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Surfaces, Orbifolds, and Dominance

Banff International Research Station for Mathematical Innovation and Discovery

Consider the set of all triangulations of a convex $(n+3)$-gon. These triangulations are related to one another by diagonal flips, and the graph defined by these flips is the 1-skeleton of the familiar $n$-dimensional polytope known as the associahedron. The $n$-dimensional cyclohedron is constructed analogously using centrally-symmetric triangulations of a regular $(2n+2)$-gon, with relations given by centrally-symmetric diagonal flips. Modding out by the symmetry, we may equivalently view the cyclohedron as arising from ``triangulations" of an orbifold: the $(n+1)$-gon with a single two-fold branch point at the center.\\ In this talk we will introduce orbifold-resection, a simple combinatorial operation which maps the ``once-orbifolded" $(n+1)$-gon to the $(n+3)$-gon. More generally, orbifold-resection maps a triangulated orbifold to a triangulated surface while preserving the number of diagonals and respecting adjacencies. This induces a relationship on the signed adjacency matrices of the triangulations, called dominance, which gives rise to many interesting phenomena. For example, the normal fan of the cyclohedron refines that of the associahedron; work is in progress to show that such fan refinement holds generally in the case of orbifold-resection. If time allows, we will explore other dominance phenomena in the context of the surfaces-and-orbifolds model.

Mathematics; Combinatorics; Group theory and generalizations

video/mp4

Author affiliation: North Carolina State University

2017-11-12

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63588

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/