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Symmetric Union Presentations for Ribbon Links
(Alex Zupan's 2022 project)

A knot is a closed loop in 3-dimensional space, while a link is the union of several of these components.  An active area of research in low-dimensional topology involves investigating connections between topological spaces in dimensions 3 and 4.  One such avenue of inquiry examines the types of surfaces that a 3-dimensional knot can bound in 4-dimensional space.  A knot is slice, for example, if it bounds a disk in 4-dimensional space, and ribbon if it bounds an immersed disk with only a special type of self-intersection (ribbon intersections) in 3-space.  A nice exercise is to show that every ribbon knot is slice, but the converse is the well-known and longstanding slice-ribbon conjecture.

 

In 2006, Christoph Lamm initiated the study of symmetric union presentations of ribbon knots.  A symmetric ribbon presentation is a particular knot diagram that can be used to draw a restrictive type of ribbon disk.  Lamm also conjectured that every ribbon knot admits a symmetric union presentation and gathered data in support of this conjecture, which remains open today.  In this project, we will investigate a related concept, symmetric ribbon presentations for links.  Students will examine data to make the appropriate definition of a symmetric ribbon presentation, and then they will investigate the question of whether every ribbon link admits a such a presentation.  This difficult problem for knots may even be more tractable in the more general setting of links.

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A ribbon disk for a knot in 3-space:

ribbon.jpg
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