Given a plabic graph on R^2, we can conormal lift its zig-zag strands to the unit cotangent bundle of R^2, obtaining a Legendrian link. If the plabic graph satisfies the "grid" condition, its Legendrian link admits a front projection in the standard contact R^3. We study certain Legendrian invariant moduli space of flag configurations associated with a grid plabic link, and construct a natural (quasi-)cluster structure on this moduli space using Legendrian weaves. In particular, we prove that any braid variety associated with (beta Delta) for a 3-strand braid beta admits cluster structures with an explicit construction of initial seeds. In this talk, I will introduce the theoretical background and describe the basic combinatorics for constructing Legendrian weaves and cluster structures from a grid plabic graph. This is based on a joint work in progress with R. Casals.
Speaker(s): Daping Weng (UC Davis)