Searching for slice-ribbon counterexamples

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Primitive people are said to have believed that the world was flat. This was a reasonable assumption based on local data. Low-dimensional topologists study 3- and 4-dimensional spaces known as manifolds in order to understand the possible geometry and topology of the physical universe and spacetime that we inhabit. For the purposes of this project, a knot is a smooth closed curve --- a loop --- in 3-dimensional space. Knots are fundamental objects which appear in many areas of science. In particular they are the building blocks for low-dimensional topology: every 3- or 4-dimensional manifold can be described using a Kirby diagram involving one or more knots. Understanding surfaces in 4-dimensional space whose boundary is a given knot is also fundamental for 4-dimensional topology, and is closely related to the study of line fields in nature such as magnetic field lines and fluid vortices.This ambitious project will develop, and implement in a computer program, a new approach to producing knotted 2-dimensional spheres in 4-dimensional space, and slices of those 2-spheres which are knotted circles in 3-dimensional space which bound 2-dimensional disks in 4-space. A longstanding conjecture due to Fox predicts that such slice knots in fact bound a special kind of disk called a ribbon disk which can be seen in 3-dimensional space. Our aim is to settle this conjecture by finding a slice knot which cannot bound a ribbon disk.