Four-dimensional Knot Theory, Math 297B
This course is an introduction to knot theory from a 4-dimensional point of view.
A knot is a circle embedded in 3-space which is considered trivial (or unknotted) if it bounds a disk embedded in 3-space. Four-dimensional knot theory studies a weaker question, namely whether the knot is slice, which means that it bounds a disk embedded in 4-space.
After introducing the basic notions, including ways of picturing disks in 4-space, we will define the Alexander polynomial of a knot in terms of the fundamental group of the knot complement. We will explain how the Alexander polynomial is related to a disk in 4-space.
This will then be generalized to a noncommutative setting, using solvable quotients of the knot group and the Ore-localization of their group rings. Finally, we'll see how traces on von Neumann algebras can be used to extract real numbers that have to vanish for all slice knots but are nonzero for many simple knot types.
The course is based on the paper
Knot concordance, Whitney towers and von Neumann signatures
by Tim Cochran, Kent Orr and Peter Teichner
Prerequisites: Linear algebra and some knowledge of the fundamental group.
Winter quarter 2001, class starts Jan. 10 and meets
Mo. and We. 2:00-3:30, APM 7218.