On Weakly Reducible SD-Splittings of Inner Genus 1

Abstract:

      Let $(M; H_{1},H_{2};F_{0})$ be a SD-splitting for bordered 3-manifold $M$. The splitting is reducible (weakly reducible, respectively) if there exist essential disks $D_1\subset H_1$ and $D_2\subset H_2$ such that $\partial D_1,\partial D_2\subset F_0$ and $\partial D_1=\partial D_2$ ($\partial D_1\cap\partial D_2=\emptyset$, respectively). A SD-splitting $(M; H_{1},H_{2};F_{0})$ for bordered 3-manifold $M$ is of inner genus 1 if $F_0$ is a punctured torus. In the present paper, we show that a weakly reducible SD-splitting of inner genus 1 is either reducible or bilongitudional.