Comment by matheist
10 hours ago
To be precise, the mean curvature and metric are the same but the immersions are different (they're not related by an isometry of the ambient space).
Topologically they're the same (the example found was different immersions of a torus).
Is it the case that 'they' are simply two ways of immersing the same two tori in R^3 such that the complements in R^3 of the two identical tori are topologically different?
If so, isn't this just a new flavor of higher-dimensional knot theory?