Knot-quiver correspondence: a brief review

TL;DR

The note surveys the knot-quiver correspondence, establishing that the coloured HOMFLY-PT generating series of a knot equals the partition function of a symmetric quiver after a precise change of variables, thereby equating knot invariants with DT data of quivers. It interprets this link physically via open topological strings and LMOV/BPS invariants, and develops practical tools such as quiver diagonalization and the use of m-loop quivers to efficiently compute DT invariants and their LMOV counterparts. It also extends the correspondence to knot complements through the FK invariants $F_K(x,q)$, showing how quantum A-polynomials and quiver data align, with concrete examples for the unknot and figure-eight. The paper highlights equivalence structures among quivers (unlinking and permutohedra) and points to rich future connections between quiver combinatorics, low-dimensional topology, and categorification of 3-manifold invariants.

Abstract

This note is an overview of the knot-quiver correspondence, which relates symmetric quivers and their partition functions, a.k.a. motivic Donaldson-Thomas generating series, to quantum invariants of knots and links in $S^3$.

Figures (5)

• Figure 1: Knots $3_1$, $4_1$, $5_1$ and $5_2$ (generated by KnotScape)
• Figure 2: Unlinking in a nutshell: removal of a pair of arrows in a symmetric quiver (as shown on the bottom) amounts to skeining the corresponding pair of holomorphic discs (top), resulting in an extra node and a loop.
• Figure 3: Examples of permutohedra $\Pi_n$, shown as planar graphs.
• Figure 4: Permutohedra graphs for knots $5_1,7_1,9_1$ (from top to bottom). Every vertex corresponds to an equivalent quiver, while the edge between two vertices corresponds to a transposition of a pair of arrows (different colours correspond to different transpositions).
• Figure 5: A real-life model of $S^3\setminus 4_1$ by Henry Segerman.

Theorems & Definitions (1)

• Conjecture 5.1.1