Data

• Title: On detection probabilities of link invariants
• Authors: Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz and Victor L. Zhang
• Status: preprint. Last update: Wed, 5 Feb 2025 03:03:57 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 link invariants insensitive to oriented mutation decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular to quantum invariants such as the Jones or HOMFLYPT polynomials. We also use a big data approach to analyze several borderline cases (e.g. integral Khovanov or HOMFLYPT homologies), where

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.