After a week off for Thanksgiving, the Geometry Seminar returns with a double header, with talks starting at 3:30.
Abstract: We review homology theories of links and graphs, focusing on Khovanov link and chromatic graph homology and relations between them.
Title: When does added symmetry shift rigid structures to flexible structures?
Abstract: For finite frameworks with graph G in dimensions 2 and 3, we have necessary conditions for rigidity; |E|=2|V|−3 in the plane (Laman’s Theorem) and |E|=3|V|−6 in 3-space (Maxwell’s condition). Recently, work by a group of researchers has given modified necessary counts for orbits of finite symmetric frameworks, whose failure guarantees symmetry generic frameworks are flexible. The most striking case, visible in a number of classical examples, is generically isostatic frameworks in 3-space which become flexible with half-turn symmetry.
Several recent papers have given necessary (and sometimes sufficient) conditions for periodic generic frameworks to be infinitesimally rigid. Building on these two foundations, recent work with Bernd Schulze (TU Berlin) and Elissa Ross (York University) has examined necessary conditions for rigidity of periodic frameworks with added symmetry. Again, there are circumstances, such as inversive symmetry in a crystal which convert the count for generic rigidity into an orbit count which guarantees flexibility.
We will present an overview of these results, with a few animations and tables, as well as the core technique of orbit rigidity matrices. We conclude with an array of unsolved problems. Related papers are on the arXiv.