Gabai's property R theorem is:


If the 0-surgery manifold of a knot $K$ is homeomorphic to $S^1\times S^2$, then $K$ is the unknot.


Recently, 3-manifold topology has been developed rapidly by Agol, Wise and many other mathematicians.


Is there an another simple proof for property R conjecture?



Simple proof for property R conjecture
Property R was reproved by Gordon and Luecke in the course of solving the knot complement problem - see [Corollary 3.2](http://www.ams.org/journals/jams/1989-02-02/S0894-0347-1989-0965210-7/home.html). They prove the stronger result (as did Gabai) that zero-frame surgery on a knot is [irreducible](http://en.wikipedia.org/wiki/Prime_decomposition_(3-manifold)) (hence cannot be $S^1\times S^2$). 


Gabai actually proved something slightly stronger, which might account for the increased difficulty of the proof, namely that there is a taut finite-depth foliation which intersects the boundary of the knot complement transversely in a foliation of the boundary torus by longitudes. Capping off this foliation with a foliation by disks in the solid torus gives a taut foliation of the zero-framed surgery, hence irreducibility. 


Gordon and Luecke use many of the techniques of Gabai (thin position, Scharlemann cycles and generalizations), but omit the foliations and sutured manifold hierarchies. A technical simplification to the argument was [subsequently made by Walter Parry](http://www.ams.org/mathscinet-getitem?mr=1039537). 


