Visualizing a Deformation Retraction

It's known that there exists a deformation retraction from the space $\mathbb{R}^3$\ $S^1$ to $S^2 \wedge S^1$, and I thought I had a visualization for it, but now it seems discontinuous. Can anyone help out with describing (or even better constructing explicitly) this map?

Thanks!

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The space $\mathbb R^3\setminus S^1$ is obtained by rotating a closed half-plane with a puncture in it. The latter space can be retracted onto semi-circle plus a diameter. The rotation of semi-circle plus a diameter creates $S^2$ plus a diameter. Now it is time to lose the rotational symmetry: we contract the Eastern hemisphere into a point (sorry, guys). The result is $S^2\wedge S^1$.