 
Summary: RESEARCH BLOG 3/10/04
Yesterday,Michael Khovanov gave a talk here at UIC. He explained
his definition of a graded homology for links Hi,j
, whose Euler char
acteristic gives the Jones polynomial. He discovered this by trying
to "categorify" the definition of the Jones polynomial via the Kauff
man bracket. Since his discovery, he and others have found cate
gorifications of other quantum invariants of knots. Khovanov plot
ted log(rk( i,j Hi,j
)) versus the hyperbolic volume, and notes that
the plot looks roughly linear (see his paper). He was inspired to do
this by Dunfield's experiment, plotting log JK(1) vs. volume. The
correlation looks much better in Khovanov's plot, especially for non
alternating knots, but this seems quite mysterious. I speculate that it
might also be interesting to do the same plot with the volume of the
2fold branched cover of the knot instead.
After Khovanov's talk, Paul Seidel described recent work of Jacob
Rasmussen. Rasmussen gives a completely combinatorial proof of Mil
nor's conjecture. The 4ball genus of a knot K S3
is the minimal
