So I know the fact that the join of $S^1$ and $S^1$ is homeomorphic to the 3-sphere, but I’m having trouble “seeing” this. I’d prefer something that appeals to geometric intuition, but more formal abstract arguments are welcome as well! I want to invoke the unit quaternions here, but I can’t think of a particularly simple way to do it.
If this question is too “soft” I sincerely apologize.
We discussed here a little while ago the fact that $S^3$ can be described as the result of gluing two solid tori along their boundaries.
To get a geometric reason for the homeomorphism $S^1*S^1\cong S^3$ you can look at how the two constructions are related: notice that two two tori have two central $S^1$s inside them…
(Alternatively: try to see what $S^1*[0,1]$ is, notice that you can describe $S^1$ as two copies of $[0,1]$ identified along the boundaries, and see how you can use this description on one of the two factors of $S^1*S^1$.)
(Another alternative: if you construct $S^1*S^1$ by doing identifications on $S^1\times[0,1]\times S^1$, cut the latter space in two parts $S^1\times[0,1/2]\times S^1$ and $S^1\times[1/2,1]\times S^1$, look at what the identifications do on each half, and then glue back the two parts)