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.

11R23, 57K10, 57K14

arithmetic statistics, arithmetic topology, topological Iwasawa invariants, knot theory, analogies between number theory and topology

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Département de Mathématiques et de Statistique, Université Laval, Pavillion Alexandre-Vachon, 1045 Avenue de la Médecine, Québec, QC, G1V 0A6, Canada

Department of Mathematics, University of British Columbia, Vancouver BC, V6T 1Z2, Canada