Writhe of center vortices and topological charge: An explicit example

The manner in which continuum center vortices generate topological charge density is elucidated using an explicit example. The example vortex world-surface contains one lone self-intersection point, which contributes a quantum 1/2 to the topological charge. On the other hand, the surface in question is orientable and thus must carry global topological charge zero due to general arguments. Therefore, there must be another contribution, coming from vortex writhe. The latter is known for the lattice analogue of the example vortex considered, where it is quite intuitive. For the vortex in the continuum, including the limit of an infinitely thin vortex, a careful analysis is performed and it is shown how the contribution to the topological charge induced by writhe is distributed over the vortex surface.