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Permutohedra for knots and quivers

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Piotr Sulkowski
Warsaw University
Die, 13/07/2021 - 14:00 - 15:30


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. I will explain that this phenomenon is generic rather than exceptional. First, I will present
conditions that characterize equivalent quivers. Then I will 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 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.

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