Fig. 2: Typical loops carrying nontrivial or trivial topological invariants.
a–c Loops carrying nontrivial topological invariants Z1, Z2 and Z3, respectively, which are the generators of the group [Eq. (4)]. The dashed lines with arrow denote quotient maps, i.e., gluing of identified points. d The loop formed by the concatenation αβα‘β‘ encloses the NIP (NIP: nondefective intersection point), which carries the topological invariant Z1Z3. Point A’ in panels a–d denotes the basepoint. e–g Evolution of a loop carrying trivial topological charge. e A loop without touching ELs (EL: exceptional line) is confined within a specific region and is trivial. f Moving the loop l in panel e upwards along the black arrow direction, we see that it becomes a product of paths l1 and l2. Both l1 and l2 are trivial loops in the quotient space M, and thus the loop as their product is also trivial. g Stretching the loop along the black arrow direction in panel f, we obtain that the loop crosses EL1 and becomes a product l1l3l4l5 of paths. The path l4, similar to l1 and l2, corresponds to a trivial loop in the quotient space M. The paths l5 and l3 are oriented in opposite directions (labeled by the arrows) and are homotopic to α and α−1, respectively (Fig. 1a). The path product l1l3l4l5 is thus trivial.
