I'm not sure what sense you're using "the same" in. Eg a cylinder can be transformed into a mobius strip (in 3d space) by cutting and gluing, but if we allow on

I'm not sure what sense you're using "the same" in. Eg a cylinder can be transformed into a mobius strip (in 3d space) by cutting and gluing, but if we allow only self-intersection, this can't be done. To me this shows that they're not "the same".