  We investigate the solution landscape of a reduced Landau--de Gennes model
for nematic liquid crystals on a two-dimensional hexagon at a fixed
temperature, as a function of $\lambda$---the edge length. This is a generic
example for reduced approaches on regular polygons. We apply the high-index
optimization-based shrinking dimer method to systematically construct the
solution landscape consisting of multiple defect solutions and relationships
between them. We report a new stable T state with index-$0$ that has an
interior $-1/2$ defect; new classes of high-index saddle points with multiple
interior defects referred to as H class and TD class; changes in the Morse
index of saddle points with $\lambda^2$ and novel pathways mediated by
high-index saddle points that can control and steer dynamical pathways. The
range of topological degrees, locations and multiplicity of defects offered by
these saddle points can be used to navigate through complex solution landscapes
of nematic liquid crystals and other related soft matter systems.
