'_n\) beginning at \(w\).

In fact, \(h_t\) preserves centers, and as \(\varepsilon \rightarrow 0\) we have \(\rho(h_{t - \varepsilon}) \rightarrow 0\). By Lemma 3, this is unwoven for \(\varepsilon\) small, so \(h_0 = w\) is unwoven as well, and we are done. □


Because it will be useful later, we explicitly record the full strength of this argument: let \(X_\varepsilon \subset \cW'_n\) be the subspace of realizations such that \(d(c,c') > \varepsilon \rho(w)\) for all \(c,c'\).

Lemma 5. The paths \(h_t\) together form a deformation retract of \(\cW'_n\) onto \(X_\varepsilon\) which preserves the centers and normals of each component.

From this perspective, Lemma 3 can be viewed as saying that \(X_\varepsilon\) is connected, which together with Lemma 5, immediatley proves the proposition.

How we’ll prove the linear case

In the next post, I’ll introduce weaves which are preserved by arbitrary isometric group actions, recovering linear weaves along the way. The key insight will be that preservation under isometric group actions depends only on component centers and normals; hence Lemma 5 continues to apply. In another post, we’ll prove an equivariant version of Lemma 3, hence proving that linking detects the equivariant unweave. This will recover linear weaves as a special case.