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The PD code [(2, 3, 1, 4), (4, 1, 3, 2)] seems to map to a non-unique knot diagram. I can describe the following two Hopf links with different orientations with this same PD code. As I understand it, while a link diagram does not have a unique PD code, a given PD code should map to just one knot diagram. Am I missing something?

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On the 2-sphere these two diagrams are equivalent. Any region can be chosen to contain the point at infinity, and this choice does not affect the PD code.

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