Topological techniques are used to study the motions of systems of point vortices in the infinite plane, in singly-periodic arrays, and in doubly-periodic lattices. Restricting to three vortices with zero net circulation, the symmetries are used to reduce each system to a one-degree-of-freedom Hamiltonian. The phase portrait of the reduced system is subdivided into regimes using the separatrix motions, and a braid representing the topology of all vortex motions in each regime is computed. This braid also describes the isotopy class of the advection homeomorphism induced by the vortex motion. The Thurston–Nielsen theory is then used to analyze these isotopy classes, and in certain cases strong implications about the chaotic dynamics of the advection can be drawn. This points to an important mechanism by which the topological kinematics of large-scale, two-dimensional fluid motions generate chaotic advection.