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Abstract
This dissertation explores bridge n-sections of knotted surfaces in 4-manifolds by defininginvariants that measure the complexity of their topology and by giving geometric constructions
that determine Lagrangian surfaces in 4-manifolds under certain conditions.
First, we present an elegant geometric construction that generates all triple grid diagrams.
A triple grid diagram is a combinatorial representation of certain knotted surfaces in CP2,
which can determine a Lagrangian surface under specific conditions. Previously, only a few
examples of triple grid diagrams were known, and no efficient method for producing examples
existed. Our key insight is to approach the search by focusing on the abstract, Tait-colored
cubic graph Γ that we aim to realize through a triple grid diagram, rather than conditioning
on the grid size n. From this viewpoint, the problem reduces to pure linear algebra and can
be solved efficiently in polynomial time, as compared to testing pairs of permutations, which
requires O((n!)2)-time. This work is a collaboration with Lambert-Cole.
Second, we define invariants L∗ and LP of bridge n-sections of knotted surfaces in 4-
manifolds. These invariants measure the complexity of knotted surfaces in terms of paths in
the dual cut complex and pants complex, respectively, of a closed, oriented surface. These
invariants generalize previous L-invariants for trisected 4-manifolds. We establish fundamental
properties of these invariants, distinguish them from previous L-invariants, and compute them
for several explicit examples. Additionally, we prove that these invariants detect unknotted
surfaces in certain 4-manifolds. This is joint work with Aranda, Blackwell, Karimi, Kim,
Meyer and Pongtanapaisan.
Third, we introduce quadruple grid diagrams for the 4-section of CP1 × CP1, analogous to
triple grid diagrams for CP2. Due to their highly symmetric nature, we modify the construction
to define quasiquadruple grid diagrams. The additional flexibility of quasiquadruple grid
diagrams has the ability to determine embedded Lagrangian surfaces in CP1 × CP1 under
certain conditions, following an adapted version of the similar construction for triple grid
diagrams. Furthermore, we generalize the notion to define geometric n-tuple grid diagrams
for toric sympletic manifolds. This work is joint with Fushida-Hardy and Wakelin.