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2. Stratifications of braid varieties

Given a positive braid, the front projection of a Legendrian link which is its rainbow closure has an augmentation variety with a ruling stratification into $(\mathbb{C}^k)\times (\mathbb{C}^*)^l$ pieces. This stratification is the same as the one coming from the cluster variety.


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      Problem 2.1.

      If we consider instead a braid variety, ie. a Legendrian associated to a $(-1)$-closure, does this still hold?


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          Problem 2.2.

          For more general Legendrians, is the augmentation variety cluster?


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              Problem 2.3.

              If $X$ is a general cluster variety, and $T\subset X$ is a cluster torus. What can we say about $X - T$ (Eg. what are the irreducible components, intersections, etc.)?


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                  Problem 2.4.

                  What can we say if $X$ is the cluster variety of a $(-1)$-closure, but $T$ comes from some weave other than the obvious/natural one?

                      Cite this as: AimPL: Cluster algebras and braid varieties, available at http://aimpl.org/clusterbraid.