Cluster structures of Legendrian cablings
최고관리자
2026-02-25
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분야
콜로퀴움
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날짜
20260424 ~ 20260424시간
15:00 ~ 17:00 -
장소
IBS-CGP
초청자
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연사
Byung Hee An
소속
Kyungpook National University
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제목
Cluster structures of Legendrian cablings
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소개 및 안내사항
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).
