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Consider the two rings that this magician is holding in his hands:

enter image description here

How would you describe that configuration in topological terms?

From a knot-theory standpoint, I would say that the rings form a two-component link that is equivalent to a Hopf link. I'm no knot-theory specialist, though; they're may be a simpler way of putting it.

I'm wondering what technical term is used in topology to describe that type of relationship between such a pair of sets.

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I suspect that the picture does not properly represent the case you have in mind. The two half link will shrink down in to two points and the middle link is a circle. –  Fly by Night May 3 '13 at 18:03
    
I'll change the picture to more accurately represent what I have in mind. –  Jubobs May 3 '13 at 18:04

1 Answer 1

up vote 2 down vote accepted

"Two (solid, unknotted) tori embedded in $\Bbb R^3$ are linked iff their complement has abelian fundamental group" works. You need to mention the ambient space in some way because without it they're just two separate tori with no relationship with eachother.

New attempt:

If we have one ring $R$, then the other ring is linked with it if it not continuously deformable to a point within $\Bbb R^3\setminus R$. (In this case it's easier if the rings are thought of as circles, rather than tori.)

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Thanks. Is there anything less esoteric for a non-specialist? What if you relax the condition that the two sets are perfect tori (they could be deformed tori)? –  Jubobs May 3 '13 at 18:15
    
@Jubobs Well, that depends on what you mean by "deformed". If you just mean that they are infinitely thin (i.e. a curve) in places, then there is no differnce. If one of the tori has a knot on it, then things aren't that simple. Just for a single torus / circle, if it has a knot, then the fundamental group of the complement can be complicated. If you throw in another torus and weave them together, then I assume it can be pretty ugly. –  Arthur May 3 '13 at 18:19
    
After some research, I think what I meant was using two "2-manifolds of genus 1" instead of two tori. Still, I was hoping there would be a simple term to describe that configuration. –  Jubobs May 3 '13 at 18:23
    
It seems to me that the OP is asking for a precise definition of linkedness and this answer is giving a characterization of linkedness using algebraic invariants. –  Pete L. Clark May 3 '13 at 18:24
    
@Jubobs After Pete's input, I edited my answer. See if you like that one better. –  Arthur May 3 '13 at 18:30

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