  We explore the structure of the space of quasisymmetric configurations
identifying them by their magnetic axes, described as 3D closed curves. We
demonstrate that this topological perspective divides the space of all
configurations into well-separated quasisymmetric phases. Each phase is
characterized by the self-linking number (a topological invariant), defining
different symmetry configurations (quasi-axisymmetry or quasi-helical
symmetry). The phase-transition manifolds correspond to quasi-isodynamic
configurations. By considering some models for closed curves (most notably
torus unknots), general features associated with these phases are explored.
Some general criteria are also built and leveraged to provide a simple way to
describe existing quasisymmetric designs. This constitutes the first step in a
program to identify quasisymmetric configurations with a reduced set of
functions and parameters, to deepen understanding of configuration space, and
offer an alternative approach to stellarator optimization that begins with the
magnetic axis and builds outward.
