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This post is under construction. This post is a reformulation of the previous post in the series, intended for a general audience. See the bottom of the first post in the series for a table of contents.

A “hello world” to a non-mathematical audience

Hello world! This blog is usually written for my own benefit, as an algebraic topologist; hence the rest is almost certainly not accessible to non-mathematicians. Eventually, once I’ve worked out enough technology to rigorously verify the RIM, I will make a document summarizing this for a general audience. We aren’t there yet, but I want to be able to refer to this, so I’m putting it here.

The name of the game is precision; after writing down a precise definition for weaves, one can prove rigorous/unarguable statements, such as providing a list of all weaves satisfting a condition (such as all unit weaves of size at most 4, which is probably more-or-less the RIM). In pursuit of this, the post today will state some (almost) precise definitions, and ask some precise questions.

A back-reference to unit weaves

This section will be particularly imprecise and terse. A helpful intuitive read is the first section here. The gist is that we’d like to define unit weaves, i.e. weaves who do not replicate in any directions, in such a way that we have a hope of counting them.

A helpful simplification (which will be made universally, until we’ve solved the simple case) is to set all inner diameters to 1, and all wire diameters to infinitesimal (so that AR is approximately \(\infty\)). We can then define a unit weave: a unit weave realization is a collection of geometrically embedded circles in 3-dimensional space, and an equivalence of two realizations is a physical manipulation of one to the other, i.e. an interval of time’s worth of such realizations which moves continually from the first realization to the second.

This equivalence partions the set of unit weaves into a number of parts; one of these parts is called a unit weave. In essence, Möbius 2 is a unit weave corresponding with any of the realizations of two circles liked to each other.

A weave has n components if it is composed of n circles; a central conjecture in this theory is that there are finitely many weaves with n circles.

However, unit weaves are not really what chain mail is about; we’d at least like to make a chain! Weaves replicating in a line are harder to define, but it’s still doable.

A first definition of linear weaves

In this section, we’ll define the notion of weaves which are woven into a chain, called linear weaves. The central observation will be that such a weave is essentially the same information as an infinitely long chain, which repeats after a finite number of rings.

To use this definition, we need a notion of infinite weaves:

Definition 1. An infinite weave realization is a collection of infinitely many non-intersecting circles in 3-space; two such realizations are equivalent if one can be manipulated into the other, at all times remaining an infinite weave realization.

An infinite weave is a part in the associated partition of the set of infinite weave realizations.

These are obviously huge: there will certainly be infinitely many of them!

Luckily, we’ll impose periodicity.

Recall that a translational symmetry of 3-space is gotten by shifting all points by a fixed distance. We say that such a symmetry permutes an infinite weave realization if it takes each ring perfectly onto another ring, and every ring is hit by this procedure.

Definition 2. An \(n\)-periodic liner weave realization is an infinite weave realization such that there exists a translational symmetry permuting the rings, partitioning them into precisely \(n\) categories. Two such realizations are equivalent if there is a manipulation making them equivalent as infinite weave realizations, \(n\)-periodic at each time.

An \(n\)-periodic linear weave is a part in the associated partition of the set of \(n\)-periodic linear weave realizations. A linear weave *is an infinite weave which is \(n\)-periodic for some \(n\), up to periodic equivalence. That is, an \(n\)-periodic linear weave is a weave with units of size \(n\).

We can similarly define semiperiodic weaves as being infinite weaves preserved by a symmetry of 3-space which is a combination of translation and a perpendicular rotation and/or reflection; we say an \((r,n)\)-semiperiodic weave if the symmetry becomes a translation when applied \(r\) times.

Lastly, a weave is said to be simply \(n\)-periodic if units of its \(n\)-periodicity connect only to adjacent units, and similarly for semiperiodic weaves.

Finite-type gradings and periodicity

A grading on a set is an assignment of a natural number to each element of the set; the \(n\)th part is the elements of value \(n\). A grading is finite type if the \(n\)th part is finite for each \(n\).

The set of linear weaves is graded by periodicity: the \(n\)th part is the set of weaves with minimal (periodic) units of size \(n\). We ask the following conjectures:

Conjecture A. The size grading on the set of unit weaves is finite type.

Conjecture B. The simple periodicity grading on the set of linear weaves is finite type.

For an example of a finite type grading, we could define a different set; for instance, let \(W'\) be the set of unit weaves resulting from repeatedly kinging and mobiusing the one ring. One might call these generalized mobius weaves. Using toroidal composition, it is not too hard to show the following:

Proposition 3. The size grading on \(W'\) is finite type.

The same strategy shows that, if there are finitely many RIM level \(n\) weaves, then the size filtration on weaves gotten by toroidal composition of the one ring by elements of RIM level at most \(n\) is finite type.

The full strength of this argument, using toroidal decomposition, is that it suffices to prove that the size grading on atoroidal weaves is finite type.