TITLE: Closed, orientable surface whose genus is very hard to find intuitively
QUESTION [7 upvotes]: I'm introducing the Classification Theorem on closed and orientable surfaces in a talk on (intuitive) topology, and to motivate it I'd like an example of an embedding of a surface in $\mathbb{R}^3$ whose genus is very hard to find. The requirements are:

Closed and orientable 2-manifold
Difficult to mentally bend and twist (even allowing "ghostly passing-through") into any familiar surface (e.g. connected sum of $g$ tori)

I allow "ghostly passing-through" so that knotting won't hinder the investigation of homeomorphisms. But after allowing that, some nice examples becomes easy:

(Image from "An Introduction to Topology" by E. C. Zeeman)
The above example would have been hard, but once ghostly passing-through is allowed, It's easy to pull out the internal tunnels and show that it is homeomorphic to a triple torus.
The difficulty of finding the surface's genus will also help me to motivate my introduction of the Euler characteristic.

REPLY [3 votes]: The great dodecahedron and small rhombihexahedron may be viewed as closed, orientable surfaces whose genera are difficult to see geometrically. (Each has genus $4$.)