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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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And to answer the question that wasn't asked, knotted and unknotted circles do have the same fundamental group. – MJD Oct 6 '12 at 22:49
The practice of referring to the "fundamental group of the knot" rather than the fundamental group of its complement is conventional sloppiness. – Brad Oct 6 '12 at 23:04
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 $S^3$. – Tabes Bridges Oct 13 '12 at 20:49

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