Extensions of Heegaard Floer Homology and Applications to Topology

Dartmouth College, Hanover NH

Investigators

Abstract

Low-dimensional topology studies the shapes of spaces in dimensions one through four, and has applications ranging from physics and cosmology (the shape of the universe) to biochemistry (understanding the behavior of knotted DNA). Closely related to the study of 3- and 4-dimensional spaces is the study of knots (loops tied in space). In the early 2000s, Ozsvath and Szabo developed a package of powerful invariants for knots and 3- and 4-dimensional spaces, generally known as Heegaard-Floer homology. Heegaard-Floer homology has since taken a major place in low-dimensional topology, and has helped obtain a lot of new results and settle numerous old conjectures. The PI and her collaborator have developed a new algebraic technique for studying the variant for knots (knot Floer homology), by cutting a knot into pieces called tangles, and studying the individual pieces and their gluing. This NSF funded project seeks to develop further this new tool called tangle Floer homology, and use it to study knot theory problems and to better understand the relation between various knot invariants. The PI will aim to involve undergraduate and graduate students in aspects of the project that have combinatorial and computational nature, and will continue to promote mathematics to a broader audience. Specific components of the project are to: 1) fully develop the ''minus" version of tangle Floer homology and introduce new concordance invariants, as well as extend tangle Floer homology to integer coefficients; 2) understand and develop the connections between knot Floer homology and quantum algebra, in order to relate existing knot invariants as well as obtain new ones; 3) use tangle Floer homology to carry out computations in knot Floer homology and to study problems on knot and link concordance, mutation, string links, and periodic knots.