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Suppose $f$ and $g$ are two embeddings of $S^1$ in $R^3$ or in $S^3$. How do I show whether they form a link or not?

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    Well... it is a difficult problem! :) You can try to evaluate some linking invariants, which may tell you there two circles are linked, like the linking number (wikipedia will tell you about it) and others, but in general pretty much everything can fail.2011-03-28
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    Do you have any further information about f and g?2011-03-28
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    @Mariano: there are computable complete linking invariants. See Zare's response.2011-03-30

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