Augmentation curve for torus knots

Alexander Hock; Sergey Shadrin

Home/Archive/arXiv

arXiv · 2610.21585

Augmentation curve for torus knots

Alexander Hock·Sergey Shadrin

View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove a version of topological recursion for the correlation differentials of colored HOMFLY-PT polynomials of torus knots on the corresponding augmentation curves, following the proposal of Gu, Jockers, Klemm, and Soroush. To this end, we explicitly construct the spectral curve from the so-called auxiliary curve of Brini, Eynard, and Mariño and prove that it coincides with the augmentation curve for every torus knot with coprime $(r,s)$. Finally, our version of topological recursion proves the higher-genus closed-string prediction of the Aganagic--Vafa conjecture for all torus knot augmentation curves.

Explore related subjects

math-ph hep-th math.AG math.GT math.MP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Alexander Hock, Sergey Shadrin. 2026-10-15. Augmentation curve for torus knots. https://arxiv.org/abs/2610.21585

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Local conservation laws, symmetries and Hamiltonian structure of three-dimensional incompressible Euler flow

All smooth local conservation laws, symmetries and adjoint symmetries of three-dimensional homogeneous incompressible Euler flow are classified in the primitive velocity--pressure variables. Explicit dependence on time and position and arbitrary finite orders of mixed time and spatial derivatives are allowed. Up to trivial currents, the conservation laws consist of incompressibility, linear and angular momentum, kinetic energy, helicity and generalised time-dependent momentum. Every local adjoint symmetry is represented by a multiplier of these currents. Every finite-order local symmetry is equivalent on solutions to a classical point symmetry, resolving a conjecture of Olver. An explicit local first-order Hamiltonian structure of the primitive system is also established in the geometric theory of partial differential equations. It maps the energy multiplier to time translation; its kernel on local adjoint symmetries is spanned by the helicity and incompressibility multipliers. The proof first excludes higher-order symmetries by comparing their responses to transverse and harmonic characteristic directions in compatible derivative data. The Hamiltonian map and its kernel then determine all local adjoint symmetries from the symmetry classification. Exact multiplier identities and the correspondence between multipliers and current classes complete the conservation-law classification.

math-ph↗

arXiv2026

Generalized Frobenius Structures on Hurwitz Spaces, Their Open Extensions and Associated Principal Hierarchies

We construct generalized Frobenius structures on Hurwitz spaces of arbitrary genus using extended primary differentials. Through contour integrals involving the Cauchy kernel, we derive explicit dispersionless Lax representations for the principal hierarchies of these structures and their almost-dual counterparts. We also construct solutions to the open WDVV equations for both the original and almost-dual structures, and derive explicit principal hierarchies for the resulting rank-one extensions. Finally, we show that these constructions admit Shramchenko deformations.

math-ph↗

arXiv2026

The dynamics through the first two Lagrange necks in the small-mass restricted three-body problem

We study the planar circular restricted three-body problem (PCR3BP) for sufficiently small mass ratios, from low energies to slightly above the second Lagrange value. We show that retrograde orbits around the primaries and Lyapunov orbits at the necks form the binding of a finite energy foliation. We construct these foliations using the theory of pseudoholomorphic curves in symplectizations developed by Hofer, Wysocki and Zehnder. Slightly above the second Lagrange value, the dynamics is transverse to families of disks bounded by the retrograde orbits, and cylinders connecting the retrograde orbits to the Lyapunov orbits, forming a $3-2-3-2$ foliation. The proof uses the rotating Kepler problem and Hill's lunar problem to control the singular limit as the mass ratio $μ\to 0$. As a consequence of the foliation, both Lyapunov orbits have infinitely many homoclinic orbits.

math-ph↗