Permutohedra for knots and quivers

TL;DR

This paper explores the non-bijective nature of the knots-quivers correspondence, revealing that equivalent quivers form permutohedral families and are interconnected through complex graph structures.

Contribution

It systematically characterizes when quivers are equivalent and shows they form permutohedra, linking these structures to dual 3d theories and knot homologies.

Findings

Equivalent quivers are characterized by specific conditions.

Families of equivalent quivers form permutohedra.

Graphs of equivalent quivers relate to dual 3d theories.

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 parameterized by vertices of a graph made of several permutohedra glued together. These graphs can be also interpreted as webs of dual 3d $\mathcal{N}=2$ theories. All these results are intimately related to properties of homological diagrams for knots, as well as to multi-cover skein relations that arise in counting of holomorphic curves with boundaries on Lagrangian branes in Calabi-Yau three-folds.