A bulge surface in a holographic spacetime is a non-minimal extremal surface that occurs between two locally minimal surfaces homologous to a given boundary region. The python’s lunch conjecture of Brown et al. relates the bulge area to the complexity of the state. We study the geometry of bulges in a variety of classical spacetimes, and discover a number of surprising features that distinguish them from more familiar extremal surfaces such as Ryu-Takayanagi surfaces: they spontaneously break spatial isometries, both continuous and discrete; they are sensitive to the choice of boundary infrared regulator; they can self-intersect; and they probe entanglement shadows, orbifold singularities, and compact spaces such as the sphere in AdS_p x S^q. These features imply, according to the python’s lunch conjecture, novel qualitative differences between complexity and entanglement in the holographic context. We also find, for extended systems such as black branes and multi-boundary wormholes, that the complexity displays a subtle non-extensitivity.
