Abstract
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 $\mathcal{N}=2$ 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.
Highlights
Knots and quivers play an important role in high energy theoretical physics
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 interpreted as webs of dual three-dimensional N 1⁄4 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
Summary
Knots and quivers play an important role in high energy theoretical physics. Knots often arise in the context of topological invariance and can be related to physical objects—such as Wilson loops, defects, and Lagrangian branes—in gauge theories and topological string theory. Every permutohedron arises from a particular pattern of transpositions of elements of quiver matrices, or equivalently from some particular way of writing a generating function of colored superpolynomials for a given knot. There are typically several ways of writing a generating function of colored superpolynomials, which lead to different permutohedra connected by the quivers they share The other hand, these conditions can be expressed in terms of multicover skein relations that arise in the counting of holomorphic curves with boundaries on a Lagrangian brane in Calabi-Yau three-folds These connections provide a new link between homological invariants of knots, Gromov-Witten theory, and moduli spaces of quiver representations. If the number of equivalent quivers is large (say over 1 000) we provide a function which just counts the number of quivers and gives a list of symmetries
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