Order-detection, representation-detection, and applications to cable knots

Given a 3 3 -manifold M M with multiple incompressible torus boundary components, we develop a general definition of order-detection of tuples of slopes on the boundary components of M M . In parallel, we arrive at a general definition of representation-detection of tuples of slopes, and show that these two kinds of slope detection are equivalent—in the sense that a tuple of slopes on the boundary of M M is order-detected if and only if it is representation-detected. We use these results, together with new “relative gluing theorems,” to show how the work of Eisenbud-Hirsch-Neumann, Jankins-Neumann and Naimi can be used to determine tuples of representation-detected slopes and, in turn, the behaviour of order-detected slopes on the boundary of a knot manifold with respect to cabling. Our cabling results improve upon work of the first author and Watson, and in particular, this new approach shows how one can use the equivalence between representation-detection and order-detection to derive orderability results that parallel known behaviour of L-spaces with respect to Dehn filling.