I have a solid torus $T$ and a curve (or knot) $C$ that winds 2-times around the torus (parallel to the longitude). Can I say that $[C] = 2 \in \mathbb{Z} \cong \pi_1(T)$, that is the curve represents 2 in the fundamental group? Can I take the fundamental group without a basepoint, since I want for $C$ do not necessarily go through the basepoint? In a paper I've heard the expression a "conjugacy class of $\pi_1(T)$", if C' represents an element in this conjugacy class, how is it different than $C$?


migrated from mathoverflow.net Mar 13 at 9:00

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  • $\begingroup$ The expression in the paper was probably referring to the conjugacy class of an element in $\pi_1 T$. Regarding the difference between the conjugacy classes and just elements in the fundamental group there are two issues: how change-of-basepoints affects fundamental group, and also the Hurewicz theorem about the map $\pi_1 X \to H_1 X$. $\endgroup$ – Ryan Budney Mar 13 at 1:55
  • $\begingroup$ Depending on the choice of isomorphism $\pi_1(T)\cong \mathbb{Z}$ your curve could also represent $-2$. $\endgroup$ – Greg Friedman Mar 13 at 5:02

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