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 |
| RBGdddd | RGBdddd |
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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.
The lantern relations is really beautiful, isn’t it? It has been discovered twice. The most recent discoverer was Dennis Johnson (who also coined the name). I asked Dennis about it, and he has no idea how he came up with it. Amazingly, long after Dennis wrote it down someone (I’m not sure who — maybe Benson Farb?) found the lantern buried deep in an old paper of Dehn.
A cute (ex post facto) way of discovering the lantern is as follows. In your picture above, cap all three “interior” boundary components with punctured discs. The mapping class group of the 4-holed sphere then becomes the 3-strand braid group. Sticking your finger on one of the punctures, you can drag it around and get an embedding of the fundamental group of a twice-punctured disc into the 3-strand braid group. The lantern relation then becomes the following obvious relation : going first around one puncture and then around the other is the same as going around both of them at the same time! If you meditate on this a bit, you will see how to lift this picture to the 4-holed sphere.