Property G and the $4$--genus

TL;DR

This paper investigates conditions under which knots in 3-manifolds preserve Thurston norm and fiberedness properties after zero surgery, linking these properties to the knot’s smooth 4-genus in a specific 4-manifold setting.

Contribution

It establishes that a knot has Property G if its smooth 4-genus in a certain 4-manifold is less than its Seifert genus, generalizing previous results.

Findings

If the smooth 4-genus is less than the Seifert genus, the knot has Property G.

Property G holds when the smooth 4-genus is zero, even in arbitrary closed 3-manifolds.

The result applies to rational homology spheres and certain 4-manifold constructions.

Abstract

We say a null-homologous knot $K$ in a $3$--manifold $Y$ has Property G, if the properties about 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.