If the circle and any knot are homeomorphic as topological spaces, why do they have different fundamental groups?
Fundamental group of a knot
4
$\begingroup$
algebraic-topology
1 Answers
5
The knot group isn't the fundamental group of the knot, it's the fundamental group of the complement of the knot.
-
0This 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