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
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-