Dehn surgery and A-polynomial for knots

Abstract

The Property P conjecture states that the 3-manifold Yr obtained by Dehn surgery on a non-trivial knot in S3 with surgery coefficient r ∈ Q has the non-trivial fundamental group (so not simply connected). Recently Kronheimer and Mrowka provided a proof of the Property P conjecture for the case r = ±2 that was the only remaining case to be established for the conjecture. In particular, their results show that the two phenomena of having a cyclic fundamental group and having a homomorphism with non-cyclic image in SU(2) are quite different for 3-manifolds obtained by Dehn fillings. In this paper we extend their results to some other Dehn surgeries via the A-polynomial, and provide more evidence of the ubiquity of the above mentioned phenomena.