  We study finite graphs embedded in oriented surfaces by associating a
polynomial to it. The tools used in developing a theory of such graph
polynomials are algebraic topological while the polynomial itself is inspired
from ideas arising in physics. We also analyze a variant of these polynomials
for colored embedded graphs. This is used to describe the change in the
polynomial under basic graph theoretic operations. We conclude with several
applications of this polynomial including detection of certain classes of
graphs and the connection of this polynomial with topological entanglement
entropy.
