Gluons and Knots
Abstract
We argue that the scattering amplitude of a single minus helicity gluon with arbitrarily many positive helicity gluons is topological. The amplitude is constant in chambers of kinematic space, its resonances are located at walls and its factorization is determined by wall-crossing. Yangian invariance dictates the amplitude to be determined by a 2d null Wilson loop that can be lifted into a positive and transverse 3d link. By reinterpreting wall-crossing and the Weinberg soft theorem as skein relations and Reidemeister moves of the subknots, we construct a state-counting description consisting of adding the `corners' of their HOMFLY-PT polynomials. Beyond the classical limit, we argue that this provides a conjectured but precise connection to topological field and string theory. The new formula is rigorously derived via a Lean-formalized dependency graph (DAG) as detailed in a companion paper, laying out the precise assumptions including Reidemeister III invariance. It is further tested exhaustively, including novel `star' kinematics where the amplitude and state count are shown to become Catalan numbers, in agreement with knot theory literature.
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BibTeX
@misc{guevara2026gluons,
title = {{Gluons and Knots}},
author = {Alfredo Guevara},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10673},
eprint = {2610.10673},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10673},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gluons and Knots
AU - Alfredo Guevara
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10673
UR - https://arxiv.org/abs/2610.10673
AB - We argue that the scattering amplitude of a single minus helicity gluon with arbitrarily many positive helicity gluons is topological. The amplitude is constant in chambers of kinematic space, its resonances are located at walls and its factorization is determined by wall-crossing. Yangian invariance dictates the amplitude to be determined by a 2d null Wilson loop that can be lifted into a positive and transverse 3d link. By reinterpreting wall-crossing and the Weinberg soft theorem as skein relations and Reidemeister moves of the subknots, we construct a state-counting description consisting of adding the `corners' of their HOMFLY-PT polynomials. Beyond the classical limit, we argue that this provides a conjectured but precise connection to topological field and string theory. The new formula is rigorously derived via a Lean-formalized dependency graph (DAG) as detailed in a companion paper, laying out the precise assumptions including Reidemeister III invariance. It is further tested exhaustively, including novel `star' kinematics where the amplitude and state count are shown to become Catalan numbers, in agreement with knot theory literature.
ER -