Homological Invariants in Low Dimensional Topology

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This National Science Foundation award supports a project in low dimensional topology, an area of mathematics that studies shapes of three- and four-dimensional spaces. Interestingly, the fundamental problem of understanding and classifying shapes of spaces is more difficult in these lower dimensions compared to higher dimensions. In fact, investigating some of the new phenomena that happen in these dimensions require the use of more modern invariants, or in other words, quantities associated to shapes that can distinguish between those with different properties. This subject is also closely related to knot theory, that is focused on studying the shapes of knotted circles in three-dimensional spaces. Low dimensional topology and knot theory have various implications in physics (quantum theory), cosmology (the shape of the universe), chemistry (molecular knots) and biology (knotting of DNA and DNA-protein interactions). This project aims to study the shapes of three- and four-dimensional spaces and configurations of knotted circles and surfaces in them using an invariant defined by the PI and her collaborator. In another direction, the PI investigates the relationship between different invariants for knotted circles. The PI plans to organize seminars and conferences and involve undergraduate students in the combinatorial and computational aspects of this project. Heegaard Floer homology is a collection of algebraic invariants for low dimensional objects (e.g. 3- and 4-manifolds, knots, links, etc.), defined by counting holomorphic disks. In particular, different types of such invariants have been introduced for 3-manifolds with boundary. For example, Eftekhary and the PI defined tangle Floer homology as a generalization of (minus) Heegaard Floer homology. In one direction, this project aims to (1) study embeddings of graphs and homology cobordism group of homology cylinders using tangle Floer homology and (2) introduce invariants for concordances and Seifert surfaces, and get bounds for unknotting number using cobordism maps for tangle Floer homology. In a different direction, the PI intends to (3) give a computationally effective way for detecting handlebodies using bordered Floer homology (different extension of Heegaard Floer invariants for 3-manifolds with boundary defined by Lipshitz-Ozsvath-Thurston) with the hope of providing a new way for detecting homotopically ribbon fibered knots. This project also proposes to study the properties of an invariant for knots and links, called symplectic sl(n) homology. This invariant is conjecturally equal to sl(n) homology, and the PI plans to (4) investigate spectral sequences for specific involutions in symplectic sl(n) homology and establish a better understanding of this conjecture.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.