This is an example I had in mind when writing the question! But Ryan pointed out (elsewhere) that, to find diffeomorphisms between such objects, one would not follow the Morse theory proof, but would rather use a direct construction following from a choice of triangulation (handle decomposition), at least in dimension 3. So I started to doubt it... maybe there actually is a (combinatorial) procedure to find explicit diffeomorphisms between two given examples...

This is a great answer. Are there bad examples which actually occur in algebra, analysis, geometry, or topology, or are they artificial constructions?

I will promote the tag "quantum topology" as being superior in this context to "quantum algebra", for reasons which you mentioned. Vassiliev invariants are not quantum algebra (although for knots, quantum invariants and Vassiliev invariants basically coincide), but they are quantum topology!

This is nice! Can you edit-in what the upshot is? That the torsion classes in $H_1 K$ represent global points in l which are not in C, among other things...

Put another way, we expect topology to be governed by algebra in dimension $n\geq 5$, but not in dimensions 3 and 4, because of the Whitney embedding theorem.

This is an interesting question!

Can you give a reference for that? Integrally, I expect some sort of Minkowski units would be needed, but I don't know whether even that is enough.

What are you asking?

Kevin Lin- I would argue that nontriviality of the trefoil is deeper, because nontriviality of the Hopf link can be detected by the Gauss linking integral, which was known earlier and is less deep than anything needed to prove nontriviality of the trefoil.