Decompositions and diagrams of symplectic surfaces in Weinstein domains

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.