 
Summary: RESEARCH BLOG 5/11/04
CORNELL TOPOLOGY FESTIVAL
Last weekend was the Cornell Topology Festival. Kronheimer talked
about his work with Mrowka solving property P. In fact, they prove
that if K S3
is a knot, then 1(K(1/n)) has a nontrivial representa
tion to SU(2), n = 0 (where K(1/n) means 1/n Dehn surgery on K).
This implies that the character variety of 1(S3
K) SL2(C) is non
trivial, since SU(2) SL(2, C). To see this, note that H1(K(1/n)) = 0.
Thus, any nontrivial representation 1(K(1/n)) SL2(C) must have
noncyclic image. This implies that the character variety of 1(S3
K)
distinguishes K from the unknot U, since 1(S3
 U) = Z. This was
known for hyperbolic and torus knot complements, but not for general
nontrivial knots (which must be satellites).
In fact, this result implies that the Apolynomial distinguishes non
trivial knots from the unknot. Nathan Dunfield and Stavros Garoufa
lidis are writing up this result, as well as Boyer and Zhang indepen
