RESEARCH BLOG 5/11/04 CORNELL TOPOLOGY FESTIVAL 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 non-trivial 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 non-trivial representation 1(K(1/n)) SL2(C) must have non-cyclic 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 non-trivial knots (which must be satellites). In fact, this result implies that the A-polynomial 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- Collections: Mathematics