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This post is under construction. See the previous post in this series for context, or the bottom of the first in the series for a table of contents.

Back to primitive notions of toroidal composition

Let \(\cW^+_*\) be the graded space of labelled weaves with a distinguished component. Then, the beginning perspective of toroidal composition was that of kinging; there is a kinging map

$$\cW^+_* \rightarrow \cW_{*+1}^{\Sym}.$$

The first genuine extension of this idea came from the idea that we can Möbius, or in fact replace the given component with any toroidal weave, yielding a map

$$\cW^+ \otimes \cW^\vee \rightarrow \cW^{\Sym}.$$

This was made nicer by noting that we can toroidally compose on every component at the same time, throwing away the distinguished component, yielding a map

$$\cW^{\Sym} \circ \cW^\vee \rightarrow \cW^{\Sym}.$$

In the last post, we exhibited this after \(\pi_0\) as the action map of a right \(W^\vee\)-module, for the analogous operad structure on \(W^\vee\). The hard part of this was the toroidal decomposition result, which vaguely resembled a primary decomposition for this structure.

In this post, we declare the beginning of a similar sort of operation, this time concerning orbits:

Orbits

In chainmaille, we frequently refer to orbits, where a ring is forced into a position where it bounds a disk punctured by components which it is not linked to. The smallest instance of this is the orbital unit, where there is an “orbiting ring” which is linked to no other ring, yet the weave is seen to be primary (in fact, atoroidal), as the orbiting ring can’t “escape” the linking ;indeed, it would need to slip another ring through its center, but they have the same radius.

Until recently, in common practice, only orbits of this form were known. This work is inspired by (and in some early sense caused) the discovery of a new weave with orbits, called Moon; it is so named because it is the smallest example where a ring orbits a ring interaction which is itself an orbit. This weave was missing from the RIM until September 2021.

In fact, this weave came about from a thought process similar to toroidal composition: let \(F_\varepsilon\) be a \(\varepsilon\)-thickening of an embedded \(S^1 \vee S^1\) made of coplanar unit circles. Let \(R_\varepsilon\) be the intersection of \(F_\varepsilon\) with a unit ball centered at the wedge point. Lastly, let \(\cW^{\fig}\) be the space of weaves \(w\) together with a distinguished point, around which the associated unit ball has intersection with \(w\) contained in \(R_\varepsilon\).

We make a few brave leaps: first, by “orbiting” the figure 8, we have an operation

$$\cW^{\fig}_* \xrightarrow{\text{orb}} \cW_{*+1}.$$

In fact, there is a natural extension

$$\cW^{\fig} \otimes \cW^\vee \xrightarrow{\text{orb}} \cW,$$

which orbits with a given toroidal weave. This is not so useful of an operation, as evidenced by the following claim.

Claim 1. By distinguishing the new orbit, \(\text{orb}\) may be enhanced to a map

$$\cW^{\fig}_* \xrightarrow{\text{orb}} \cW_{*+1}^+.$$

Then, the extended orbit map factors through the unextended orbit map followed by toroidal composition:

$$\cW^{\fig} \otimes \cW^\vee \xrightarrow{\text{orb} \otimes \operatorname{id}} \cW^+ \otimes \cW^\vee \rightarrow \cW.$$

Second, if we define \(\cW^{\fig n}\) to be the space of weaves with \(n\) distinguished places for orbiting, we in fact get an operation

$$\cW^{\fig n}_* \xrightarrow{\text{orb}} \cW_{*+n}.$$

with a simil.ar factorization to the claim.

Last, we must say something with content; in the case of \(M_2\) as a figure 8 weave, we can actually upgrade the image of orbiting to itself have a figure 8 subweave:

Question A. Can this always be done for weaves which themselves fit into a figure 8?

The answer to this is probably yes.

Question B. Can this always be done for figure 8 subweavs?

This is almost certainly not true; it requires movement of all of the rings in the fig 8 subweave, which seems unlikely to work. An even more ambitious question follows:

Question C. Is there a decomposition result for orbitals; that is, is there a complete invariant given by a combination of combinatorial data and weaves without orbits?