Joel Foisy

A Survey of Intrinsically Linked and Intrinsically Knotted Graphs

Abstract:

Take 6 points in space, and connect every possible pair of points by non-intersecting arcs. In the 1980s, Conway-Gordon and Sachs proved that no matter how the points are connected, two non-splittably linked loops will form. We say that the complete graph on six vertices is intrinsically linked. Conway and Gordon also proved that the complete graph on seven vertices is intrinsically knotted. Mathematicians have since attempted to classify all intrinsically linked and intrinsically knotted graphs. In the 1990s, Robertson, Seymour and Thomas classified the complete set of "minor-minimal" intrinsically linked graphs. Their proof is difficult, and intrinsically knotted graphs have been even more difficult to classify.

In this talk, we will survey some known results and open questions about intrinsically linked and intrinsically knotted graphs. There will be a lot of pictures.

Biography:

Joel Foisy was introduced to mathematics research while a student at Williams College, participating in the SMALL Geometry group under Frank Morgan. He went on to obtain his doctorate in mathematics in 1996, studying geometric topology at Duke University under John Harer. Since 1996, he has been teaching at SUNY Potsdam. For 16 summers, he has had the privilege of working with students in a summer REU program, held jointly by SUNY Potsdam and Clarkson University. Most of those summers have been spent studying intrinsically linked and knotted graphs.