Legendrian graphs, invariants and DNA topology

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The goal of this project is to broaden our understanding of many topological aspects of spatial graphs, and make it possible to answer larger questions about its external three-dimensional space (or three-space) and biological mechanisms. A graph is a set of points called vertices and curves between the points called edges. A spatial graph is a graph that is sitting in three-space. A spatial graph can be very complicated and knotted. Spatial graph theory is a recent field which grew out of knot theory, the study of knotted circles in three-space. In addition to contributing to our understanding of connections between spatial graphs and their surrounding three-space, this project has direct applications to biological questions. Knot theory has become important within the life sciences, in the field of DNA Topology. Most of the DNA that is worked with in laboratories is in small circular molecules that are called plasmids. Circular DNA also appears in nature. Plasmids can naturally be modeled with knots. With every cell division the DNA is copied in a process called replication. This is a complicated process that involves a number of enzymes. After replication the two new plasmids are linked (or catenated) and sometimes knotted. The reason for knotting is not completely understood. In the middle of replication DNA forms the more complicated structures, that of a knotted graph, this is an area of DNA topology that has yet to be investigated by mathematicians. This project will have immediate contributions to our biological understanding of DNA replication. In particular, the PI will focus on three main projects centering around spatial graphs: First, the PI will study a number of aspects of Legendrian graphs. Central results in contact geometry use Legendrian graphs in their proofs. Thus this opens an exciting new area with great potential. These projects will increase the overall understanding of Legendrian graphs, further increasing their potential to use Legendrian graphs as a tool to understand 3-manifolds, and potentially distinguish contact structures. Second, the PI will develop new invariants and better understand existing invariants of spatial graphs and knots, including graph Floer homology. Third, the PI will begin classification of unknotting number one theta-graphs to understand unknotting in DNA replication intermediates.