A survey of knots and quivers

TL;DR

This survey reviews the development of knot polynomials, their categorification, and the emerging connections between knots and quivers, highlighting physical and geometric insights into knot invariants and their unification.

Contribution

It provides a comprehensive overview of knot homologies, the knot-quiver correspondence, and the physical and geometric perspectives that unify various knot invariants.

Findings

Unification of knot homologies via superpolynomials.

Insight into the knot-quiver correspondence and BPS state integrality.

Analysis of the 3d $ ext{N}=2$ theory $T[Q_K]$ related to quivers.

Abstract

This survey explores knot polynomials and their categorification, culminating in the homological invariants of knots. We begin with an overview of classical knot polynomials, progressing towards the superpolynomial and its role in unifying various knot homologies. Along the way, we provide physical and geometric insights into the unification of the $sl(N)$ Khovanov-Rozansky and the knot Floer homology. We then turn our attention to the intriguing correspondence between knots and quivers, examining how this perspective sheds light on the integrality of BPS states encoded in the Labastida-Mari\~no-Ooguri-Vafa (LMOV) invariants. We will further investigate the knot-quiver correspondence from a physics and geometric side and study the 3d $\mathcal{N}=2$ theory $T[Q_K]$ for the quivers.