I wish to verify the following statement (which comes from Fox, "A Quick Trip Through Knot Theory", although that is probably not important).

"$\Gamma=\pi\_1 (M)=\langle x, a \mid a^2x=xa\rangle$ so the homology of $M$ is infinite cyclic."

So, I need to find the Abelianization of the fundamental group. Using the relations I get

$$y\_1:=[x,a]=x^{-1}ax,\qquad y\_2:=[a,x]=x^{-1}a^{-1}x$$

generate $[\Gamma,\Gamma]$. Now thinking of $\Gamma/[\Gamma,\Gamma]$ as left cosets I find

$$xy\_1=ax,\qquad ay\_1=ax^{-1}ax$$
$$xy\_2=a^{-1}x,\qquad ay\_2=ax^{-1}a^{-1}x$$

but I don't see how this is infinite cyclic. Using various combinations of the relations don't seem to get me to a single coset. Is there some trick I am missing related to $[\Gamma,\Gamma]$ apparently being the conjugates of $a$? Or did I just screw up something else?