The same PD code seems to yield two different knot diagrams of the Hopf link

Question

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?

Answer 1

Yes, Hopf links have an orientation ambiguity when using the increasing-index orientation convention for unoriented PD codes. A sure way to avoid this ambiguity is to switch to oriented PD codes, where you could write

PD[Xm[2,3,1,4], Xm[4,1,3,2]]

to unambiguously mean the first diagram in your linked image. There are details in the KnotTheory package's documentation: http://katlas.org/wiki/Planar_Diagrams

Another option is to have degree-2 nodes to disambiguate the orientation. For example,

[(3,4,1,5), (5,1,6,2), (2,3), (6,4)]

Now there are at least three indices in each component.

This ambiguity also arises when a link is an obvious connect sum with a Hopf link. It's a known problem that Scott Morrison, one of the authors of KnotTheory, once warned me about.

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For illustration, here are the four ways of interpreting orientations for the PD code [(2,3,1,4), (4,1,3,2)]: