genus

The genus of a net is here defined as follows. Imagine the net to be composed of edges of finite width. Take a primitive cell and join edges going in the + [uvw] direction to their partners in the -[uvw] direction. One then obtains a "handlebody" with n handles. n is the genus. For a 3-periodic net the genus must be greater than or equal to 3. Minimal nets (Beukeman, A; Klee, W. E., Z. Kristallogr. 1992, 201, 37-51) have genus = 3. If there are v vertices and e edges per unit cell then the genus is 1 + e - v. (The genus is the cyclomatic number of the quotient graph, see the cited reference.)

A useful observation is that the genus of the net of a tiling is equal to the genus of the net of the dual tiling. (If the tiling has f faces and p tiles, the net of the dual tiling has genus 1 + f + p. But for a tiling v - e + f - p = 0. See e.g Coxeter, H. S. M. Regular Polytopes 3rd ed. Dover, New York 1973. page 172).

Warning. In graph theory the genus of a finite graph is quite different. It is the minimum number of handles that must be added to the plane to allow an embedding of the graph without any edge intersections. Using this definition, the genus of any plane graph is zero. However using our definition given above, the genus of a periodic net that is the labyrinth system of a periodic minimal (or other) surface, is the same as that of the surface itself. In this last connection see Fischer, W.; Koch, E. Genera of minimal balance surfaces. Acta Crystallogr. 1989, A45, 726-732.