Spheres can knot in 4 dimensions

Geometry Topology Student Seminar
Wednesday, September 6, 2023 - 2:00pm for 1 hour (actually 50 minutes)
Skiles 006
Sean Eli – Georgia Tech – seaneli@gatech.edu
Sierra Knavel

You are probably familiar with the concept of a knot in 3 space: a tangled string that can't be pushed and pulled into an untangled one. We briefly show how to prove mathematical knots are in fact knotted, and discuss some conditions which guarantee unknotting. We then give explicit examples of knotted 2-spheres in 4 space, and discuss 2-sphere version of the familiar theorems. A large part of the talk is practice with visualizing objects in 4 dimensional space. We will also prove some elementary facts to give a sense of what working with these objects feels like. Time permitting we will describe know knotted 2-spheres were used to give evidence for the smooth 4D Poincare conjecture, one of the guiding problems in the field.