  We present how to construct elliptically fibered K3 surfaces via Weierstrass
models which can be parametrized in terms of Wilson lines in the dual heterotic
string theory. We work with a subset of reflexive polyhedras that admit two
fibers whose moduli spaces contain the ones of the $E_{8}\times E_{8}$ or
$\frac{Spin(32)}{\mathbb{Z}_{2}}$ heterotic theory compactified on a two torus
without Wilson lines. One can then interpret the additional moduli as a
particular Wilson line content in the heterotic strings. A convenient way to
find such polytopes is to use graphs of polytopes where links are related to
inclusion relations of moduli spaces of different fibers. We are then able to
map monomials in the defining equations of particular K3 surfaces to Wilson
line moduli in the dual theories. Graphs were constructed developing three
different programs which give the gauge group for a generic point in the moduli
space, the Weierstrass model as well as basic enhancements of the gauge group
obtained by sending coefficients of the hypersurface equation defining the K3
surface to zero.
