The principal investigator will use tools from Heegaard Floer homology to study phenomena in low-dimensional topology. One goal is to better understand the knot concordance group, in particular the difference between the smooth and topological categories. Concordance invariants coming from Floer homology, such as d-invariants and epsilon, are well-suited for this task. Specifically, the principal investigator aims to show that the subgroup of the smooth concordance group generated by topologically slice knots contains an infinite rank summand. A different approach to concordance is via bordered Floer homology. In the same sense that knot Floer homology categorifies the Alexander polynomial, one may hope that the bordered Floer homology of a Seifert surface categorifies the Seifert form. Many classical concordance invariants can be defined in terms of the Seifert form, and the project will investigate how these invariants manifest themselves within the bordered invariants.
The structure of knotted curves in space is intimately related to our understanding of the shape of 3- and 4-dimensional space. Thinking of time as the fourth dimension, the study of knot concordance becomes a question about the evolution of knots over time. By putting different restrictions on how knots may evolve, one obtains different definitions of the complexity of a knot. These measures of complexity have applications from the very small (e.g., the behavior of knotted strands of DNA) to the very large (e.g., the shape of the universe).
