I'm trying to compute the deformation retract of the complement in $\mathbb{R}^3$ of two disjoint circles that are in a common plane.
I've found the following site that explains with detail the case of one circle (the argument is the same as Hatcher's book). The problem I'm suppossed to solve is left as trivial in all the books/sites I've found but I don't know how to start with it. I know that the result is $$(\operatorname{S}^2\vee\operatorname{S}^1)\vee(\operatorname{S}^2\vee\operatorname{S}^1).$$
I't will be sufficient for me a qualitative description of the deformation retract since I want to obtain some intuition on the stuff (It's the first time I study algebraic topology and the goals of my course is to see the ideas of the signature with naturality).
If you understand the case with one circle, it follows the exact same process. Deformation retract to a prism surrounding the two circles. Imagine a plane that splits this prism into two pieces, one containing each circle. As in the first visualization, if we restrict to one of the cubes we have a cube minus a circle. We can deformation retract this, fixing the shared wall, to a cube with a line going from one edge to the opposite. Since the deformation retraction fixes the shared wall, you can do this for both cubes at once. Now contract the shared wall to a point and we have a wedge of two spheres with lines on the inside of each. Pick a path on each sphere between the points of contact of this line and contract it, then we arrive at $(S^2 \vee S^1) \vee (S^2 \vee S^1)$.