Through joint and individual projects, Livingston, Orr and Turaev investigate smooth and topological concordance and cobordism of knots in dimension three, quantum topology and the theory of virtual knots and strings. The relation of cobordism fundamentally underlies the topology of manifolds, dating from the seminal discoveries of Pontrjagin, Thom, Fox and Milnor. The principal goal of the proposal is to deepen the recently introduced geometric and algebraic methods in the theory of knot cobordism in dimension three and to extend this theory to a wider class of topological objects including virtual knots and virtual strings. The novelty of our approach lies in a new combination of methods and techniques developed in classical knot theory and in their extension to cobordism and concordance of virtual knots and links. This should lead to a better understanding of the fundamental topological features of knots, links, 3- manifolds, and 4-manifolds. Principal Investigators anticipate a better understanding of the interplay between the fundamental groups of 3-manifolds and 4- manifolds. The proposed study should open new perspectives for applications of methods of knot theory and low-dimensional topology in other areas of mathematics and theoretical physics, including Teichmuller theory, topological quantum field theory, and combinatorial theory of words.
Classical knot theory attempts to understand and classify closed loops in three dimensional space. How many essential ways can circles sit in space, and how do we distinguish these mathematically? Knots model genetic structures and chemical bonds. Deformations of knots through four dimensional space encode the fundamental underpinnings of four dimensional space through the surgery theory classification tools for four manifolds.
