Lantern relation

[1] The general form of the lantern relation involves seven Dehn twists in the mapping class group of a disk with three holes,[1][2] as shown in the figure on the right.

According to the relation, where DA, DB, and DC are the right-handed Dehn twists around the blue curves A, B, and C, and DR, DS, DT, DU are the right-handed Dehn twists around the four red curves.

Note that the Dehn twists DR, DS, DT, DU on the right-hand side all commute (since the curves are disjoint, so the order in which they appear does not matter.

Depending on the setting, some of the Dehn twists appearing in the lantern relation may be homotopic to the identity function, in which case the relation involves fewer than seven Dehn twists.

The lantern relation is used in several different presentations for the mapping class groups of surfaces.