Exotically knotted disks and complex curves

Abstract: This paper studies properly embedded surfaces in the 4-ball that are exotically knotted (i.e., topologically but not smoothly isotopic), and leverages this local phenomenon to study surfaces in larger 4-manifolds. The main results provide a new construction of exotically knotted surfaces, including exotic slice surfaces of all genera in the 4-ball and exotic closed surfaces in larger 4-manifolds. The construction is well-suited to the complex and symplectic settings, providing the first examples of exotically knotted complex curves and symplectic 2-spheres. Along the way, we articulate some diagrammatic tools for constructing symplectic surfaces and complex curves. We also use local knotting to investigate the geography problem for knot groups, constructing the first examples of exotically knotted surfaces in closed, simply connected 4-manifolds whose knot groups contain nonabelian free subgroups, hence are not expected to be "good" groups in the sense of surgery theory.