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The 2023 Ribbon Knots Project

This is a brief introduction to the 2023 Polymath Jr project that will be run by Jeffrey Meier and Alexander Zupan.

A knot is a closed loop in 3-dimensional space, considered up to smooth deformations that can bend and twist (but not break) the knot.  Every knot is the boundary of a surface in 3-space, and one way we can study knots is to understand the various surfaces they bound.  For example, the unknotted loop is the only knot that bounds an embedded disk, but there are lots of different knots bounding disks that can contain self-intersections, and when all of these self-intersections are of a particular form (as shown in the example below), the disk is called a ribbon disk, and the knot is called a ribbon knot.  In this project, we will examine connections between the minimal number of self-intersections in any ribbon disk bounded by a ribbon knot K, called the ribbon number of K, and other knot invariants, such as the knot determinant and Alexander polynomial of K.  No previous experience in knot theory or topology is necessary, but it may be helpful for participants to be familiar with basic linear algebra.

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