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Slice and ribbon knots

Student-Run Applied & Math Seminar

Speaker: Carson Rogers, UC Davis
Location: 2112 MSB
Start time: Wed, Apr 23 2014, 1:10AM

One approach to understanding the geometry of knots lies in studying the surfaces that they bound in space. In the simplest case, a knot K bounds a disk embedded in 3-dimensional space, in which case it is unknotted. Thus, in some next, the "next best thing" to the unknot is a knot that, while sitting in S^3, bounds a nicely embedded disk in the 4-dimensional unit ball. Such knots are known as slice knots, as they are precisely the knots that arise as "slices" of 2-dimensional spheres embedded in R^4. Such spheres are related to substantial open problems in 4-dimensional topology, so topologists would like to have a more concrete description of slice knots. Ideally, we want to find a nice way to detect sliceness directly from a planar diagram of a knot. As it turns out, nearly all slice knots bound immersed disks in 3-dimensional space that have the nicest possible type of self-intersections, called "ribbon singularities." Naturally, such knots are called ribbon knots. All ribbon knots are slice, as the singularities of a ribbon disk can be resolved in 4-dimensional space to obtain an embedded disk. However, it is not known if all slice knots are ribbon. The slice-ribbon conjecture holds that this is the case, and has been open since it was first posed in the 1960's. In this talk, I aim to give the audience a good idea of how to think about these kinds of knots, and outline the progress that has been made on the problem. If time allows, I will also describe the relevant research that I am involved in.