Data

• Title: On detection probabilities of link invariants
• Authors: Tuomas Kelomäki, Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz and Victor L. Zhang
• Status: preprint. Last update: Sat, 29 Nov 2025 21:42:22 UTC
• Code and (possibly empty) Erratum: Click and Click
• ArXiv link: https://arxiv.org/abs/2509.05574
• LaTex Beamer presentation: Slides1, Slides2

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.

A few extra words

Here is the essentially self-explanatory main result:

Maybe even more importantly, we also provide big data evidence that this phenomena is true in more generality.