Permutohedra for knots and quivers

The knots-quivers correspondence states that various characteristics of a knot are encoded in the corresponding quiver and the moduli space of its representations. However, this correspondence is not a bijection: more than one quiver may be assigned to a given knot and encode the same information. In this work we study this phenomenon systematically and show that it is generic rather than exceptional. First, we find conditions that characterize equivalent quivers. Then we show that equivalent quivers arise in families that have the structure of permutohedra, and the set of all equivalent quivers for a given knot is parametrized by vertices of a graph made of several permutohedra glued together. These graphs can be also interpreted as webs of dual three-dimensional N =2 theories. All these results are intimately related to properties of homological diagrams for knots, as well as to multicover skein relations that arise in the counting of holomorphic curves with boundaries on Lagrangian branes in Calabi-Yau three-folds.