Taut foliations from knot diagrams

We prove that knots in $S^3$ satisfying certain diagrammatic properties are persistently foliar. As a consequence, all their non-trivial surgeries support coorientable taut foliations. All non-fibered two-bridge knots, as well as many pretzel and Montesinos knots, satisfy these diagrammatic properties. More generally, our result applies to arborescent knots defined by weighted planar trees with more than one vertex, provided that all weights have absolute value greater than one and at least one weight has absolute value greater than two. We also use this result to provide sufficient conditions for closures of 3-braids, and more generally odd-strand braids, to be persistently foliar. The ideas involved in the proof extend to links: as an application, we show that for all surgeries $M$ on the Borromean rings one has that $M$ is not an $L$-space if and only if it supports a coorientable taut foliation.