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This post is under construction. Let \(G\) be a group acting on \(\RR^3\) through isometries. This note serves to note that the previous post concerning linking is naturally \(G\)-equivariant: the (non-equivariant) linking graph, which is weaker than the \(G\)-equivariant linking graph, detects the \(G\)-equivariant unweave.

See the bottom of the first post in the series on chain mail mathematics for a table of contents.

The linking graph is equivariant and graded

Our construction of the linking graph was manifestly equivariant; that is, we supplied a function \(\cW \rightarrow \cbr{\mathrm{Graphs \; in \; } \RR^3}\), which happens to be equivariant for the evident \(\GG\)-action on the target space.

By Elmendorf’s theorem, this data is equivalent to a natural transformation

$$L^G:\cW^G \rightarrow \cbr{G\mathrm{-equivariant\;graphs\;in\;}\RR^3}.$$

To provide a grading to this graph is equivalent to upgrading this to a natural transformation over \(\pi_0 \Set^{BG}\). In fact, this is easy; the map \(|-|:\cbr{G\mathrm{-equivariant\;graphs\;in\;}\RR^3} \rightarrow \pi_0 \Set^{BG}\) is given by taking the underlying set of vertices and associated \(G\)-action, which is evidently natural and makes the required diagram commute.

(Non-equivariant) linking detects the \(G\)-unweave

Theorem 1. Let \(X\) be the space of \(G\)-weaves of component set \(S\) such that \(L(w)\) is trivial, i.e. no pairs of components are linked. Then, \(X\) is connected.

The first step is already done: let \(\iota: X_\varepsilon \hookrightarrow X\) denote the subspace of weave realizations such that the radius \(\rho(w)\) is at most \(\varepsilon d(c,c')\) for all components \(c,c'\). Then, the non-equivariant proof constructed paths from weave relizations in \(X\) to those in \(X_\varepsilon\);these paths preserve centers and normals, hence they are \(G\)-equivariant. Together, these form a deformation retract of \(X\) onto \(X_\varepsilon\), so \(\iota\) is a homotopy equivalence, and we can just prove that \(X_\varepsilon\) is connected.

The proof that \(X_\varepsilon\) is connected is not too hard: first note that, by separately orienting the rings within their respective \(\varepsilon \operatorname{min}_{c,c'} d(c,c')\) balls, we have the following:

Lemma 2. There is an equivalence \(X_\varepsilon \simeq \prn{\RRP^2}^n \times \Map(S,\RR^3)^G\).

Having proved this, the first factor is obviously connected, so we just have to prove the same for the second. This is the key geometric input: for an orbit \(G/H \simeq \langle s \rangle \subset S\), write \(R := S \backslash \prn{\langle s \rangle}\), so that

$$\begin{align*} \Map(S,\RR^3)^G &\simeq \Map(\langle s \rangle, \RR^3 - R)^G\\ &\simeq \prn{\RR^3 - R}_H\\ &\simeq \RR^3_H - R_H. \end{align*}$$

Hence it is sufficient to prove that the complement of a discrete set of points in the 3-manifold \(\RR^3_H\) is (path-)connected. This is true for any manifold of dimension 2 or higher; indeed, any path in \(\RR^3_H\) can be isotoped to one which doesn’t intersect \(R_H\), by a routine general position argument. □