In-Person and Online Talk

We show that geometric, topological, and combinatorial complexities of certain surfaces in CP^2 are related:  We prove that a positive genus surface in CP^2 that minimizes genus in its homology class is isotopic to a complex curve if and only if it admits a hexagonal lattice diagram, a special type of shadow diagram in which arcs meets only at bridge points and tile the central surface of the standard trisection of CP^2 by hexagons.  This allows us to give a purely combinatorial reformulation of the symplectic isotopy problem in CP^2.