# On detection probabilities of link invariants

arXiv: https://arxiv.org/abs/2509.05574 · HTML: https://mlsanity.com/paper/2509.05574/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 52 references that are in the map.
- First posted on arXiv 6 Sep 2025.

## Builds on (most cited first, 4 of 4)

- [Big Data Approaches to Knot Theory: Understanding the Structure of the Jones Polynomial](https://mlsanity.com/paper/1912.10086/) (2019)
- [Big data approach to Kazhdan-Lusztig polynomials](https://mlsanity.com/paper/2412.01283/) (2024)
- [Mapper-type algorithms for complex data and relations](https://mlsanity.com/paper/2109.00831/) (2021)
- [Big data comparison of quantum invariants](https://mlsanity.com/paper/2503.15810/) (2025)

## Built on by (most cited first, 2 of 2)

- [On knot detection via picture recognition](https://mlsanity.com/paper/2510.06284/) (2025)
- [RL unknotter, hard unknots and unknotting number](https://mlsanity.com/paper/2603.07955/) (2026)

## Abstract (arXiv)

We prove that, for many standard link invariants, both the proportion of distinct invariant values and the detection probability among prime alternating links with at most n crossings decay exponentially in n, with an explicit universal rate. In fact, almost every such link belongs to an invariant fiber whose size is itself exponential in n. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. The companion website gives a much more detailed view of the data, including complete distributions of fiber sizes, separate alternating and non-alternating data, and topological data analysis.