Knot exterior with all possible meridional essential surfaces

Since the work of Haken and Waldhausen, it is common to study 3-manifolds, and knot exteriors in particular, by their decomposition into submanifolds. A very important class of surfaces used in these decompositions are the essential surfaces. A particularly interesting occurrence is the existence of knots with the property that their exteriors have closed essential surfaces of arbitrarily high genus, which were originally given on a classic paper by Lyon. In this talk we will show the first examples of a stronger phenomenon: We show the existence of infinitely many knots where each exterior contains meridional essential surfaces of independently unbounded genus and number of boundary components. In particular, we construct examples of knot exteriors where each of which has all possible compact orientable surfaces embedded as meridional essential surfaces.