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Let's consider the following bipartite cubic planar non-simple graph

$\hskip2.3in$enter image description here

Looking at the orientation of the edges around the vertices, it is obvious that left and right are oriented opposite.

Now blow up every edge like in a ribbon or fat graph. Including the change of orientation the resulting graph looks like:

$\hskip1.7in$enter image description here

where I stuck to the convention that I flipped every fat graph edge in the same direction. The resulting knot is a trefoil.

Further, the bicubic planar graphs can be related to Riemann surfaces (see here and references therein). Is there a relation between the Riemann surfaces and the knot?

How does the topology of the graphs' Riemann surface relate to its knot representation?

Concerning the construction of Riemann Surfaces, I can't judge which is the most promising. I collected a bunch of constructions (any additional ones are welcome here):

but I can't judge which one fits best to knots. If I'm forced to pick, I'll pick Hurwitz.

Any help is appreciated...

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    $\begingroup$ The computation of the rank and Ihara zeta function for this graph seem to be irrelevant to the actual question. I recommend editing that out and adding a self-contained description of the construction of a Riemann surface that you have in mind. $\endgroup$
    – j.c.
    Commented May 17, 2018 at 14:52
  • $\begingroup$ @j.c. I removed the rank computation, but I'm not able to choose a specific construction of a Riemann surface... $\endgroup$
    – draks ...
    Commented Apr 30, 2020 at 22:07
  • $\begingroup$ Suppose that the initial cubic plane graph has a cycle of odd length. Then the fat graph will not be orientable. So the surface is not a Riemann surface. So, I don't understand the actual question here... $\endgroup$
    – Sam Nead
    Commented Aug 3, 2020 at 10:48
  • $\begingroup$ @Sam I edited my question, accordingly... $\endgroup$
    – draks ...
    Commented Aug 4, 2020 at 7:54
  • $\begingroup$ The question is not clear to me. It is true that for any planar graph you can associate a knot diagram, using the algorithm you described. But I am not sure what you are asking. $\endgroup$
    – Henry
    Commented Aug 4, 2020 at 10:01

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