GEOTOP-A | Knotted and branching defects in ordered media | Tamas Kalman 

I will discuss a homotopy classification of the global defect in ordered media, with a particular emphasis on the example of biaxial nematic liquid crystals. These are systems in which the order parameter space is the quotient of the 33-sphere 𝑆3S 3 by the quaternion group 𝑄Q, and an important feature of them is that disclination lines may branch and form graphs. Therefore as a model, I will consider continuous maps from complements of spatial graphs to 𝑆3/𝑄S 3/Q modulo a certain equivalence relation, and find that the equivalence classes are enumerated by the six subgroups of 𝑄Q. Via monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of 𝑄Q; once one passes to planar diagrams, these labels can be refined to elements of 𝑄Q associated to each arc. The same classification scheme applies not only in the case of 𝑄Q but also to arbitrary groups. This research is joint with Yuta Nozaki, Yuya Koda, and Masakazu Teragaito.