It is my understanding that the knot group of two "unlinked" links is a free group on two generators. So, in theory, first image below should yield such a group if interpreted with the tools of the second image.
First image:
Second image:
The problem is, I seem to end up with the presentation $\langle x,y,z| \ xzx^{-1}y^{-1}\rangle$ which I do not believe is a presentation for $F_2$, though I could be wrong about that. I'm very new to this content and I've only recently worked through the Hatcher exercise showing how the presentation arises.
Thanks
The relation $xzx^{-1}y^{-1}=1$ means $z=x^{-1}yx$. This means any word involving $z$ can be rewritten in terms of only $x$'s and $y$'s, so the group is generated by $x$ and $y$. The relation can be rewritten with respect to these generators as $x(x^{-1}yx)x^{-1}y^{-1}$, but that is just $1$, so the group is just the free group generated by $x$ and $y$.