On Lagrangians of Oakley-Usher type

The first main theme of this paper is to study symplectic (un-)knottedness of Lagrangian submanifolds which are not tori. Our construction of such Lagrangians is inspired by the generalized Polterovich Lagrangians of Oakley–Usher and relies on an adaption of symplectic reduction methods going back to Chekanov–Schlenk and McDuff’s probes to the case of non-abelian group actions. We compare Oakley–Usher Lagrangians to their (un-)knotted cousins to find that, depending on the situation, they are: (1) not diffeomorphic, (2) diffeomorphic, but not Lagrangian isotopic; (3) Lagrangian isotopic, but not Hamiltonian isotopic, (4) Hamiltonian isotopic. This relationship between them frequently changes when the ambient space is compactified. The second main theme is an in-depth study of monotone Lagrangian tori obtained from appyling our constructions to the three-dimensional quadric $Q^3$. Computing the versal deformation of the $\mathrm{\Psi }$-invariant of Shelukhin–Tonkonog–Vianna, we find a torus which is knotted in a strong sense: it is not Hamiltonian isotopic to the Biran-lift of any Vianna torus in the two-dimensional quadric. Furthermore, we investigate enumerative properties of the Oakley–Usher torus in $Q^3$ and prove its non-displaceability, settling an open question asked by Oakley–Usher. We relate this to mirror symmetry by proving a split-generation result for the monotone Fukaya category of $Q^3$, which is inspired by, and can be compared to a similar result of Abouzaid–Diogo for cotangent bundles of the sphere.