Topological Iwasawa invariants and Arithmetic Statistics

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.