Topological Iwasawa invariants and Arithmetic Statistics

TL;DR

This paper explores topological analogues of Iwasawa invariants in link complements, providing explicit detection criteria, statistical results on their distribution, and conjectures linking topology with arithmetic properties.

Contribution

It introduces explicit criteria for detecting topological Iwasawa invariants and establishes statistical results on their distribution in 2-bridge links, connecting topology with arithmetic statistics.

Findings

Density of 2-bridge links with vanishing μ-invariant and λ-invariant 1 is (1 - 1/p)

Unconditional proof of statistical distribution of invariants for fixed prime p

Conjecture that the μ-invariant vanishes for almost all 2-bridge links

Abstract

Given a prime number $p$, we study topological analogues of Iwasawa invariants associated to $\mathbb{Z}_p$-covers of the $3$-sphere that are branched along a link. We prove explicit criteria to detect these Iwasawa invariants, and apply them to the study of links consisting of $2$ component knots. Fixing the prime $p$, we prove statistical results for the average behaviour of $p$-primary Iwasawa invariants for $2$-bridge links that are in Schubert normal form. Our main result, which is entirely unconditional, shows that the density of $2$-bridge links for which the $\mu$-invariant vanishes, and the $\lambda$-invariant is equal to $1$, is $(1-\frac{1}{p})$. We also conjecture that the density of $2$-bridge links for which the $\mu$-invariant vanishes is $1$, and this is significantly backed by computational evidence. Our results are proven in a topological setting, yet have arithmetic significance, as we set out new directions in arithmetic statistics and arithmetic topology.