Eleftherios Kastis *note unusual room*

Spanning disks in triangulated surfaces and rigidity on few locations

Given a triangulation T of a surface S, we say that a cycle is cellular if its topological interior is homeomorphic to the plane. A closed disc determined by such a cellular cycle is called spanning if it contains all the vertices of T. A natural question is to consider when a triangulated surface contains a spanning disc. It turns out that minimum face-width, i.e. the minimum number of intersections between T and any noncontractible curve in S, is a necessary condition.

In this talk, we will show that triangulated surfaces with sufficiently high face-width always admit spanning discs. The initial motivation for our work comes from rigidity theory. Kiraly's recently proved that if A is generic set on the plane and |A|≥26, then for every a planar Laman graph G=(V,E), there exists a placement p:V→A, such that (G,p) is infinitesimally rigid. Hence, applying our main theorem, we immediately obtain an analogous result for triangulations of arbitrary surfaces with sufficiently large facewidth.

This is joint work with K. Clinch, S. Dewar, N. Fuladi, M. Gorsky, T. Huynh, A. Nixon and B. Servatius.