File:Separating loops.jpg

Nonseparating loops on an open surface. (a) Patch with three topological defects (i.e., handles). (b) The wavefront intersection is used to identify a handle on the surface. Note that the remaining faces in the defect are connected and that two associated nonseparating loops are identified: choosing the “green” loop corrects the topology by cutting the handle, while choosing the “yellow” one fills the corresponding hole. The magnified region illustrates the wavefront intersection with two triangles (dashed region) intersecting in a single vertex. (c) The front intersection indicates the potential presence of a handle in the defect. In this example, the rest of the faces are connected but not within the defect: this wavefront evolution can be used to locate only one single nonseparating loop, which, in this case, corresponds to filling the hole. We note that other wavefront evolutions (i.e., starting from other seed faces) will often produce two nonseparating loops, and not always one single nonseparating loop. (d) A wavefront intersection does not always imply that a nonseparating loop exists, as this example illustrates: the remaining faces are not connected. Therefore, we continue evolving the front.