Let me give a worked-out example: The following cubic planar [non-simple](http://mathworld.wolfram.com/SimpleGraph.html) graph

$\hskip2.3in$[![enter image description here][1]][1]

has the adjacency matrix $A=\pmatrix{0&3\\3&0}$. The reciprocal of Ihara's $\zeta$ function can be evaluated 
$$
\frac{1}{\zeta_G(u)}={(1-u^2)^{2-1}\det(I - Au + 2u^2I)}\\
={(1-u^2) (4u^4-5u^2+1)}
$$

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][5]. Including the change of orientation the resulting graph looks like:

$\hskip1.7in$[![enter image description here][2]][3]

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][7] 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)][7]:

* Grothendieck's [Dessin D'Enfants][11]
* Makover's [Approach][12]
* Hurwitz's [Way][13]
* Nieser's [Method][14]

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...


  [1]: https://i.sstatic.net/eQYHS.png
  [2]: https://i.sstatic.net/ixMNDm.jpg
  [3]: http://www.win.tue.nl/~vanwijk/seifertview/tutorial5.htm#trefoilseifert
  [4]: https://en.wikipedia.org/wiki/Trefoil_knot#Invariants
  [5]: https://ncatlab.org/nlab/show/ribbon+graph
  [6]: https://math.stackexchange.com/q/1626800/19341
[7]: https://math.stackexchange.com/q/3424936/19341

[11]: https://math.stackexchange.com/a/2709419/19341
[12]: https://math.stackexchange.com/q/1626800/19341
[13]: https://math.stackexchange.com/q/3034043/19341
[14]: https://math.stackexchange.com/q/2047672/19341