A criterion for satellite 1-genus 1-bridge knots

Let $K$ be a knot in a closed orientable irreducible 3-manifold $M$. Suppose $M$ admits a genus 1 Heegaard splitting and we denote by $H$ the splitting torus. We say $H$ is a $1$-genus $1$-bridge splitting of $(M,K)$ if $H$intersects $K$ transversely in two points, and divides $(M,K)$ into two pairs of a solid torus and a boundary parallel arc in it. It is known that a $1$-genus $1$-bridge splitting of a satellite knot admits a satellite diagram disjoint from an essential loop on the splitting torus. If $M=S^3$ and the slope of the loop is longitudinal in one of the solid tori, then $K$ is obtained by twisting a component of a $2$-bridge link along the other component. We give a criterion for determining whether a given $1$-genus $1$-bridge splitting of a knot admits a satellite diagram of a given slope or not. As an application, we show there exist counter examples for a conjecture of Ait Nouh and Yasuhara.