Invariants and Constructions of Knotted Surfaces in 4-Manifolds

This dissertation explores bridge n-sections of knotted surfaces in 4-manifolds by defining invariants 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 [doublestruck C doublestruck P superscript 2], 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 [gamma] 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!)[superscript 2])-time. This work is a collaboration with Lambert-Cole. Second, we define invariants L* and L[superscript P] 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 [doublestruck C doublestruck P superscript 1] x [doublestruck C doublestruck P superscript 1], analogous to triple grid diagrams for [doublestruck C doublestruck P superscript 2]. 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 [doublestruck C doublestruck P superscript 1] x [doublestruck C doublestruck P superscript 1] 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. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]