The Lantern Relation II: The Fake Lantern Relation
Gotta love orientations. Handedness, order of composition, sign conventions. Thought I’d put the end results of the Real and the Fake Lantern Relations side by side. Shown again is the four punctured sphere with three colored annuli on it.
Let’s say R, G, B denote right handed Dehn twists along the red, green, and blue curves and that dddd denotes left handed Dehn twists along the boundary components. We’re composing from left to right.
|The Real Lantern Relation||The Fake Lantern Relation|
In the last row, I picked out the arc that had the simplest looking image in the Fake Lantern Relation to compare with the corresponding one in the Real Lantern Relation. With a bit of eyeballin’ you can see the arc in the Real is isotopic to the identity. Here’s a simplified picture of the arc in the Fake.