Fundamental group of the complement of $S^1 \cup Z \subset \mathbb{R^3}$

Question

I want to calculate the fundamental group of the complement of this space:

This is the complement of $S^1 \cup Z \subset \mathbb{R^3}$ where $S^1$ is the unit circle in the $xy$-plane and $Z$ is the vertical axis.

---

My idea is to homotopy retract the space onto a more familiar space which we know the fundamental group.

I am having trouble 'visualizing' what the space could be and would like some help

---

Edit - I think it may be the torus, since the two generating loops have a similar structure. I could use help showing this explicitly

Answer 1

Recall that $\mathbb{R}^3$ is homeomorphic to $D^3$, by contracting it radially. Under this map, $Z$ goes to a vertical line through the ball, and $S^1$ stays right where it is (maybe smaller, depending on your parameterization).

Then jam your thumbs through the $z$-axis, and you'll get a solid torus minus an equatorial circle inside it. Then you can thin down the walls (or inflate the circular gap, depending on how you want to think of it) to get a torus. I sketched something out in MS Paint to illustrate this: