16.02.2026, 16.00-18.00 MMF (ЦИСИ) room 305

Serge A. LAWRENCE: What does tell us the action of the automorphism group of a graph on the set of triangulations of a given surface with this graph?

By judiciously using the Pólya's Enumeration Theorem, also known as the Pólya-Redfield Theorem, we can extract not only the number of such triangulations from the action of the automorphism group of a given graph on the set of triangulations of a given surface with this graph, but also write out all these triangulations (with labeled vertices!) explicitly. In particular, we will show how to explicitly write out all twelve triangulations of the torus with the complete 4-partite graph K_2,2,2,2. (This graph is remarkable in that it is the graph of a 4-dimensional hyperoctahedron.) The beauty of the construction becomes simply intriguing when this graph is thought of algebraically, namely, as the Cayley graph of the quaternion group Q_8. Although the emphasis will be on algebra and combinatorics, the geometric component will be illustrated.