On double Hurwitz numbers with completed cycles

In this paper, we collect a number of facts about double Hurwitz numbers, where the simple branch points are replaced by their more general analogues: completed (r+1)-cycles. In particular, we give a geometric interpretation of these generalized Hurwitz numbers and derive a cut-and-join operator for completed (r+1)-cycles. We also prove a strong piecewise polynomiality property in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural ELSV/GJV-type formula, that is, an expression in terms of some intrinsic combinatorial constants that might be related to the intersection theory of some analogues of the moduli space of curves. The structure of these conjectural 'intersection numbers' is discussed in detail.