We prove relative versions of the symplectic capping theorem and sufficiency of Giroux’s criterion for Stein fillability and use these to study the 4-genus of knots. More precisely, suppose we have a symplectic 4-manifold $X$ with convex boundary and a symplectic surface $\Sigma$ in $X$ such that $\delta\Sigma$ is a transverse knot in $\delta X$. In this paper, we prove that there is a closed symplectic 4-manifold $Y$ with a closed symplectic surface $S$ such that $(X, \Sigma)$ embeds into $(Y, S)$ symplectically. As a consequence we obtain a relative version of the symplectic Thom conjecture. We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in $\mathbb S^3$. Further, we give a criterion for quasipositive fibered knots to be strongly quasipositive.
Volume 130, 2020
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