Tangent unit-vector fields: nonabelian homotopy invariants, the Dirichlet energy and their applications in liquid crystal devices
We compute the infimum Dirichlet energy, E(H), of unit-vector fields defined on an octant of the unit sphere, subject to tangent boundary conditions on the octant edges and of arbitrary homotopy type denoted by H. The expression for E(H) involves a new topological invariant – the spelling length – associated with the (nonabelian) fundamental group of the n-times punctured two-sphere. These results are then used for the modelling of the Post Aligned Bistable Nematic (PABN) device, designed by Hewlett Packard Laboratories. We provide analytic approximations for the experimentally observed stable equilibria in the PABN device and propose novel topological and geometrical mechanisms for bistability or multistability in prototype liquid crystal device geometries. This is joint work with Jonathan Robbins, Maxim Zyskin and Chris Newton.