  In the complement of a hyperbolic Montesinos knot with 4 rational tangles, we
investigate the number of closed, connected, essential, orientable surfaces of
a fixed genus $g$, up to isotopy. We show that there are exactly 12 genus 2
surfaces and $8\phi(g - 1)$ surfaces of genus greater than 2, where $\phi(g -
1)$ is the Euler totient function of $g - 1$. Observe that this count is
independent of the number of crossings of the knot. Moreover, this class of
knots form an infinite class of hyperbolic 3-manifolds and the result applies
to all such knot complements.
