The Whitney immersion is a Lagrangian sphere inside the four-dimensional symplectic
vector space which has a single transverse double point of Whitney self-intersection
number
.
This Lagrangian also arises as the Weinstein skeleton of the complement of a binodal
cubic curve inside the projective plane, and the latter Weinstein manifold is thus the
“standard” neighbourhood of Lagrangian immersions of this type. We classify the
Lagrangians inside such a neighbourhood which are homologically essential, and
which are either embedded or immersed with a single double point; they are shown to
be Hamiltonian isotopic to either product tori, Chekanov tori, or rescalings of the
Whitney immersion.
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