The proposal deals with algebraically defined invariants of links and link cobordisms, known as link homology theories. In a typical instance, such a theory assigns bigraded homology groups to a link and a homomorphism of groups to a link cobordism. Various quantum invariants of links gain a novel interpretation as the Euler characteristics of these theories. Link homology theories can be though of as four-dimensional topological quantum field theories, restricted to links in the 3-space and link cobordisms. Applications of link homology include a combinatorial proof of the Milnor conjecture, efficient estimates of the Thurston-Bennequin invariant of knots, and nonexistence of taut foliations on many double branched covers of knots. The proposal's aim is to deepen understanding of known link homology theories and their interrelations, including the ones between Khovanov homology and knot Floer homology, find more applications of link homology to low-dimensional topology, discover link homology theories categorifying a variety of quantum link invariants, extend knot Floer homology to tangles, find categorifications of quantum 3-manifold invariants, and explore the relations of link homology to various branches of mathematics, including homological algebra and representation theory. The principal investigators will place a significant emphasis on experimental aspects of the theory, including writing efficient programs to compute link homology.
A link is a collection of knotted circles located in the usual space that we live in, and 3-manifolds are geometric object modeled on that space. One of the main problems in studying links and manifolds is their classification, that is, finding methods to tell them apart from each other. From the very beginning, algebra played an important role in distinguishing links and manifolds, since algebraic objects are in general easier to compare to each other than geometric ones. Categorification, also known as link homology, is a procedure of replacing a known link invariant with a family of algebraic objects that significantly enhance the original invariant. This procedure was recently developed by the lead principal investigator, who found categorifications of several polynomial invariants of links. These categorifications are relatively easy to describe, but rather challenging to compute for a given link. The aim of this project is to better understand existing link homology theories and their interrelations, as well as to find new ones. The principal investigators also plan to write efficient programs for computing link homology. Link homology is a young and quickly growing field. It lies on the crossroads of research in 3- and 4-dimensional topology, symplectic topology, homological algebra, and representation theory. The recent explosion of interest in link homology, its structure and applications, is likely to continue in the foreseeable future.
