Balanced curves, quasimorphisms, and the equator conjecture

We construct a new infinite-dimensional family of homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms of the two-sphere. Moreover, for any constant $K$ less than the total area of the sphere, we produce unbounded homogeneous quasimorphisms that vanish on any map supported on some disk of area at most $K$. As an application, we prove an analogue of the equator conjecture, namely that the space of equators equipped with any choice of quantitative fragmentation metric has infinite diameter. To prove our results, we introduce tools that draw inspiration from the theory of curve graphs used to study mapping class groups.