Why can't a knot be embedded in a 2-sphere and what types of knot are allowed on a simplex?

Question

I've never studied knot theory so the following proof sorts of trip me up. Please have a look at the questions below.

Here are my questions:
1. Why are knots can't be embedded in $S^2$?
2. What is the final stage of dividing the ball into two that we'd get stuck? Initially I thought it'd be something like a layer of simplices covering the "outer" side of the tunnel, but that doesn't seem right.
3. Also, my guess is that the contradiction at the last sentence of the proof arises simply because a simplex has no edge e contained in its interior. Is this correct?

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Edit: here's the definition of constructibility that we use.

Answer 1

1. Any closed curve embedded in $S^2$ is a trivial knot. You can consider this as a corollary of the Jordan curve theorem.
2. Note that by definition any constructible complex has only finitely many simplices. Therefore any decreasing sequence of non-empty subcomplexes of maximal dimension has to reach the stage where there are only one, or two, simplices. The contradiction then comes from Point 3 below.
3. You are right, the contradiction is that a simplex has no edge in its interior.