This course on Khovanov homology and surfaces in four-manifolds aims to teach students about the applications of gauge theory, exotic smooth structures, the differential on Khovanov homology, the Rasmussen invariant, and various constructions of homotopy 4-spheres. The course covers topics such as surfaces in 4-manifolds, slice genus, knot trace, Gluck twists, and special RBG links. The teaching method includes lectures, examples, computer experiments, and proofs of old results. This course is intended for individuals interested in advanced topics in mathematics, particularly in the field of topology and knot theory.
Overview
Syllabus
Intro
Outline
Four dimensions
Exotic smooth structures
Applications of gauge theory
Surfaces in 4-manifolds
Surfaces in B4
Slice genus
The differential on Khovanov homology
More on Khovanov homology
The Rasmussen invariant
New proofs of old results
A new application
The knot trace
New applications
A possible approach to SPC4
Gluck twists
A negative result
A more positive result
Another construction of homotopy 4-spheres
Knots with the same 0-surgeries
Special RBG links
Slides From a special RBG link we obtain a knot ko by sliding Gover Runtil
An example
Computer experiments
Possibly slice knots
More examples
Taught by
Joint Mathematics Meetings
