On detection probabilities of link invariants
Geometric Topology
2025-12-02 v2 Machine Learning
Quantum Algebra
Abstract
We prove that the detection rate of n-crossing alternating links by many standard link invariants decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. We also use a big-data approach to analyze knots and provide evidence that, for knots as well, these invariants exhibit the same asymptotic failure of detection.
Keywords
Cite
@article{arxiv.2509.05574,
title = {On detection probabilities of link invariants},
author = {Tuomas Kelomäki and Abel Lacabanne and Daniel Tubbenhauer and Pedro Vaz and Victor L. Zhang},
journal= {arXiv preprint arXiv:2509.05574},
year = {2025}
}
Comments
15 pages, many figures, revision containing stronger results, comments welcome