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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.

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closed as not a real question by Andreas Blass, Ryan Budney, Bill Johnson, Matthew Kahle, Andy Putman Jan 4 '12 at 20:03

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

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This question strikes me as much too broad and open-ended for MO. –  Andreas Blass Jan 4 '12 at 18:10
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I agree with Andreas, you're asking for a survey is a huge chunk of near all of mathematics. For example, does pattern recognition fall into (3)? –  Ryan Budney Jan 4 '12 at 18:31
    
So people do something like this by computing the homotopy type of enriched mapping spaces in the relevant model category. But it is certainly done in more generality. Also, your equivalence relation is implicitly isotopy in an appropriate category. Have you looked up what Embedding theory is? –  Sean Tilson Jan 5 '12 at 0:32
    
By the way, classical knot theory studies knots up to ambient isotopy, not up to isotopy. –  Douglas Zare Jan 5 '12 at 2:28
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1 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.

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