I'm having trouble solving problem 12 from Section 1.2 in Hatcher's "Algebraic Topology".
Here's the relevant image for the problem:
I'm trying to find $\pi_1(R^3-Z)$, where $Z$ is the graph shown in the first figure. The answer (according to the problem statement) is supposed to be $\langle a,b,c| aba^{-1}b^{-1}cb^\epsilon c^{-1}=1\rangle$, where $\epsilon=\pm 1$.
I attempted to use Van Kampen's theorem, using a cover of two open sets, depicted in the lower image. The first open set is the area above the bottom horizontal line, minus the graph, and the second open set is the region below the top horizontal line, minus the graph. The intersection is the area in between the two horizontal lines minus the graph.
Call the top set $A$, and the bottom set $B$. With applications of Van Kampen, I got that $\pi_1(A)\cong \pi_1(B)\cong Z*Z*Z$, and that $\pi_1(A \cap B)\cong Z*Z*Z*Z*Z$. After using Van Kampen's theorem again, I got that $\pi_1(R^3-Z)\cong Z*Z$, which is wrong.