The Lantern Relation
(Edited… this now has the true lantern relation. Thanks Saul.)
Look at the four punctured sphere with the three annuli on it.
The lantern relation says that the composition of right handed Dehn twists about the red, blue, then green annuli is the same as right handed Dehn twists about each of the boundary components. Or instead, right handed Dehn twists about the red, blue, then green annuli followed by left handed Dehn twists about the boundary components is the identity.
Let’s see this in action…. We’ll also see that the identity can look pretty messed up. And it can look pretty when messed up.
Hmmm… troubles embedding the video… watch it here.
Continue on for stills of each stage.