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Decompositions and diagrams of symplectic surfaces in Weinstein domains

  • Román Aranda
  • Patricia Cahn
  • Agniva Roy
  • Melissa Zhang
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Abstract

We introduce combinatorial and diagrammatic methods for representing properly embedded symplectic surfaces in 4-dimensional Weinstein domains. We show that positive ascending surfaces, which include complex curves in Stein domains and multisections of Lefschetz fibrations, can be placed in bridge position with respect to Islambouli--Starkston's bisection-with-divides structure on the Weinstein domain. We algorithmically relate various decompositions of such surfaces, including transverse banded unlink diagrams, quasipositive factorizations, bridge bisections with divides, shadow diagrams (curves on surfaces), and pointed monodromy factorizations. We also develop a new way to present branched covers of Weinstein domains along positive ascending surfaces, which, combined with work of Loi--Piergallini, recovers Islambouli--Starkston's result that every compact Weinstein domain admits a bisection with divides.

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Cite

BibTeXRIS
BibTeX
@misc{aranda2026decompositions,
  title = {{Decompositions and diagrams of symplectic surfaces in Weinstein domains}},
  author = {Román Aranda and Patricia Cahn and Agniva Roy and Melissa Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00163},
  eprint = {2609.00163},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00163},
}
RIS
TY  - UNPB
TI  - Decompositions and diagrams of symplectic surfaces in Weinstein domains
AU  - Román Aranda
AU  - Patricia Cahn
AU  - Agniva Roy
AU  - Melissa Zhang
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.00163
UR  - https://arxiv.org/abs/2609.00163
AB  - We introduce combinatorial and diagrammatic methods for representing properly embedded symplectic surfaces in 4-dimensional Weinstein domains. We show that positive ascending surfaces, which include complex curves in Stein domains and multisections of Lefschetz fibrations, can be placed in bridge position with respect to Islambouli--Starkston's bisection-with-divides structure on the Weinstein domain. We algorithmically relate various decompositions of such surfaces, including transverse banded unlink diagrams, quasipositive factorizations, bridge bisections with divides, shadow diagrams (curves on surfaces), and pointed monodromy factorizations. We also develop a new way to present branched covers of Weinstein domains along positive ascending surfaces, which, combined with work of Loi--Piergallini, recovers Islambouli--Starkston's result that every compact Weinstein domain admits a bisection with divides.
ER  -