On detection probabilities of link invariants
Tuomas Kelomäki, Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz, Victor L. Zhang
TL;DR
The paper investigates how effective classical, quantum, and categorified knot invariants are at distinguishing alternating links, showing that many invariants are invariant under oriented mutation and thus have detection rates that decay exponentially with crossing number, yielding $Q(n)^{\%}_{AL}\in O(\delta^{n})$ and zero detection probability. It combines a rigorous mutation-based counting framework with large-scale knot-data experiments to assess practical performance, finding that polynomial and homological invariants perform similarly on knots. The results reveal fundamental limitations of widely used invariants for knot classification and suggest that distinguishing power requires mutation-sensitive or fundamentally different invariants. These insights have implications for both theoretical understanding and computational approaches in knot theory.
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.