Property G and the 4-genus.

We say a null-homologous knot K in a 3-manifold Y has Property G, if the Thurston norm and fiberedness of the complement of K is preserved under the zero surgery on K. In this paper, we will show that, if the smooth 4-genus of K\times \{0\} (in a certain homology class) in (Y\times [0,1])\#N\overline {\mathbb CP^2}, where Y is a rational homology sphere, is smaller than the Seifert genus of K, then K has Property G. When the smooth 4-genus is 0, Y can be taken to be any closed, oriented 3-manifold. [ABSTRACT FROM AUTHOR]