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If the circle and any knot are homeomorphic as topological spaces, why do they have different fundamental groups?

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The knot group isn't the fundamental group of the knot, it's the fundamental group of the complement of the knot.

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    This seems a good place to note that, for such purposes, a knot $K$ should really be thought of in terms of a pair $(X,K)$, where $X$ is typically $\mathbb{R}^3$ or $S^3$.2012-10-13