JSJ decompositions of knot exteriors, Dehn surgery and the L-space conjecture

In this article, we apply slope detection techniques to study properties of toroidal 3-manifolds obtained by performing Dehn surgeries on satellite knots in the context of the L-space conjecture. We show that if K is an L-space knot or admits an irreducible rational surgery with non-left-orderable fundamental group, then the JSJ graph of its exterior is a rooted interval. Consequently, any rational surgery on a composite knot has a left-orderable fundamental group. This is the left-orderable counterpart of Krcatovich's result on the primeness of L-space knots, which we reprove using our methods. Analogous results on the existence of co-orientable taut foliations are proved when the knot has a fibred companion. Our results suggest a new approach to establishing the counterpart of Krcatovich's result for surgeries with co-orientable taut foliations, on which partial results have been achieved by Delman and Roberts. Finally, we prove results on left-orderable p/q-surgeries on knots with p small.