Property G and the 4-genus

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