Question

1) Classic Knotting problem: Classify embeddings of circle into 3D Euclidean space up to isotopy. http://en.wikipedia.org/wiki/Knot_theory

2) General topological knotting problem: Classify embeddings of one topological space into another up to isotopy. http://www.map.him.uni-bonn.de/index.php/High_codimension_embeddings:_classification

3) General knotting problem: Classify embeddings of some kind of one structure into another up to whatever equivalence relation is considered interesting.

Question: What examples have been studied of classifying embeddings of the third kind ?

For example has there been much work done on classifying embeddings of one metric space into another up to isometry ?

Edit: To try to make this more specific and more like a knotting problem - it shouldn't be any old equivalence relation but one that arises from morphisms of the larger space to itself and finding that in doing so, some embeddings can be transformed into each other and others cannot.

The more like a knotting problem a situation is the better, for example if there are an infinite number of equivalence classes, some measure of complexity with the simplest class being termed the unknot.

Answer 1

I am not sure why this is labeled "category-theory", but embeddings of finite metric spaces in Euclidean spaces have been studied quite intensively. One classic (but hard to obtain) reference is "Embeddings and Extensions in Analysis" by Wells and Williams. From a topological standpoint, these spaces have been studied by Kapovich and Millson, and by Allen Knudson and Jean-Claude Haussmann. If you go back to your question (3), the study of configuration spaces of collections of points or disks have been studied quite intensively (our own Matt Kahle is a major contributor). In the algebraic geometric category, studying, e.g, rational curves in varieties is very much an example of (3). So, there is an enormous variety of results of this ilk.