Permutohedra for knots and quivers

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.

arXiv comment72 pages, 36 figures

References in corpus (18)

• Lectures on 0/1-polytopes
• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Knot Invariants and Topological Strings
• Super-A-polynomial for knots and BPS states
• Cohomological Hall algebra of a symmetric quiver
• Volume Conjecture: Refined and Categorified
• 3d analogs of Argyres-Douglas theories and knot homologies
• Knots-quivers correspondence
• BPS states, knots and quivers
• Topological strings, strips and quivers
• Donaldson-Thomas invariants, torus knots, and lattice paths
• 3d-3d Correspondence Revisited
• $\hat{Z}$ at large $N$: from curve counts to quantum modularity
• Multi-cover skeins, quivers, and 3d $\mathcal{N}=2$ dualities
• Quivers for 3-manifolds: the correspondence, BPS states, and 3d $\mathcal{N}$=2 theories
• Higher rank $\hat{Z}$ and $F_K$
• Rational links and DT invariants of quivers
• Branes, quivers and wave-functions