Branches, quivers, and ideals for knot complements

TL;DR

This paper extends the $F_K$ invariant, explores the knots-quivers correspondence, and connects $A$-polynomials with quiver representations, providing new insights into knot invariants, 3d-5d theories, and 3-manifold invariants.

Contribution

It introduces a generalized $F_K$ invariant associated to $A$-polynomial branches and links it to quiver generating series, expanding the framework of knot invariants and their algebraic structures.

Findings

Explicit expressions for $F_K$ invariants of simple knots.

Representation of $F_K$ invariants as quiver generating series.

Generalization of the quantum $a$-deformed $A$-polynomial and its classical limit.

Abstract

We generalize the $F_K$ invariant, i.e. $\widehat{Z}$ for the complement of a knot $K$ in the 3-sphere, the knots-quivers correspondence, and $A$-polynomials of knots, and find several interconnections between them. We associate an $F_K$ invariant to any branch of the $A$-polynomial of $K$ and we work out explicit expressions for several simple knots. We show that these $F_K$ invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using $R$-matrices. We generalize the quantum $a$-deformed $A$-polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter $a$, and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide $t$-deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d $\mathcal{N}=2$ theory $T[M_3]$ and to the data of the associated modular tensor category $\text{MTC} [M_3]$.