Collaborative Research: Algebraic Structures and Cohomology Theories Associated to Knottings

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New state-sum invariants for knots in 3-dimensional space and knotted surfaces in 4-dimensional space were defined, in a state-sum form, by the principal investigator and collaborators, using self-distributive operations called quandles and their colorings of knot and surface diagrams. The weights of the state-sum are derived from quandle cohomology theories. A number of applications to various properties of knots and surfaces have been discovered. The project investigates relationships among quandles, Lie algebras, coalgebras, crossed modules and their cohomology theories in order to develop applications such as manifold invariants. It also proposes to use geometric and diagrammatic methods to analyse specific categorifications, quantum groups, and cohomology theories.A knot is a circle situated in space. Surfaces in four-dimensional space can also be knotted. Knot theory studies such knotted circles and surfaces, and has provided models and applications to DNA theory, molecular configurations, and physics. Knot diagrams drawn on a piece of paper, and numerical quantities that are easily computable from diagrams, have been extensively used in knot theory. The principal investigators and their collaborators have developed algebraic systems from the knot diagrams that give a close reflection of the visual representations of knots. The algebra of these diagrams and related versions concisely encode deep connections among knots and physical systems. The current project develops new connections between the algebraic system of diagrams and other established algebraic systems (Lie algebras and crossed modules) that are closely associated with the standard model in physics. The techniques will also be applied in the context of categorification --- a process by which identity is replaced by an instruction of how to identify.