Knots in 3-space are called concordant if they cobound an embedded annulus in the product of 3-space with a closed interval. The set of concordance classes forms an abelian group; this project investigates the structure of this concordance group. There are three major facets to the project: (1) Developing a better understanding of known concordance invariants and discovering new invariants, (2) Applying these invariants to classify important families of knots, and (3) Using these results to unravel the underlying structure of the concordance group.
The perspective of classical physics is three-dimensional, focusing on the three spatial dimensions in which we live. Modern physics has a higher-dimensional perspective, for instance including time to create a four-dimensional model of the universe. In the 1960s it was recognized that four-dimensional spaces can be built by gluing together simpler pieces; the way these pieces are glued together can be represented by knots, with the complexity of the resulting space reflected in the complexity of the knots that occur. This proposal is focused on understanding knotting from a four-dimensional perspective. In particular, new tools will be developed, and these will be applied to complete our four-dimensional analysis of low- crossing number knots.
Professor Kirk and Livingston's recent work, during the years 2010 through June, 2014, supported by NSF grant 1007196, includes the publication of roughly 15 papers in refereed journals. One other paper has been accepted for publication and five are currently under submission. Kirk and Livingston have continued to serve as advisors to graduate students: during the period of support, two students received Ph.D. degrees, three others are carrying out beginning research, and a sixth is starting advanced study. The website "KnotInfo," with development supported by the grant, provides knot theoretic content to users at up to 100 domestic and international educational institutions each month. Many of the most significant results of Professors Kirk and Livingston's research concern topics in low-dimensional geometric topology. In their joint published work, they have addressed problems in the study of four-dimensional aspects of classical knotting captured by the general notion of "knot concordance." Their jointly written papers include one paper written with Hedden, another with Herald, and a third with Collins. These address different aspects of the problem of distinguishing knots (up to concordance) in what until now had been especially intractible situations. The latest work with Collins completes a classification result for knots with eight or fewer crossings. In work done independently of his joint work with Livingston, Professor Kirk has undertaken research studying low dimensional manifolds via geometric techniques such as Chern Simons theory and symplectic structures (in papers with Hedden and with Baldridge) and via algebraic methods studying representation spaces (in papers with Herald-Hedden and with Fukumoto-Pinzon). In joint work with Baldridge, Kirk introduced a new symplectic surgery method and used it to construct new families of symplectic Calabi-Yau 6-manifolds. In work done independently from that with Kirk, Professor Livingston has completed research with Hedden-Ruberman and Hedden-Kim which has explored large new families of knots that, from a purely topological perspective, are similar to the unknot (formally, are slice knots) but from the perspective of differential topology and concordance are independent. Work with Gilmer examined the role of non-orientable surfaces, such as the Mobius band, in knot theory. His most recent work, being done jointly with Borodzik, is exploring the application of methods from smooth topology and knot concordance to studying algebraic curves and their singularities. The website KnotInfo provides the values of roughly 70 knot invariants for the nearly 3,000 prime knots of 12 or fewer crossings. These values are updated to reflect current research. As an example of usage, during April, 2014, researchers and students at roughly 50 domestic colleges and universities accessed the site; international usage included 15 colleges in Japan, 12 in France, and eight in the UK.
