Cluster structures of Legendrian cablings
최고관리자
2026-02-25
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Category
Seminar
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Date
20260424 ~ 20260424Time
15:00 ~ 17:00 -
Place
IBS-CGP
Host
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Speaker
Byung Hee An
Affiliation
Kyungpook National University
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Subject
Cluster structures of Legendrian cablings
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Notice
Title: Cluster structures of Legendrian cablings
Speaker: Byung Hee An(Kyungpook National University)
Abstract:
It is known that both augmentation varieties and moduli spaces of constructible sheaves admit A and X cluster structures, respectively, for a certain class of Legendrian knots and links. The first known such a class is a positive braid closure by Shen-Weng, which can be seen as a cabling (or a satellite) along the boundary of the Lagrangian disk properly embedded in the four-ball.
In the first part of the talk, we generalize this viewpoint to the class of cablings along the boundary of compact orientable surfaces L regarded as a Lagrangian in its cotangent bundle and consider algebraic and geometric objects associated with those Legendrians such as augmentation varieties AugL(β), moduli spaces of sheaves ShL(β), and so on. We show that both AugL(β) and ShL(β) admit cluster structures under certain conditions.In the second part, we consider a companion link S(λ,β) of a positive braid closure λ with the cabling data β. When λ admits a Lagrangian filling L, then we discuss how AugL(β), Aug(S(λ,β)) and Aug(λ) (or ShL(β), Sh(S(λ,β)) and Sh(λ)), and their cluster structures are related to each other.In particular, we also provide an algorithmic way in terms of N-graphs to describe the relationship between Lagrangian fillings of β over L, and S(λ,β), λ over the disc.This is a joint work in progress with Youngjin Bae (INU).
