Generation of sutured manifolds

Given a compact, oriented, connected surface $F$, we show that the set of connected sutured manifolds $(M,\gamma )$ with $R_{\pm }(\gamma )\cong F$ is generated by the product sutured manifold $(F,\mathrm{\partial }F)\times [0,1]$ through surgery triads. This result has applications in Floer theories of $3$–manifolds. The special case when $F=D^2$ or $S^2$ has been a folklore theorem, which has already been used by experts before.