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Definition 1. Two links are equivalent if they are ambient isotopic in $\mathbb{R^3}$.

Definition 2. Two links are equivalent if they are ambient isotopic in ${S^3}$.

Are these two definitions (essentially) the same?

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These definitions are the exact same, because generically an isotopy sweeps a (locally) $2$-dimensional locus, and in a three-dimensional space it can always be made to avoid a particular point. All you have to do is call that point $\infty$.

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