(Directory and $Path initialization not included.)

Needs["KnotTheory`"]

A simple link, albeit with ambiguous orientation:

hopfGC={{1,-2},{2,-1}};

I ask for its planar diagram:

hopfPD=PD[GaussCode@@hopfGC]

But that’s impossible. Here are those crossings...

... and I try to connect them ...

... but 3 and 4 are trapped on opposite sides.

DrawMorseLink finds out:

DrawMorseLink[hopfPD]

Jones doesn’t panic, justs gives wrong stuff:

Factor[Jones[hopfPD][t]]

Same link, in hopes of coaxing:

hopfGC2={{1,-2},{-1,2}};

Still wrong:

hopfPD2=PD[GaussCode@@hopfGC2]

DrawMorseLink[hopfPD2]

Jones is still wrong, but different.

Factor[Jones[hopfPD2][t]]

But isn’t the Hopf link in the big list?

AllLinks[2,Alternating]

hopf0=Link[2,Alternating,1]

This is good, but perhaps only because it was precomputed...

PD[hopf0]

... and this looks okay, until you realize we saw it before.

GaussCode[hopf0]

Nice picture ... until you follow the orientation!  (Two praying mantises locked in a death-kiss?)

DrawMorseLink[hopf0]

This was run in Mathematica 10, but the same sorts of things were present in 9 (and even 8, I believe). The same behavior occurs in a “version of March 22, 2011, 21:10:4.67737”.

Other trivial links suffer similarly. Two overlapping unlinked components can even produce the same impossible planar diagram:

PD[GaussCode[{1,2},{-1,-2}]]

PD[GaussCode[{1,-2},{-1,2}]]