Title: Weighted Cycles on Weaves
Abstract: Weaves were first introduced by Casals and Zaslow as a graphical tool to describe a family of Legendrian surfaces living inside the 1-jet space of a base surface. Casals, Gorsky, Gorsky, Le, Shen, and Simental later generalized weaves to all Dynkin types such that the original weaves for Legendrian surfaces belong to Dynkin type A, and they use weaves of general Dynkin types to describe the cluster structure on braid varieties. In my previous joint work with Casals, we gave a topological interpretation of the cluster structures associated with weaves of Dynkin type A by associating the quiver with intersections of certain 1-cycles on surfaces and associating cluster variables with merodromies (parallel transports) along dual relative 1-cycles. In this talk, I will generalize this topological interpretation to all general Dynkin types by introducing a new diagrammatic object called “weighted cycles” and constructing an intersection pairing between them. I will define the merodromy along a weighted cycle and explain how to describe cluster variables using merodromies. If time allows, I will also mention a connection to quantum groups and skein algebras.