For positive braids with full Demazure product, the ruling, weave, Deodhar, and sheaf decompositions of the associated variety coincide, and cluster variables can be computed from Morse complex sequences.
Braid loops with infinite monodromy on the Legendrian contact DGA
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Decompositions of augmentation varieties via weaves and rulings
For positive braids with full Demazure product, the ruling, weave, Deodhar, and sheaf decompositions of the associated variety coincide, and cluster variables can be computed from Morse complex sequences.