Thursdays 11:00-12:30 in room 7421.
Organisers: Justin Roberts, Peter Teichner
We'll read a paper by Princeton graduate student Jacob Rasmussen, on
"Khovanov homology and the slice genus". It is a breakthrough paper,
proving the so called "Milnor conjectures" by purely combinatorial
means. These conjectures predict the 4-dimensional genus of torus
knots, and their only known proof involves Gauge theory, or more
precisely, Seiberg-Witten theory.
Rasmussen gives a proof using only methods from combinatorial skein
theory and a certain spectral sequence. He obtains a concordance
invariant for knots, which is absolutely mind blowing. It can show that
certain knots with trivial Alexander polynomial are not smoothly slice,
even though by Freedman's theorem they are topologically slice. Hence
the invariant sees the difference between smooth and topological
phenomena in dimension four.
As usual, the talks in this seminar will be given by the participants,
with two survey lectures at the beginning given by Justin and Peter.
Rasmussen's paper, supplemented by some survey articles, will be used
as reference for later talks.
The main paper is:
Khovanov homology and the slice genus, by Jacob Rasmussen
Two other papers needed in the proof are:
On Khovanov invariant for alternating links, by Eun Soo Lee
The support of the Khovanov's invariants for alternating knots, by Eun
Soo Lee
Two useful surveys on Khovanov homology are:
On Khovanov's categorification of the Jones polynomial, by Dror
Bar-Natan
Remarks on definition of Khovanov homology, by Oleg Viro
Finally, there is a potential connection with the gauge theory
invariants described in:
Heegaard diagrams and holomorphic disks, by Peter Ozsvath and Zoltan
Szabo
No specific background will be assumed of the audience. We invite all
interested grad students, visitors and faculty to attend.
Provisional schedule:
April 1: Peter Teichner Survey/organisational
meeting
April 8: Justin Roberts The Jones polynomial and TQFT
April 15: Ben Cooper Khovanov homology for knots
April 22: Tom Fleming Lee homology for knots
April 29: Gregg Musiker Khovanov homology and Reidemeister
moves
May 6: Henning Hohnhold Spectral sequences
May 13: Li Yu The Khovanov-to-Lee spectral sequence
May 20: Jana Comstock Rasmussen's invariant
May 27: Sean Raleigh Estimates for the 4-ball genus
June 3: Ryan Blair/Mike Gurvich Applications to the genus
of positive knots
justin@math.ucsd.edu
