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Description
Triangulating a surface means finding a subcomplex of a simplex that is homeomorphic to the surface. Vertex-minimal triangulations of closed surfaces have been characterized in classical work of Jungerman and Ringel.
The corresponding problem for cubes has been much less studied. Notably, Coxeter found surfaces in the $d$-cube of maximal possible genus and Schulz gave bounds on the dimension of the cube required to realize a particular surface as a subcomplex. These latter bounds are tight for orientable surfaces and nonorientable surfaces of even demigenus $k \geq 12$, while for surfaces of odd demigenus they may be off by one.
In the cubical case, minimizing the embedding dimension is not equivalent to minimizing the number of vertices, and finding vertex-minimal cubical realizations of surfaces remains poorly understood. We provide new theoretical bounds for this problem and, using computational methods, give a complete enumeration of connected closed surfaces in the 5-cube. We find that there are 2690 isomorphism classes of such surfaces. As a consequence, we obtain the minimal f-vectors of these surfaces in the 5-cube and complete Schulz's characterization for the even demigenus case, while discovering some new examples in the process.
