This proposal aims to study knot and link concordance using tools from Heegaard Floer homology, a powerful set of techniques in low dimensional topology introduced in 2000 by Ozsváth and Szabó. In its short lifespan, Heegaard Floer homology has already been used to solve numerous problems in topology, many of which had resisted mathematicians' efforts for decades. Specifically, it has provided new insight into knot theoretic questions related to the Seifert genus, fibered knots, Legendrian and transversal knots, the 4-ball genus of knots and the smooth concordance order. Knot concordance, introduced by Fox and Milnor in the 1960s, spans the whole spectrum (of dimensions) of low dimensional topology: concordance studies knots, 1-dimensional objects in 3-dimensional spaces, up to an equivalence relation (concordance) which is defined using surfaces and 4-dimensional manifolds. Despite many efforts, very little is known about the structure of the concordance group. This project will apply tools from Heegaard Floer theory to a number of aspects of knot concordance. We will study smooth concordance of topologically slice knots, concordance of two-component links, torsion in the knot concordance group. We will also generalize the current Heegaard Floer techniques by utilizing higher fold branched covers.
Topology is the study of abstract spaces, called manifolds, of various dimensions. Low dimensional topology focuses on spaces of dimensions up to four. Knot theory is the study of knottings of loops in 3-dimensional space. With a long history reaching back to the 19th century, it is fundamental to many areas of mathematics and physics, and also has applications in biology and chemistry. Knot concordance encompasses the study of a series of questions about knots and surfaces bounded by knots. Heegaard Floer techniques grew out of a synergistic interaction between mathematics and physics, specifically gauge theories from quantum physics. Research in the last 30 years has shown that low dimensional topology is in many ways more mysterious than topology in higher dimensions. The study of knot concordance reflects much of this complexity inherent in low dimensions, and advances in our understanding of concordance are likely to have far-reaching impact in topology.
