I have been told that there's a theorem about closed connected curves in 3D:
Theorem. Any closed connected curve in 3D is equivalent to one of the following:
And the two curves above are not equivalent.
Questions:
What does "equivalent" mean here? Does it mean diffeomorphism?
Are there references for the theorem?
I think something has been lost in between the actual mathematics and what you've been told.
Perhaps the equivalence relation is isotopy of simple closed curves, in which case what you were told is wrong, the trefoil knot (depicted in a fattened version in the first link) being a counterexample. The subject of knot theory is, in essence, the study of isotopy classes of simple closed curves in 3D.
Perhaps the equivalence relation is isotopy of unknotted simple closed curves, in which case what you were told is also wrong, because those two things you've depicted are equivalent to each other, and in fact there is only one isotopy class of unknotted simple closed curves.
I can make a few other guesses, but each of the guesses I can think of leads to a different version of the same conclusion that what you were told is wrong: either the two things you've depicted are equivalent to each other; or there are many, many, many additional things which are inequivalent to both of the ones depicted.
As you can see, I'm shooting in the dark here, so your best recourse is to track down your source and see what the real intention was.