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 graph has three faces, so the rank of $G$ is [$\chi(G)=2$](https://math.stackexchange.com/questions/582616/of-face-and-circuit-rank). The reciprocal of Ihara's $\zeta$ function can be evaluated 
$$
\frac{1}{\zeta_G(u)}={(1-u^2)^{\chi(G)-1}\det(I - Au + 2u^2I)}\\
={(1-u^2) (4u^4-5u^2+1)}
$$

**EDIT**: Then $\zeta_G(u)=\frac{(1-u^2)^{-1} }{(4u^4-5u^2+1)}=\prod_p (1-u^{L(p)})$ with the product running over prime paths $p$ and $L(p)$ being their lenghts. The $-1$ in the numerator's exponent $(1-u^2)^{-1}$ is due to $|V|-|E|=2-3=-1$...

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][6] 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?

  [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