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 8φ(g - 1) surfaces of genus greater than 2, where φ(g - 1) is the Euler totient function of g - 1. Observe that this count is independent of the number of crossings of the knot. This is the first infinite family of 3-manifolds where such counts are precisely known, but simply not zero for all large g.

The author would like to thank his advisor Nathan Dunfield for this interesting question and for the guidance and stimulating discussions throughout. The author would like to thank the referee for their helpful comments and suggestions. Lastly, the author would also like to thank Chaeryn Lee for helpful discussions, reading an early draft of this paper, and giving helpful advice to improve it. This work was partially supported under US National Science Foundation grant DMS-1811156.