Persistently Foliar Composite Knots

A knot $\kappa$ in $S^3$ is persistently foliar if, for each boundary slope, there is a co-oriented taut foliation meeting the boundary of the knot complement transversely in a foliation by curves of that slope. For rational slopes, these foliations may be capped off by disks to obtain a co-oriented taut foliation in every manifold obtained by Dehn surgery on that knot. We first show that any composite knot with a persistently foliar summand is persistently foliar and that any nontrivial connected sum of fibered knots is persistently foliar. As an application, it follows that any composite knot in which each of two summands is a fibered, alternating or Montesinos knot is persistently foliar.