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adding details on the intended construction
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Scott McKuen
Scott McKuen
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Update: Model Theory Background Let me say a bit more about the model theory background to address Ryan's comment: take an at-most countably-infinite collection of finitely-generated structures (such as all finite graphs, all f.g. torsion-free Abelian groups, or all finite sets that are linearly ordered) up to the appropriate idea of isomorphism for that structure. Given three properties on that collection - "hereditarity", "joint embedding", and "amalgamation" - Fraïssé showed that there is a unique (up to isomorphism) countable structure (the Fraïssé limit) that

  1. admits every member of the collection as a substructure and has no other non-isomorphic finitely-generated-substructures (the technical term is the original collection is the age of the Fraïssé limit.)

  2. in which any isomorphism between two of its f.g. substructures extends to an automorphism of the entire structure

Hereditarity: the original collection is closed under taking finitely-generated substructures of its members.

Joint Embedding: for any $A, B$ in the collection, there is a common $C$ in the collection into which they both embed.

The property that avoids Ryan's trivial limit is the third:

Amalgamation: given $A_1, A_2$ in the collection, with a common substructure $B$ and embeddings $f_1:B \rightarrow A_1$ and $f_2:B \rightarrow A_2$, there is a $C$ in the collection and embeddings $g_1:A_1 \rightarrow C$, $g_2:A_2 \rightarrow C$ where $g_1 \circ f_1 = g_2 \circ f_2$.

So, for example, if $A_1$ is the Hopf link, and $A_2$ is the unknot linked in some way with a right trefoil, putting them side-by-side would be a joint embedding, but an amalgamation would have to treat one of the unknots in $A_1$ as the same unknot in $A_2$. So Joel's comment is right about what I want - this process builds a structure that has every finite link relating to every other finite link in all possible ways.

"The amalgamation property" is also the model-theorist's response to "why is the limit of finite linear orders $\mathbb{Q}$ rather than $\mathbb{Z}$?"

In the context of model theory, the definition of embedding and substructure depends critically on the choice of logical language for the structure, but the construction of a chain of embeddings using amalgamation leading to the final limit object does not depend on the underlying language. My thought was to drop the logical part (for now), and proceed analogously replacing "finitely generated substructure" with "sublink with finitely many components".

Update: Model Theory Background Let me say a bit more about the model theory background to address Ryan's comment: take an at-most countably-infinite collection of finitely-generated structures (such as all finite graphs, all f.g. torsion-free Abelian groups, or all finite sets that are linearly ordered) up to the appropriate idea of isomorphism for that structure. Given three properties on that collection - "hereditarity", "joint embedding", and "amalgamation" - Fraïssé showed that there is a unique (up to isomorphism) countable structure (the Fraïssé limit) that

  1. admits every member of the collection as a substructure and has no other non-isomorphic finitely-generated-substructures (the technical term is the original collection is the age of the Fraïssé limit.)

  2. in which any isomorphism between two of its f.g. substructures extends to an automorphism of the entire structure

Hereditarity: the original collection is closed under taking finitely-generated substructures of its members.

Joint Embedding: for any $A, B$ in the collection, there is a common $C$ in the collection into which they both embed.

The property that avoids Ryan's trivial limit is the third:

Amalgamation: given $A_1, A_2$ in the collection, with a common substructure $B$ and embeddings $f_1:B \rightarrow A_1$ and $f_2:B \rightarrow A_2$, there is a $C$ in the collection and embeddings $g_1:A_1 \rightarrow C$, $g_2:A_2 \rightarrow C$ where $g_1 \circ f_1 = g_2 \circ f_2$.

So, for example, if $A_1$ is the Hopf link, and $A_2$ is the unknot linked in some way with a right trefoil, putting them side-by-side would be a joint embedding, but an amalgamation would have to treat one of the unknots in $A_1$ as the same unknot in $A_2$. So Joel's comment is right about what I want - this process builds a structure that has every finite link relating to every other finite link in all possible ways.

"The amalgamation property" is also the model-theorist's response to "why is the limit of finite linear orders $\mathbb{Q}$ rather than $\mathbb{Z}$?"

In the context of model theory, the definition of embedding and substructure depends critically on the choice of logical language for the structure, but the construction of a chain of embeddings using amalgamation leading to the final limit object does not depend on the underlying language. My thought was to drop the logical part (for now), and proceed analogously replacing "finitely generated substructure" with "sublink with finitely many components".

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Scott McKuen
Scott McKuen
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What is known about links with a countably-infinite number of tame components?

I want to be clear about my phrasing, but I'm not a topologist, so when I say "knot" or "link" I mean the equivalence class under ambient isotopy of an embedding of the circle into $\mathbb{R}^3$.

For a while I have been looking for references to what I've taken to calling the "Fraïssé link" - this thing would have (an instance of) every tame link occurring as a subset, and you would build it the same way that the model-theoretic Fraïssé construction builds the rational numbers from amalgamations of finite linear orders, or the random graph from finite graphs. However, when applying Fraïssé's construction, just ignore the model-theoretic question of what language you're using, or what a structure on a knot is. Just use "is a sublink" to replace the idea of a (logical) embedding, isotoping components where you need to to make the amalgamation property work.

I believe the chain-construction part of Fraïssé's work still succeeds as a plain-old union of sets this way. I also think the construction should commute with passing from a knot-instance to a knot, so realizations of this big snarly thing in $\mathbb{R}^3$ should all be ambient-isotopic to each other, too, but I'm not certain about that. If so, this would be an example of the kind of object I'm asking about in my question: links with a non-finite number of components, but the lowest-level components are ordinary knots.

This does not look like the usual sense of "wild knot" that people talk about, but if I'm wrong please let me know. Searches for "universal homogeneous link" or things like that have not led me to anything that looks similar to this object. Same with casual discussions with some model theorists. It's natural enough that I suspect I'm just missing the right terminology.

Ultimately, I'd like to find the right language to do some model theory with this object - in particular, a language that can express the Reidemeister moves as axioms, that works with the Fraïssé construction. But my specific request is whether knot theorists have noted any results about this particular object, or about any other links that have a countably-infinite number of tame components?