Existence of inductive weaves with only Y-tree Lusztig cycles for a given braid

Determine, for an arbitrary positive braid β, whether there exists an inductive weave from β to δ(β)=w0 in which all Lusztig cycles are Y-trees. Establishing existence of such an inductive weave is a necessary condition for β to be cycle decomposable via successive cycle deletions of Lusztig cycles.

Background

The paper studies decompositions of braid and augmentation varieties using weaves, rulings, and related structures. A central theme is how to obtain decompositions by deleting cycles (Lusztig cycles) in weaves, a process that corresponds to performing certain topological moves on the associated Legendrian surface.

A braid is called cycle decomposable if one can start from a suitable (inductive) weave and repeatedly perform cycle deletions to produce a full decomposition of the braid variety. A necessary condition for such a process is that the inductive weave has all Lusztig cycles of Y-tree type. The authors point out that, for a given braid, it is not clear whether such an inductive weave exists, highlighting a concrete gap in current understanding.