The majority of 3-manifold theory studies sub-manifolds of \(M\) (often surfaces) to gain information about \(M\). Among these, the most relevant ones are the “incompressible” surfaces, which, in intuitive terms, are properly embedded surfaces that cannot be further simplified while remaining non-trivial. Their significance to 3-manifolds is analogous to the significance of essential simple curves to surfaces. In this talk, I will present my results on classifying orientable incompressible surfaces with non-empty boundary in a hyperbolic mapping torus with fibers homeomorphic to a 4-punctured sphere. By examining the transverse intersections of an incompressible surface with the fibers, we see that they yield a certain path in the arc complex of a 4-punctured sphere, thus reducing the problem to a combinatorial one. Our result generalizes the similar theorems proved by Floyd, Hatcher, and Thurston for incompressible surfaces in punctured torus bundles and 2-bridge link complements.