Tomaž Pisanski: Abstract polygonal complexes (with an application to synthetic biology)

Povzetek. In this talk we briefly outline the theory of abstract polygonal complexes and their representations. Our definition partially overlaps with the definition of a polygonal complex introduced recently by Pellicer and Schulte (2010, 2013) but is limited to the finite case. It also generalizes their definition in three respects: one the one hand it enables the study of maps on surfaces with boundary and on the other it separates the combinatorial structure of the polygonal complex from its geometric representation. Finally, we relax their strong condition that the boundary of each face is a cycle in the skeleton graph, with no repeated edges or vertices.

The theory enables a uniform approach to maps on surfaces and their polyhedral representations. In particular, the relationship between automorphism groups of abstract polygonal complexes and the symmetry groups of their geometric representation is pointed out. Flag graphs, symmetry type graphs in connection with some operations on polygonal complexes will be indicated. The talk will be illustrated by computer implementation of most important features of this theory. The emphasis will be on polygonal complexes specifying surfaces. A surprising application to synthetic biology will be given.