Constructing Lagrangians from triple grid diagrams
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917125807931392 |
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| author | Blackwell, Sarah Gay, David T. Lambert-Cole, Peter |
| author_facet | Blackwell, Sarah Gay, David T. Lambert-Cole, Peter |
| contents | Links in $S^3$ can be encoded by grid diagrams; a grid diagram is a collection of points on a toroidal grid such that each row and column of the grid contains exactly two points. Grid diagrams can be reinterpreted as front projections of Legendrian links in the standard contact 3-sphere. In this paper, we define and investigate triple grid diagrams, a generalization to toroidal diagrams consisting of horizontal, vertical, and diagonal grid lines. In certain cases, a triple grid diagram determines a closed Lagrangian surface in $\mathbb{CP}^2$. Specifically, each triple grid diagram determines three grid diagrams (row-column, column-diagonal and diagonal-row) and thus three Legendrian links, which we think of collectively as a Legendrian link in a disjoint union of three standard contact 3-spheres. We show that a triple grid diagram naturally determines a Lagrangian cap in the complement of three Darboux balls in $\mathbb{CP}^2$, whose negative boundary is precisely this Legendrian link. When these Legendrians are maximal Legendrian unlinks, the Lagrangian cap can be filled by Lagrangian slice disks to obtain a closed Lagrangian surface in $\mathbb{CP}^2$. We construct families of examples of triple grid diagrams and discuss potential applications to obstructing Lagrangian fillings. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_16404 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Constructing Lagrangians from triple grid diagrams Blackwell, Sarah Gay, David T. Lambert-Cole, Peter Geometric Topology Symplectic Geometry 57K43 (Primary), 57K33 Links in $S^3$ can be encoded by grid diagrams; a grid diagram is a collection of points on a toroidal grid such that each row and column of the grid contains exactly two points. Grid diagrams can be reinterpreted as front projections of Legendrian links in the standard contact 3-sphere. In this paper, we define and investigate triple grid diagrams, a generalization to toroidal diagrams consisting of horizontal, vertical, and diagonal grid lines. In certain cases, a triple grid diagram determines a closed Lagrangian surface in $\mathbb{CP}^2$. Specifically, each triple grid diagram determines three grid diagrams (row-column, column-diagonal and diagonal-row) and thus three Legendrian links, which we think of collectively as a Legendrian link in a disjoint union of three standard contact 3-spheres. We show that a triple grid diagram naturally determines a Lagrangian cap in the complement of three Darboux balls in $\mathbb{CP}^2$, whose negative boundary is precisely this Legendrian link. When these Legendrians are maximal Legendrian unlinks, the Lagrangian cap can be filled by Lagrangian slice disks to obtain a closed Lagrangian surface in $\mathbb{CP}^2$. We construct families of examples of triple grid diagrams and discuss potential applications to obstructing Lagrangian fillings. |
| title | Constructing Lagrangians from triple grid diagrams |
| topic | Geometric Topology Symplectic Geometry 57K43 (Primary), 57K33 |
| url | https://arxiv.org/abs/2306.16404 |