The contact cut graph and a Weinstein $\mathcal{L}$-invariant
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arXiv
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| Hauptverfasser: | , , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929455087222784 |
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| author | Castro, Nickolas Islambouli, Gabriel Min, Jie Sakallı, Sümeyra Starkston, Laura Wu, Angela |
| author_facet | Castro, Nickolas Islambouli, Gabriel Min, Jie Sakallı, Sümeyra Starkston, Laura Wu, Angela |
| contents | We define and study the contact cut graph which is an analogue of Hatcher and Thurston's cut graph for contact geometry, inspired by contact Heegaard splittings. We show how oriented paths in the contact cut graph correspond to Lefschetz fibrations and multisection with divides diagrams. We also give a correspondence for achiral Lefschetz fibrations. We use these correspondences to define a new invariant of Weinstein domains, the Weinstein $\mathcal{L}$-invariant, that is a symplectic analogue of the Kirby-Thompson's $\mathcal{L}$-invariant of smooth $4$-manifolds. We discuss the relation of Lefschetz stabilization with the Weinstein $\mathcal{L}$-invariant. We present topological and geometric constraints of Weinstein domains with $\mathcal{L}=0$. We also give two families of examples of multisections with divides that have arbitrarily large $\mathcal{L}$-invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05340 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The contact cut graph and a Weinstein $\mathcal{L}$-invariant Castro, Nickolas Islambouli, Gabriel Min, Jie Sakallı, Sümeyra Starkston, Laura Wu, Angela Geometric Topology Symplectic Geometry We define and study the contact cut graph which is an analogue of Hatcher and Thurston's cut graph for contact geometry, inspired by contact Heegaard splittings. We show how oriented paths in the contact cut graph correspond to Lefschetz fibrations and multisection with divides diagrams. We also give a correspondence for achiral Lefschetz fibrations. We use these correspondences to define a new invariant of Weinstein domains, the Weinstein $\mathcal{L}$-invariant, that is a symplectic analogue of the Kirby-Thompson's $\mathcal{L}$-invariant of smooth $4$-manifolds. We discuss the relation of Lefschetz stabilization with the Weinstein $\mathcal{L}$-invariant. We present topological and geometric constraints of Weinstein domains with $\mathcal{L}=0$. We also give two families of examples of multisections with divides that have arbitrarily large $\mathcal{L}$-invariant. |
| title | The contact cut graph and a Weinstein $\mathcal{L}$-invariant |
| topic | Geometric Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2408.05340 |