nLab knots-quivers correspondence

Idea

The knots-quivers correspondence [KRSS19] is a certain identification between invariants of knots and data encoded by representations of corresponding quivers.

The correspondence has been conjectured from geometric engineering via intersecting D-brane models of both knot invariants and quiver gauge theory, motivated by comparing the framework of Ooguri-Vafa large $N$ duality [OV19] with the motivic generating series for symmetric quivers [R20][E20][KS20].

Beware that the correspondence is far from bijective: Most quivers correspond to no knot and most knots correspond to many different quivers at a time.

Related concepts

• McKay correspondence

• AGT correspondence

• 3d-3d correspondence

References

The original articles:

• Piotr Kucharski, Markus Reineke, Marko Stosic, Piotr Sułkowski, Knots-quivers correspondence, Adv. Theor. Math. Phys. 23 7 (2019) 1849-1902 [arXiv:1707.04017, doi:10.4310/ATMP.2019.v23.n7.a4]

• Piotr Kucharski, Markus Reineke, Marko Stosic, Piotr Sułkowski, BPS states, knots and quivers, Phys. Rev. D 96 121902 (2017) [arXiv:1707.02991, doi:10.1103/PhysRevD.96.121902]

Review:

• Piotr Sułkowski, Section 5 of: Quantum fields, strings, and physical mathematics, PoS 390 (2021) [arXiv:2104.03350, doi:10.22323/1.390.0042]

• Shivrat Sachdeva: A survey of knots and quivers [arXiv:2505.02059]

• Piotr Kucharski, Dmitry Noshchenko: Knot-quiver correspondence: a brief review [arXiv:2505.05668]

See also:

• Hirosi Ooguri, Cumrun Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000), 419-438 [arXiv:hep-th/9912123, doi:10.1016/S0550-3213(00)001188]

• Markus Reineke, Degenerate Cohomological Hall algebra and quantized Donaldson-Thomas invariants for m-loop quivers, [arXiv:1102.3978]

• Alexander I. Efimov, Cohomological Hall algebra of a symmetric quiver, Compositio Mathematica, 148 4 (2012) 1133-1146 [arXiv:1103.2736, doi:10.1112/S0010437X12000152]

• Maxim Kontsevich, Yan Soibelman: Cohomological Hall algebra exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Num. Theor. Phys. 5 (2011) 231-352 [arXiv:1006.2706]

Further discussion:

• Vivek Kr. Singh, S. Chauhan, A. Dwivedi, P. Ramadevi, B. P. Mandal, S. Dwivedi: Knot-Quiver correspondence for double twist knots [arXiv:2303.07036]

• Marko Stošić, Generalized knots-quivers correspondence [arXiv:2402.03066]

On relating the quivers appearing in $D=3$, $\mathcal{N}=2$ SYM and $D=4$, $\mathcal{N}=2$ SYM quiver gauge theories:

• Piotr Kucharski, Pietro Longhi, Dmitry Noshchenko, Sunghyuk Park, Piotr Sułkowski: Quivers and BPS states in $3d$ and $4d$ [arXiv:2508.09729]

• Piotr Kucharski: Quivers and BPS states in 3d and 4d, talk at CQTS (Oct 2025) [slides:pdf]