Analytical Formulas for Alignment Angles of Nematic Liquid Crystals in Multiply Connected Domains with Topological Defects

We present analytical formulas associated with the alignment angles of nematic liquid crystals in multiply connected domains with the presence of multiple topological defects. This can be done by using complex analysis techniques and special functions in multiply connected domains called Schottky-Klein prime functions derived by D. Crowdy. The boundary condition considered is a tangential anchoring of nematic liquid crystals, where the alignment angle is tangential on all boundaries. Numerical results based on the minimization of Frank free energy show that there exist a pair of stable formations of defects depending on the initial locations of the defects, i.e.,

symmetry-breaking formations.