# BPS counting for knots and combinatorics on words

@article{Kucharski2016BPSCF, title={BPS counting for knots and combinatorics on words}, author={Piotr Kucharski and Piotr Sułkowski}, journal={Journal of High Energy Physics}, year={2016}, volume={2016}, pages={1-39} }

A bstractWe discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating functions (Hilbert-Poincaré series) are solutions to those equations and reproduce generating series that encode BPS invariants. Furthermore, BPS invariants in question are…

## 18 Citations

Donaldson-Thomas invariants, torus knots, and lattice paths

- MathematicsPhysical Review D
- 2018

In this paper, we find and explore the correspondence between quivers, torus knots, and combinatorics of counting paths. Our first result pertains to quiver representation theory—we find explicit…

Nahm sums, quiver A-polynomials and topological recursion

- MathematicsJournal of High Energy Physics
- 2020

Abstract
We consider a large class of q-series that have the structure of Nahm sums, or equivalently motivic generating series for quivers. First, we initiate a systematic analysis and…

Topological strings, strips and quivers

- MathematicsJournal of High Energy Physics
- 2019

A bstractWe find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as…

Knots-quivers correspondence

- MathematicsAdvances in Theoretical and Mathematical Physics
- 2019

We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing…

Checks of integrality properties in topological strings

- Mathematics
- 2017

A bstractTests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent…

Integrality of Framing and Geometric Origin of 2-functions

- Mathematics
- 2017

We say that a formal power series ∑ anz n with rational coefficients is a 2-function if the numerator of the fraction an/p − pan is divisible by p for every prime number p. One can prove that…

On explicit formulae of LMOV invariants

- MathematicsJournal of High Energy Physics
- 2019

Abstract
We started a program to study the open string integrality invariants (LMOV invariants) for toric Calabi-Yau 3-folds with Aganagic-Vafa brane (AV-brane) several years ago. This paper is…

Topological strings, quiver varieties, and Rogers–Ramanujan identities

- Mathematics
- 2017

Motivated by some recent works on BPS invariants of open strings/knot invariants, we guess there may be a general correspondence between the Ooguri–Vafa invariants of toric Calabi–Yau 3-folds and…

Integrality of Framing and Geometric Origin of 2-functions

- Mathematics
- 2017

We say that a formal power series $\sum a_n z^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p}-p^2 a_n$ is divisible by $p^2$ for every prime number $p$. One…

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