Loading Web-Font TeX/Main/Regular
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

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.


    1. Add remark

      Problem 2.1.

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


        • Add remark

          Problem 2.2.

          For more general Legendrians, is the augmentation variety cluster?


            • Add remark

              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.)?


                • Add remark

                  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.