# Topological Iwasawa invariants and Arithmetic statistics

Cédric Dion  Département de Mathématiques et de Statistique, Université Laval,
Pavillion Alexandre-Vachon, 1045 Avenue de la Médecine, Québec, QC, Canada G1V
0A6<EMAIL_ADDRESS>and  Anwesh Ray  Department of Mathematics
University of British Columbia
Vancouver BC, Canada V6T 1Z2<EMAIL_ADDRESS>
###### 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.

###### Key words and phrases:

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

###### 2020 Mathematics Subject Classification:

11R23, 57K10, 57K14

##  1\. Introduction

Let $p$ be a fixed odd prime number and $K$ be a number field. Iwasawa studied
growth patterns of $p$-primary class groups in certain towers of number field
extensions of $K$. More precisely, a $\mathbb{Z}_{p}$-extension of $K$ is a
Galois extension $K_{\infty}$ of $K$ with Galois group
$\operatorname{Gal}(K_{\infty}/K)\simeq\mathbb{Z}_{p}$. For any positive
integer $n$, let $K_{n}\subset K_{\infty}$ be the unique extension of $K$ with
$[K_{n}\mathrel{\mathop{\mathchar 58\relax}}K]$, and let
$\operatorname{Cl}_{p}(K_{n})$ be the $p$-Sylow subgroup of the class group of
$K_{n}$. Iwasawa [Iwa73] showed that for $n\gg 0$,

(1.1) $\\#\operatorname{Cl}_{p}(K_{n})=p^{p^{n}\mu+n\lambda+\nu},$

where $\mu,\lambda\in\mathbb{Z}_{\geq 0}$ and $\nu\in\mathbb{Z}$ are Iwasawa
invariants associated to the $\mathbb{Z}_{p}$-extension $K_{\infty}/K$. The
characterization of Iwasawa invariants is the subject of much mystery and
speculation. Letting $K_{\operatorname{cyc}}/K$ be the _cyclotomic
$\mathbb{Z}_{p}$-extension_, the associated $\mu$ and $\lambda$ Iwasawa
invariants shall (in this section) be denoted $\mu_{p}(K)$ and
$\lambda_{p}(K)$.

Our primary motivation is to study the statistical distribution of such
invariants. Given a fixed prime number $p$, one is interested in the average
behavior of $\mu_{p}(K)$ and $\lambda_{p}(K)$. More precisely, we fix $n>0$,
and let $\mathcal{F}$ be an infinite family of number fields $K$ with
$[K\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]=n$, ordered by absolute
discriminant $|\Delta_{K}|$. For instance, when $n=2$, it is natural to
consider the family of all imaginary quadratic fields, or the family of all
real quadratic fields. Furthermore, it is natural in this context to impose
splitting conditions on $p$. Consider the following question.

######  Question 1.1.

Fix a prime number $p$ and $\mu,\lambda\in\mathbb{Z}_{\geq 0}$. Given a family
of number fields $\mathcal{F}$ of degree $n$, let $\mathcal{F}^{\mu,\lambda}$
be the subset of $\mathcal{F}$ consisting of number fields $K$ such that
$\mu_{p}(K)=\mu$ and $\lambda_{p}(K)=\lambda$. Given a real number $x>0$, let
$\mathcal{F}_{<x}$ be the subset of $\mathcal{F}$ of all $K$ such that
$|\Delta_{K}|<x$ and set
$\mathcal{F}^{\mu,\lambda}_{<x}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{F}^{\mu,\lambda}\cap\mathcal{F}_{<x}$. Then, what can be
said about the proportion of number fields $K$ in $\mathcal{F}$ with
$\mu_{p}(K)=\mu$ and $\lambda_{p}(K)=\lambda$, i.e., the limit

(1.2)
$\mathfrak{d}(\mathcal{F}^{\mu,\lambda})=\lim_{x\rightarrow\infty}\frac{\\#\mathcal{F}^{\mu,\lambda}_{<x}}{\\#\mathcal{F}_{<x}}?$

Note that the limit (1.2) need not exist, however, one may ask the same
questions for the upper and lower limits, defined by replacing the limit by
$\limsup$ and $\liminf$ respectively. Even for the simplest families of number
fields, i.e. imaginary/real quadratic fields, such questions have proved
difficult. On the other hand, the systematic study of arithmetic statistics
for the Iwasawa invariants of elliptic curves was initiated by the second
named author, and has gained considerable momentum in the following works:
[KR21a, KR21b, HKR21, KLR22, BKR21, Ray21, RS21].

The main goal of the present work is to study a topological analogue of
question 1.1, which has proven to be a more tractable problem. There is a deep
analogy called _arithmetic topology_ between number fields and $3$-manifolds.
This analogy asserts a mysterious parallel between primes in number fields and
knots embedded in a $3$-manifold. As a result, there are many arithmetic
phenomena for number fields that are reflected on the topological side and
vice versa. This philosophy originates from observations made by Mazur in
[Maz63], which he attributes to conversations with Mumford. For a modern
account of various aspects of this analogy, we refer to [Mor11]. Guided by
this analogy, Hillman, Matei and Morishita [HMM06] have defined analogues of
Iwasawa invariants associated to a link $\mathcal{L}$ embedded in a
$3$-sphere. Such Iwasawa invariants are used to prove an asymptotic formula
for the growth patterns of $p$-homology in a $\mathbb{Z}_{p}$-cover of $S^{3}$
that is branched along $\mathcal{L}$. We refer to [HMM06, Theorem 5.1.7] for
the precise statement. This formula has a strong resemblance to the classical
formula (1.1) due to Iwasawa. It is not difficult to prove that for any knot,
the associated $\mu$ and $\lambda$ invariants vanish (see Lemma 3.4).
Therefore, the study of Iwasawa invariants is of genuine interest only for
links with at least $2$ knot components.

In this paper, we consider the family of links in $S^{3}$ with $2$ knot
components. For such links, the $\lambda$-invariant is $\geq 1$. Such links
are presented as _2-bridge links_ , and there is an explicit parametrization
called the _Schubert form_ , which associates a rational number $a/b$ with
$(a,b)=1$, $b$ even and $0<a<2b$ to an explicit presentation which is denoted
$\mathcal{L}_{a/b}$. With respect to counting function
$\operatorname{ht}(\mathcal{L}_{a/b})\mathrel{\mathop{\mathchar
58\relax}}=\max\\{a^{2},b\\}$, our main result shows that for a density of
$(1-1/p)$ links, the $\mu$-invariant vanishes and the $\lambda$-invariant is
equal to $1$. In greater detail, an integral vector $z=(z_{1},z_{2})$ is
called admissible if $z_{1}z_{2}\neq 0$ and
$\operatorname{gcd}(z_{1},z_{2})=1$. The choice of $z$ fixes a
$\mathbb{Z}_{p}$-cover of the link complement
$S^{3}\backslash\mathcal{L}_{a/b}$. Associated with this cover are $\mu$ and
$\lambda$-invariants, $\mu_{p,a/b,z}$ and $\lambda_{p,a/b,z}$. Let
$\mathcal{N}$ be the set of all fractions $a/b$ with $(a,b)=1$, $b$ even,
$0<a<2b$. For $x>0$, set $\mathcal{N}_{<x}$ to be the subset of $\mathcal{N}$
consisting of fractions $a/b$ such that
$\operatorname{ht}(\mathcal{L}_{a/b})<x$. It is easy to see that
$\mathcal{N}_{<x}$ is finite, we set $\\#\mathcal{N}_{<x}$ to denote its
cardinality. Given a subset $\mathcal{S}$ of $\mathcal{N}$, let
$\mathcal{S}_{<x}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{S}\cap\mathcal{N}_{<x}$. We say that $\mathcal{S}$ has
density $\mathfrak{d}(\mathcal{S})$ if the following limit exists

$\mathfrak{d}(\mathcal{S})\mathrel{\mathop{\mathchar
58\relax}}=\lim_{x\rightarrow\infty}\frac{\\#\mathcal{S}_{<x}}{\\#\mathcal{N}_{<x}}.$

Denote by $\mathcal{S}^{p,z}$ the set of all fractions $a/b$ in $\mathcal{N}$
such that $\mu_{p,\mathcal{L}_{a/b},z}=0$ and
$\lambda_{p,\mathcal{L}_{a/b},z}=1$. Our main result is as follows.

######  Main Theorem (Theorem 5.4).

With respect to notation above, $\mathcal{S}^{p,z}$ has density given by
$\mathfrak{d}(\mathcal{S}^{p,z})=1-1/p$.

We stress that the above result entirely unconditional. Furthermore, in
section 6, we provide computational evidence for the vanishing of the
$\mu$-invariant for $100\%$ of links parametrized in this way, for $z=(1,1)$.

_Organization:_ In section 2, we outline the parallels between topology and
arithmetic and make preliminary observations. Section 3 is devoted to the
study of the _Alexander polynomial_ as well as certain $p$-adic analogs. In
section 4, we prove explicit criteria that relate _linking numbers_ to Iwasawa
invariants. We state and prove our main results in section 5 for the family of
$2$-bridge links. Section 6 provides examples of links with positive
$\mu$-invariant (for the total linking number covering space), and
computational data showing that one may expect that $\mu=0$ for $100\%$ (i.e.,
a density $1$ set) of $2$-bridge links.

###  1.1. Acknowledgments

The first named author’s research is supported by the Canada Graduate
Scholarships – Doctoral program from the Natural Sciences and Engineering
Research Council of Canada. The authors would like to thank Antonio Lei for
helpful comments.

##  2\. Parallels between topology and arithmetic

In this section, we introduce some preliminary notions and delineate the
parallels between topology and arithmetic. This shall motivate our
investigations in _topological Iwasawa theory_ , i.e., the Iwasawa theory of
the homology of various covers of $S^{3}$ (the $3$-dimensional sphere), that
are branched along a fixed link.

Before discussing the topological analog, we recall the classical Iwasawa
theory of a number field. For a comprehensive account, the reader is referred
to the original work of Iwasawa [Iwa73]. Fix a prime number $p$ and a number
field $K$. Let $\bar{K}$ be a choice of algebraic closure of $K$ and
$\operatorname{G}_{K}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\bar{K}/K)$ the _absolute Galois group_ of $K$.
Given a finite set of prime numbers $S$, let $K_{S}$ be the maximal extension
of $K$ contained in $\bar{K}$ in which all primes $v\notin S$ are unramified.
Set $\operatorname{G}_{K,S}$ to denote the Galois group
$\operatorname{Gal}(K_{S}/K)$. A $\mathbb{Z}_{p}$-extension is a Galois
extension $K_{\infty}/K$ such that $\operatorname{Gal}(K_{\infty}/K)$ is
isomorphic to $\mathbb{Z}_{p}$ as a topological group. Note that all primes
$v\nmid p$ of $K$ are unramified in $K_{\infty}$. The cyclotomic
$\mathbb{Z}_{p}$-extension $K_{\operatorname{cyc}}/K$ is the unique
$\mathbb{Z}_{p}$-extension contained in $K(\mu_{p^{\infty}})$. Given a
$\mathbb{Z}_{p}$-extension $K_{\infty}$, let $K_{n}$ be the unique extension
of $K$ which is contained in $K_{\infty}$, such that
$[K_{n}\mathrel{\mathop{\mathchar 58\relax}}K]=p^{n}$. Denote by
$\operatorname{Cl}_{p}(K_{n})$ the $p$-Sylow subgroup of the class group of
$K_{n}$. Iwasawa proved the following asymptotic formula for the growth of the
$p$-class groups

$\\#\operatorname{Cl}_{p}(K_{n})=p^{p^{n}\mu+n\lambda+\nu}$

for $n\gg 0$, where $\mu,\lambda\in\mathbb{Z}_{\geq 0}$, and
$\nu\in\mathbb{Z}$ are Iwasawa invariants associated to $K_{\infty}/K$. Thus
in particular, $\mu=\lambda=0$ if and only if
$\\#\operatorname{Cl}_{p}(K_{n})$ is bounded in $n$. For the cyclotomic
$\mathbb{Z}_{p}$-extension $K_{\operatorname{cyc}}/K$, it is conjectured by
Iwasawa that the $\mu$-invariant of $K_{\operatorname{cyc}}/K$ must vanish.
This was proven by Ferrero and Washington [FW79] in the specific case when $K$
is an abelian extension of $\mathbb{Q}$.

Given a $\mathbb{Z}_{p}$-extension $K_{\infty}/K$, set
$X\left(K_{\infty}\right)$ to be the inverse limit of the $p$-primary class
groups $\operatorname{Cl}_{p}(K_{n})$ with respect to norm maps. Note that
$X\left(K_{\infty}\right)$ is a $\mathbb{Z}_{p}$-module as well as a module
over $\Gamma=\operatorname{Gal}(K_{\infty}/K)$. The Iwasawa algebra associated
to $\Gamma$ is defined to be the inverse limit of finite group algebras
$\widehat{\Lambda}\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\mathbb{Z}_{p}[\Gamma/\Gamma^{p^{n}}]$. Letting
$\gamma$ be a topological generator of $\Gamma$, there is an isomorphism
$\mathbb{Z}_{p}\llbracket X\rrbracket\simeq\widehat{\Lambda}$ which identifies
the formal variable $X$ with $(\gamma-1)$. We note here that in most
expositions in Iwasawa theory, it is customary to let $\Lambda$ denote the
Iwasawa algebra, however, we follow the notation used in [HMM06] and [Mor11]
in which $\Lambda$ is reserved for the Laurent series ring $\mathbb{Z}[t^{\pm
1}]$.

Given a finitely generated and torsion $\widehat{\Lambda}$-module $M$, the
structure theorem [Law97, Theorem 13.12] states that there is a map of
$\widehat{\Lambda}$-modules

$M^{\vee}\longrightarrow\left(\bigoplus_{i=1}^{s}\widehat{\Lambda}/(p^{\mu_{i}})\right)\oplus\left(\bigoplus_{j=1}^{t}\widehat{\Lambda}/(f_{j}(X))\right)$

with finite kernel and cokernel. Here, $\mu_{i}>0$ and $f_{j}(X)$ is a
distinguished polynomial (i.e., a monic polynomial with non-leading
coefficients divisible by $p$). The characteristic ideal of $M$ is defined to
be up to a unit generated by

$f_{M}(X)\mathrel{\mathop{\mathchar
58\relax}}=p^{\sum_{i}\mu_{i}}\prod_{j}f_{j}(X)=p^{\mu}g_{M}(X),$

where $\mu=\sum_{i}\mu_{i}$ and $g_{M}(X)$ is the distinguished polynomial
$\prod_{j}f_{j}(X)$. The quantity $\mu$ is the $\mu$-invariant, which is
denoted $\mu_{p}(M)$, and the $\lambda$-invariant $\lambda_{p}(M)$ is the
degree of $g_{M}(T)$. For ease of notation, we simply set
$\mu_{p}(K_{\infty}/K)\mathrel{\mathop{\mathchar
58\relax}}=\mu_{p}\left(X(K_{\infty})\right)$ and
$\lambda_{p}(K_{\infty}/K)\mathrel{\mathop{\mathchar
58\relax}}=\lambda_{p}\left(X(K_{\infty})\right)$. We let
$K_{\operatorname{cyc}}$ be the cyclotomic $\mathbb{Z}_{p}$-extension of $K$.
Thus, the result of Ferrero and Washington states that if $K/\mathbb{Q}$ is
abelian and $p$ is any prime, then, $\mu_{p}(K_{\infty}/K)=0$. We shall simply
refer to the characteristic polynomial of $X(K_{\infty})$ as the $p$-primary
Iwasawa polynomial of $K_{\infty}/K$.

Our main focus will be to study the behavior of various topological analogues
of such Iwasawa invariants that arise naturally in the context of knot theory.
First, we explain the parallels between the Galois theory of number fields
with restricted ramification, and the topological theory of coverings of
$S^{3}$ that are branched along a link. Let $K$ be a number field and
$\mathcal{O}_{K}$ be the ring of integers.

Class field theory is based on certain duality properties of the
$1$-dimensional scheme $X=\operatorname{Spec}\mathcal{O}_{K}$. Let
$\mathbb{G}_{m,X}$ be the étale sheaf on $X$, which associates to an étale
cover $\varphi\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Spec}B\rightarrow X$, the group of multiplicative
units $\mathbb{G}_{m,X}(\varphi)=B^{\times}$. Given a _constructible sheaf_
$\mathfrak{M}$ on $X$ (see [Mil06, p.146] for the definition), set
$\mathfrak{M}^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\underline{\operatorname{Hom}}(\mathfrak{M},\mathbb{G}_{m,X})$
(where $\underline{\operatorname{Hom}}(\cdot,\cdot)$ is standard notation for
the sheaf of homomorphisms, see [Har13, Chapter 2] or _loc. cit._). We denote
by $\hat{H}^{i}(X,\mathfrak{M})$ the modified étale cohomology groups of
$\mathfrak{M}$. The Artin-Verdier duality theorem (see [Mil06, Theorem 1.8])
states that there is a canonical isomorphism
$\hat{H}^{3}(X,\mathbb{G}_{m,X})\simeq\mathbb{Q}/\mathbb{Z}$ and a
nondegenerate pairing of abelian groups

$\hat{H}^{i}(X,\mathfrak{M}^{\vee})\times\operatorname{Ext}_{X}^{3-i}(\mathfrak{M},\mathbb{G}_{m,X})\rightarrow\hat{H}^{3}(X,\mathbb{G}_{m,X})\xrightarrow{\sim}\mathbb{Q}/\mathbb{Z}.$

The above pairing is analogous to the Poincare duality pairing for a compact
oriented $3$-manifold $Y$ and foreshadows a deep resemblance between the
arithmetic properties of a number ring $\mathcal{O}_{K}$ and the topological
properties of an oriented $3$-manifold $Y$.

There is also a close analogy between embedded knots in $3$-manifolds and
primes in number rings, which we briefly explain. For further details, the
reader may refer to [Mor11, chapter 3]. Given a finite field $\mathbb{F}$, the
étale fundamental group of $\operatorname{Spec}\mathbb{F}$ is given by

$\pi_{1}(\operatorname{Spec}\mathbb{F})=\operatorname{Gal}(\bar{\mathbb{F}}/\mathbb{F})\simeq\hat{\mathbb{Z}}.$

In fact, $\operatorname{Spec}\mathbb{F}$ is the étale analogue of the
Eilenberg-Maclane space $K(\widehat{\mathbb{Z}},1)$ (see [Hat, p.87, p.365]
for the definition of this notion). On the topological side, since
$\pi_{1}(S^{1})\simeq\mathbb{Z}$ and $\pi_{i}(S^{1})=0$ for $i\geq 2$, the
circle $S^{1}$ is $K(\mathbb{Z},1)$. For every integer $n\geq 1$, there is a
unique extension $\mathbb{F}^{\prime}/\mathbb{F}$ of degree $n$ with
$\operatorname{Gal}(\mathbb{F}^{\prime}/\mathbb{F})\simeq\mathbb{Z}/n\mathbb{Z}$.
On the topological side, there is a unique $\mathbb{Z}/n\mathbb{Z}$ cover
$[n]\mathrel{\mathop{\mathchar 58\relax}}S^{1}\rightarrow S^{1}$. Identifying
$S^{1}$ with the unit circle in $\mathbb{C}$, the covering map $[n]$ maps
$z\mapsto z^{n}$. Thus the finite Galois covers of $\mathbb{F}$ are in
correspondence with finite covers of $S^{1}$.

Given a prime ideal $\mathfrak{p}$ in a number ring $\mathcal{O}_{K}$, let
$\mathbb{F}_{\mathfrak{p}}$ be the residue field at $\mathfrak{p}$. Let
$K_{\mathfrak{p}}$ be the completion of $K$ at $\mathfrak{p}$, and
$\mathcal{O}_{\mathfrak{p}}$ the valuation ring in $K$. In the diagram

(2.1)
$\operatorname{Spec}K_{\mathfrak{p}}\hookrightarrow\operatorname{Spec}\mathcal{O}_{\mathfrak{p}}\hookleftarrow\operatorname{Spec}\mathbb{F}_{\mathfrak{p}},$

the map
$\operatorname{Spec}\mathbb{F}_{\mathfrak{p}}\hookrightarrow\operatorname{Spec}\mathcal{O}_{\mathfrak{p}}$
is induced by reduction mod-$\mathfrak{p}$ and is an _étale homotopy
equivalence_ (see [AM06]). On the other hand, the map
$\operatorname{Spec}K_{\mathfrak{p}}\hookrightarrow\operatorname{Spec}\mathcal{O}_{\mathfrak{p}}$
is induced by the inclusion of $\mathcal{O}_{\mathfrak{p}}$ in
$K_{\mathfrak{p}}$.

On the topological side, the ambient $3$-manifold is assumed for simplicity to
be $S^{3}$, and there would be significant work required to extend the results
to links embedded in more general $3$-manifolds. We note that some authors do
work in slightly greater generality, for instance, in [Uek17], results are
proven for _rational homology $3$-spheres_. We come to the definition of a
_knot_ and more generally, a _link_. We refer to [Kaw96, Chapter 1] and
[Lic12, Chapter 1] for a comprehensive treatment of some of the basic notions
in knot theory, as well as definitions that are equivalent to that given
below, which we have chosen since in our estimation, it is easiest to state.

######  Definition 2.1.

A knot (in $S^{3}$) is an embedding $\iota\mathrel{\mathop{\mathchar
58\relax}}S^{1}\hookrightarrow S^{3}$ which is locally flat, i.e., locally
homeomorphic to the canonical embedding
$\mathbb{R}\hookrightarrow\mathbb{R}^{3}$ sending $x$ to $(x,0,0)$. Two knots
are equivalent if there is an isotopy $f_{t}\mathrel{\mathop{\mathchar
58\relax}}S^{3}\rightarrow S^{3}$, $0\leq t\leq 1$, taking one knot to the
other. In order to simplify notation, we shall identify the embedding
$\iota\mathrel{\mathop{\mathchar 58\relax}}S^{1}\hookrightarrow S^{3}$ with
the $\mathcal{K}\mathrel{\mathop{\mathchar 58\relax}}=\iota(S^{1})$. A _link_
(in $S^{3}$) is a union of disjoint knots
$\mathcal{L}=\mathcal{K}_{1}\cup\dots\cup\mathcal{K}_{r}$ in $S^{3}$.

Let $\mathcal{K}$ be a knot in $S^{3}$, fix a _tubular neighbourhood_ (see
[Lee00, p.137, ll.7-11] for the definition) $\mathcal{V}$ of $\mathcal{K}$,
and let $\partial\mathcal{V}$ be the boundary of $\mathcal{V}$. Note that
$\partial\mathcal{V}$ is homotopic to $S^{1}\times\mathcal{K}$. The diagram
(2.1) is analogous to

$\partial\mathcal{V}\hookrightarrow\mathcal{V}\hookleftarrow\mathcal{K}.$

Note that the inclusion $\mathcal{K}\hookrightarrow\mathcal{V}$ is a homotopy
equivalence, and $\partial\mathcal{V}$ is homeomorphic to
$S^{1}\times\mathcal{K}$. By abuse of notation, we identify $\mathcal{V}$ with
the product $S^{1}\times\mathcal{K}$. Consider the composite of maps

$\xi\mathrel{\mathop{\mathchar
58\relax}}\pi_{1}(\partial\mathcal{V})\rightarrow\pi_{1}(\mathcal{V})\xrightarrow{\sim}\pi_{1}(\mathcal{K}).$

Given $a\in S^{1}$, we refer to a loop of the form $\\{a\\}\times\mathcal{K}$
in $\partial\mathcal{V}$ as a _longitude_ of $\mathcal{K}$, and we denote the
associated homotopy class by
$\ell_{\mathcal{K}}\in\pi_{1}(\partial\mathcal{V})$. Note that this homotopy
class is independent of the choice of $a$. The image of $\ell_{\mathcal{K}}$
is a generator of $\pi_{1}(\mathcal{K})$, and for this reason,
$\ell_{\mathcal{K}}$ is an analog of a Frobenius element
$\sigma_{\mathfrak{p}}\in\operatorname{Gal}(\bar{K}_{\mathfrak{p}}/K_{\mathfrak{p}})$.
The kernel of $\xi$ is referred to as the _inertia group_ of $\mathcal{K}$. A
_meridian_ $\tau_{\mathcal{K}}$ is a generator of the inertia group of the
form $S^{1}\times\\{b\\}$. Note that this homotopy class is independent of the
choice of point $b$. The fundamental group $\pi_{1}(\partial\mathcal{V})$ is
the generated by $\ell_{\mathcal{K}}$ and $\tau_{\mathcal{K}}$, with the
single relation
$[\ell_{\mathcal{K}},\tau_{\mathcal{K}}]=\ell_{\mathcal{K}}\tau_{\mathcal{K}}\ell_{\mathcal{K}}^{-1}\tau_{\mathcal{K}}^{-1}=1$.

Let $\mathcal{L}=\mathcal{K}_{1}\cup\dots\cup\mathcal{K}_{r}$ be an _oriented
link_ , which is to say that each knot $\mathcal{K}_{j}$ is oriented. Given
distinct knots $\mathcal{K}_{i}$ and $\mathcal{K}_{j}$, let $\ell_{i,j}$ be
the _linking number_ of $\mathcal{K}_{i}$ and $\mathcal{K}_{j}$, we refer to
[Kaw96, p. 11] for a precise definition. Let $X_{\mathcal{L}}$ be the
complement of $\mathcal{L}$ in $S^{3}$, and let $x_{0}$ be a choice of base-
point in $X_{\mathcal{L}}$.

######  Definition 2.2.

The _link group_ of $\mathcal{L}$ is the fundamental group
$G_{\mathcal{L}}\mathrel{\mathop{\mathchar
58\relax}}=\pi_{1}(X_{\mathcal{L}};x_{0})$.

A comprehensive study of the analogies between fundamental groups and Galois
groups of number fields is undertaken in [VH96]. The _link group_ is the
analogue of the group $\operatorname{Gal}(K_{S}/K)$, where $S$ is a finite set
of primes and $K_{S}$ is the maximal algebraic extension of $K$ which is
unramified at primes $v\notin S$. The set of primes $S$ here is the set at
which $K_{S}$ may ramify and likewise. On the other hand, $G_{\mathcal{L}}$ is
the deck group of $\pi_{\mathcal{L}}\mathrel{\mathop{\mathchar
58\relax}}\widetilde{X}_{\mathcal{L}}\rightarrow X_{\mathcal{L}}$, the
universal cover of $X_{\mathcal{L}}$.

##  3\. The Alexander Module associated to a link in a $3$-sphere

The classical Alexander polynomial is an important topological invariant
associated to a link in a $3$-sphere (or, more generally $3$-manifold). It was
first introduced by Alexander [Ale28]. On the other hand, there is a $p$-adic
version introduced by Hillman-Matei-Morishita [HMM06] which is widely regarded
as the multi-variable topological analogue of the Iwasawa polynomial of
$\mathbb{Z}_{p}$-extension. In this setting, one may define $\mu$, $\lambda$
and $\nu$ invariants, and these invariants give a formula for the asymptotic
growth of the $p$-parts in the homology for the various finite covering spaces
occuring in a chosen $\mathbb{Z}_{p}$-cover which is branched along
$\mathcal{L}$. In this section, we recall the notion of the completed
Alexander polynomial and associated Iwasawa invariants, and explain the
relationship with $p$-homology classes occurring in a $\mathbb{Z}_{p}$-cover.

Let $\mathcal{L}$ be a link in $S^{3}$ consisting of component knots
$\mathcal{K}_{1},\dots,\mathcal{K}_{r}$, and denote the complement by
$X_{\mathcal{L}}\mathrel{\mathop{\mathchar
58\relax}}=S^{3}\backslash\mathcal{L}$. For each knot $\mathcal{K}_{i}$, we
let $x_{i}\in G_{\mathcal{L}}$ the class generated by the meridian looping
about $\mathcal{K}_{i}$ starting and ending at the base point $x_{0}$. Note
that this involves a choice of a small enough tubular neighbourhood of
$\mathcal{K}_{i}$, that is contained in $X_{\mathcal{L}}$. We set
$\pi_{\mathcal{L}}^{\operatorname{ab}}\mathrel{\mathop{\mathchar
58\relax}}\widetilde{X_{\mathcal{L}}}^{\operatorname{ab}}\rightarrow
X_{\mathcal{L}}$ to be the maximal abelian cover of $X_{\mathcal{L}}$ and
$G_{\mathcal{L}}^{\operatorname{ab}}$ the maximal abelian quotient of
$G_{\mathcal{L}}$. Then note that there is a natural isomorphism

$G_{\mathcal{L}}^{\operatorname{ab}}\simeq\operatorname{Gal}\left(\widetilde{X_{\mathcal{L}}}^{\operatorname{ab}}/X_{\mathcal{L}}\right).$

By the Hurewicz theorem, $G_{\mathcal{L}}^{\operatorname{ab}}$ is isomorphic
to $H_{1}(X_{\mathcal{L}};\mathbb{Z})$. The homology group
$H_{1}(X_{\mathcal{L}};\mathbb{Z})$ is generated by
$t_{i}=x_{i}[G_{\mathcal{L}},G_{\mathcal{L}}]$, $i=1,\dots,r$, as a free
$\mathbb{Z}$-module of rank $r$. Letting $\Lambda_{r}$ be the group algebra
$\mathbb{Z}[G_{\mathcal{L}}^{\operatorname{ab}}]$, we find that $\Lambda_{r}$
is the Laurent polynomial ring

$\Lambda_{r}=\mathbb{Z}[t_{1}^{\pm 1},\dots,t_{r}^{\pm 1}].$

######  Definition 3.1.

Let $\mathcal{L}$ be a link with $r$ components as above. The _Alexander
module_ is the relative homology group
$A_{\mathcal{L}}=H_{1}(\widetilde{X_{\mathcal{L}}}^{\operatorname{ab}},\mathcal{F}_{x_{0}};\mathbb{Z})$,
where $\mathcal{F}_{x_{0}}$ is the fibre of $x_{0}$ in
$\widetilde{X_{\mathcal{L}}}^{\operatorname{ab}}$.

Note that the Galois group
$\operatorname{G}_{\mathcal{L}}^{\operatorname{ab}}$ acts on
$\widetilde{X_{\mathcal{L}}}^{\operatorname{ab}}$ by deck transformations, and
this gives rise to action of
$\operatorname{G}_{\mathcal{L}}^{\operatorname{ab}}$ on $A_{\mathcal{L}}$. As
a result, the Alexander module and link modules are modules over
$\Lambda_{r}$. In order to define the Alexander polynomial, we will first
recall some prerequisite notions from commutative algebra.

Let $R$ be a noetherian unique factorization domain and $M$ a finitely
generated $R$-module. Let $u_{1},\dots,u_{q}$ be a finite set of generators of
$M$ as an $R$-module, and $d_{1}\mathrel{\mathop{\mathchar
58\relax}}R^{q}\rightarrow M$ the surjective homomorphism sending the $i$-th
coordinate generator $e_{i}=(0,\dots,0,1,0,\dots)$ to $u_{i}$. Let
$d_{2}\mathrel{\mathop{\mathchar 58\relax}}R^{\ell}\rightarrow R^{q}$ be an
$R$-module homomorphism such that $R^{\ell}$ surjects onto the kernel of
$d_{1}$. Note that the maps fit into a right exact sequence

$R^{\ell}\xrightarrow{d_{2}}R^{q}\xrightarrow{d_{1}}M\rightarrow{0}.$

The map $d_{2}$ is represented by a matrix $A$. The $i$-th elementary ideal
$E^{(i)}_{R}(M)$ (also called the $i$-th Fitting ideal) is the ideal generated
by all determinants of $(q-i)\times(q-i)$ submatrices in $A$. The _divisorial
hull_ of any ideal $I$ in $R$ is the intersection of all principal ideals
containing $I$. We let $\Delta_{R}^{(i)}(M)$ be the divisorial hull of
$E_{R}^{(i)}(M)$.

The maximal abelian extension
$\pi_{\mathcal{L}}^{\operatorname{ab}}\mathrel{\mathop{\mathchar
58\relax}}\widetilde{X_{\mathcal{L}}}^{\operatorname{ab}}\rightarrow
X_{\mathcal{L}}$ contains various subextensions

$\pi_{\mathcal{L},z}\mathrel{\mathop{\mathchar
58\relax}}X_{\mathcal{L},z}\rightarrow X_{\mathcal{L}}$

such that
$\operatorname{Gal}(X_{\mathcal{L},z}/X_{\mathcal{L}})\simeq\mathbb{Z}$. Here,
$z$ is an a choice of integral vector $z=(z_{1},\dots,z_{r})\in\mathbb{Z}^{r}$
such that

  * •

$\operatorname{gcd}(z_{1},\dots,z_{r})=1$,

  * •

and $\prod_{i}z_{i}\neq 0$.

An integral vector satisfying the above conditions is said to be _admissible_.
Given an admissible $z$, define a surjective homomorphism
$\sigma_{z}\mathrel{\mathop{\mathchar
58\relax}}H_{1}(X_{\mathcal{L}};\mathbb{Z})\rightarrow\mathbb{Z}$ sending
$t_{i}$ to $z_{i}$ for $i=1,\dots,r$. Let $H_{z}$ be the kernel of
$\sigma_{z}$. Note that there corresponds to $H_{z}$ a cover

$\pi_{\mathcal{L},z}\mathrel{\mathop{\mathchar
58\relax}}X_{\mathcal{L},z}\rightarrow X_{\mathcal{L}}$

whose deck group is canonically identified with
$\operatorname{G}_{\mathcal{L}}/H_{z}\simeq\mathbb{Z}$. This gives a family of
$\mathbb{Z}$-covers, and this family is infinite if $r\geq 2$. Letting
$\mathbf{1}=(1,\dots,1)$, we call $X_{\mathcal{L},\mathbf{1}}$ the _total
linking number_ covering space of $X_{\mathcal{L}}$. For ease of notation we
abbreviate $X_{\mathcal{L},z}$ to $X_{z}$, and $\pi_{\mathcal{L},z}$ to
$\pi_{z}$.

######  Definition 3.2.

The _multivariable Alexander polynomial_ associated to $\mathcal{L}$ is
defined as

$\Delta_{\mathcal{L}}(t_{1},\dots,t_{r})=\Delta_{\Lambda_{r}}^{(1)}(A_{\mathcal{L}}).$

Let $z$ be an admissible integral vector. The _reduced Alexander polynomial_
associated to $X_{z}$ is defined in
$\Lambda_{1}\simeq\mathbb{Z}[\operatorname{Aut}(X_{z}/X)]$ as

$\Delta_{L,z}(t)=\Delta_{\Lambda_{1}}^{(0)}\left(H_{1}(X_{z};\mathbb{Z})\right)=\Delta_{\mathcal{L}}(t^{z_{1}},\dots,t^{z_{r}}).$

Following [HMM06], we now define the completed Alexander module. The prime $p$
is fixed and will be suppressed in much of our notation. We let
$\widehat{G_{\mathcal{L}}}$ be the pro-$p$ completion of $G_{\mathcal{L}}$
defined as the inverse limit

$\widehat{G_{\mathcal{L}}}\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{N}G_{\mathcal{L}}/N$

where $N$ ranges over all normal subgroups of $G_{\mathcal{L}}$ with finite
$p$-power index. We enumerate the knot components in $\mathcal{L}$ by
$\mathcal{K}_{1},\dots,\mathcal{K}_{r}$ and let
$x_{i},y_{i}\in\widehat{G_{\mathcal{L}}}$ be the elements corresponding to the
meridian and longitude around $\mathcal{K}_{i}$ respectively. Then Milnor
shows in [Mil57] that there is a presentation of the form

$\widehat{G_{\mathcal{L}}}=\langle x_{1},\dots,x_{r}\mid[x_{i},y_{i}]=1\text{
for }i=1,\dots,r\rangle.$

Given a pro-$p$ group $\mathcal{G}$, the associated $p$-adic Iwasawa algebra
is the inverse limit

$\widehat{\Lambda}(\mathcal{G})\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{\mathcal{N}}\mathbb{Z}_{p}[\mathcal{G}/\mathcal{N}],$

where $\mathcal{N}$ ranges over all finite index normal subgroups of
$\mathcal{G}$. Set $\widehat{H}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{Z}_{p}^{r}$, the associated Iwasawa algebra is denoted
$\widehat{\Lambda}_{r}$, and it is easy to show that $\widehat{\Lambda}_{r}$
is isomorphic to the formal power series ring $\mathbb{Z}_{p}\llbracket
X_{1},\dots,X_{r}\rrbracket$. Here, we choose a basis
$\gamma_{1},\dots\gamma_{r}$ of $\widehat{H}$ and let $X_{i}$ be the formal
variable which corresponds to $(\gamma_{i}-1)$. Consider the quotient
$\widehat{G_{\mathcal{L}}}\rightarrow\widehat{H}$ given by sending $x_{i}$ to
$\gamma_{i}$, which in turn shows that there is a quotient
$\Lambda(G_{\mathcal{L}})\rightarrow\widehat{\Lambda}_{r}$ mapping $x_{i}$ to
$\gamma_{i}$. This map is well defined since the relations $[x_{i},y_{i}]$
must map to $0$ as $\widehat{H}$ is abelian.

We let $F$ be the (non-abelian) free group in $r$-generators
$x_{1},\dots,x_{r}$ and $\widehat{F}$ its pro-$p$ completion. Denote by
$\epsilon\mathrel{\mathop{\mathchar
58\relax}}\Lambda(\widehat{F})\rightarrow\mathbb{Z}_{p}$ the augmentation map,
sending every group-like element to $1$. It follows from the work of Ihara
[Iha86] that the Fox differential calculus extends to $\widehat{F}$. One may
define $\mathbb{Z}_{p}$-homomorphisms for $i=1,\dots,r$,

$\partial_{i}=\frac{\partial}{\partial x_{i}}\mathrel{\mathop{\mathchar
58\relax}}\Lambda(\widehat{F})\rightarrow\Lambda(\widehat{F})$

such that any element $\alpha\in\Lambda(\widehat{F})$ can be uniquely
expressed as a sum

$\alpha=\epsilon(\alpha)+\sum_{i=1}^{r}\partial_{i}(\alpha)(x_{i}-1).$

The derivatives of higher order are defined inductively as follows. Given a
sequence $I=(i_{1},\dots,i_{m})$ with $1\leq i_{j}\leq r$, set

$\partial_{I}\mathrel{\mathop{\mathchar
58\relax}}=\partial_{i_{1}}\dots\partial_{i_{m}}(\alpha).$

We refer to [HMM06, section 2] for the basic properties of $\partial_{I}$.

Denote by $\mathbb{Z}_{p}\langle X_{1},\dots,X_{r}\rangle$ the formal non-
commuting power series in $\mathbb{Z}_{p}$. An element of
$\mathbb{Z}_{p}\langle X_{1},\dots,X_{r}\rangle$ is represented as a formal
sum of monomials $X_{I}=X_{i_{1}}X_{i_{2}}\dots X_{i_{n}}$, where
$I=(i_{1},\dots,i_{n})$. For $m\geq 1$, let $I(m)$ be the ideal in
$\mathbb{Z}_{p}\langle X_{1},\dots,X_{r}\rangle$ which is generated by all
monomials $X_{i_{1}}\dots X_{i_{m}}$. Let $\mathbb{Z}_{p}\langle\langle
X_{1},\dots,X_{r}\rangle\rangle$ be the formal power series ring in $n$
noncommuting variables, defined as the completion

$\mathbb{Z}_{p}\langle\langle
X_{1},\dots,X_{r}\rangle\rangle\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{m}\left(\frac{\mathbb{Z}_{p}\langle
X_{1},\dots,X_{r}\rangle}{I(m)}\right).$

The pro-$p$ Magnus embedding $M\mathrel{\mathop{\mathchar
58\relax}}\widehat{F}\rightarrow\Lambda(\widehat{F})$ is defined by

$M(x_{i})=1+X_{i}\text{ and
}M(x_{i}^{-1})=\sum_{j=0}^{\infty}(-1)^{j}X_{i}^{j}.$

Denote by $\hat{\psi}\mathrel{\mathop{\mathchar
58\relax}}\Lambda(\widehat{G}_{\mathcal{L}})\rightarrow\widehat{\Lambda}_{r}=\mathbb{Z}_{p}\llbracket
X_{1},\dots,X_{r}\rrbracket$ the natural quotient map. Following [HMM06,
section 3], the Alexander matrix is the $r\times r$ matrix
$\widehat{P}_{\mathcal{L}}\mathrel{\mathop{\mathchar
58\relax}}=\left(\hat{\psi}(\partial_{j}[x_{i},y_{i}])\right)$ and the
completed Alexander module over $\widehat{\Lambda}_{r}$ is denoted
$\widehat{A}_{\mathcal{L}}$ and is defined to be the cokernel of the map
$\widehat{\Lambda_{r}}^{r}\xrightarrow{\widehat{P}_{\mathcal{L}}}\widehat{\Lambda_{r}}^{r}$.
Let $\widehat{\Delta}_{\mathcal{L}}$ be the greatest common divisor of all
determinants of $(r-1)\times(r-1)$ minors of $\widehat{P}_{\mathcal{L}}$. Note
that $\widehat{\Delta}_{\mathcal{L}}$ is well defined up to multiplication by
a unit in $\widehat{\Lambda_{r}}$.

Recall that $\widehat{\Lambda}$ is the Iwasawa algebra
$\widehat{\Lambda}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{Z}_{p}\llbracket X\rrbracket$ and let
$\tau_{z}\mathrel{\mathop{\mathchar
58\relax}}\widehat{\Lambda}_{r}\rightarrow\widehat{\Lambda}$ be the reducing
homomorphism defined by $\tau(X_{i})=(1+X)^{z_{i}}-1$. Also set
$\tau\mathrel{\mathop{\mathchar 58\relax}}=\tau_{\mathbf{1}}$. The reduced
completed Alexander module is defined by
$\widehat{A}_{\mathcal{L}}^{\operatorname{red}}\mathrel{\mathop{\mathchar
58\relax}}=\widehat{A}_{\mathcal{L}}\hat{\otimes}_{\widehat{\Lambda}_{r}}\widehat{\Lambda}$,
which is presented by $\tau(\widehat{P}_{\mathcal{L}})$, and set

$\tau(\widehat{\Delta}_{\mathcal{L}})\mathrel{\mathop{\mathchar
58\relax}}=\widehat{\Delta}_{\mathcal{L}}(X,X,\dots,X).$

More generally, set $\widehat{A}_{\mathcal{L}}^{z}\mathrel{\mathop{\mathchar
58\relax}}=\widehat{A}_{\mathcal{L}}\hat{\otimes}_{\widehat{\Lambda}_{r},\tau_{z}}\widehat{\Lambda}$
and

$\widehat{\Delta}_{\mathcal{L},z}\mathrel{\mathop{\mathchar
58\relax}}=\tau_{z}(\widehat{\Delta}_{\mathcal{L}})=\widehat{\Delta}_{\mathcal{L}}\left((1+X)^{z_{1}}-1,(1+X)^{z_{2}}-1,\dots,(1+X)^{z_{r}}-1\right).$

The relationship between the completed Alexander polynomial
$\widehat{\Delta}_{\mathcal{L},z}\in\mathbb{Z}_{p}\llbracket X\rrbracket$ and
the classical Alexander polynomial
${\Delta}_{\mathcal{L},z}\in\mathbb{Z}_{p}[t^{\pm 1}]$ is given by

$\widehat{\Delta}_{\mathcal{L},z}(X)={\Delta}_{\mathcal{L},z}(1+X),$

where we plug $(1+X)$ in place of $t$ (see [HMM06, (3.1.6) and (3.1.7)]). Note
that this equality is up to a unit in $\widehat{\Lambda}$. Assume for the
moment that $\widehat{\Delta}_{\mathcal{L},z}(X)\neq 0$. By the Weierstrass
preparation theorem, we may write

(3.1) $\widehat{\Delta}_{\mathcal{L},z}(X)=p^{\mu}P(X)u(X),$

where $\mu\in\mathbb{Z}_{\geq 0}$, $P(X)$ is a monic polynomial whose non-
leading coefficients are divisible by $p$, and $u(X)$ is a unit in
$\mathbb{Z}_{p}$. The $\mu$-invariant associated with
$\widehat{\Delta}_{\mathcal{L},z}(X)$ is the number $\mu$ that appears in
(3.1), and the $\lambda$-invariant is the degree of the distinguished
polynomial $P(X)$. We shall denote these $\mu$ and $\lambda$-invariants by
$\mu_{\mathcal{L},z}$ and $\lambda_{\mathcal{L},z}$ respectively. On the other
hand, if $\widehat{\Delta}_{\mathcal{L},z}(X)=0$, we set
$\mu_{\mathcal{L},z}=\lambda_{\mathcal{L},z}=\infty$. It is indeed possible
for the multivariable Alexander polynomial itself to be zero, for instance,
for L11n244, which is a link with $2$ knot components.

Note that in [HMM06], the $\mu$ and $\lambda$-invariants are associated to the
_Hosokawa polynomial_ and not the Alexander polynomial. The only difference is
that the choice of $\lambda_{L,z}$ is $(r-1)$ more than the
$\lambda$-invariant of the Hosokawa polynomial. This is only a matter of
convention, we state well-known results essentially due to Mayberry-Murasugi
[MM82], Hillman-Matei-Morishita (for $z=\mathbf{1}$) [HMM06, Theorem 5.1.7]
and later adapted to a slightly more general case (for general $z$) by
Kadokami-Mizusawa (see [KM08, Theorem 2.1]). For the rest of this section, we
shall recall these results.

Following the notation and conventions of [KM13, section 2], recall that the
choice of $z$ gives rise to an infinite cyclic cover
$\pi_{z}\mathrel{\mathop{\mathchar 58\relax}}X_{z}\rightarrow X$. Given
$n\in\mathbb{Z}_{\geq 0}$ denote by $\pi_{z}^{n}\mathrel{\mathop{\mathchar
58\relax}}X_{z,p^{n}}\rightarrow X$ the unique
$\mathbb{Z}/p^{n}\mathbb{Z}$-subcover of $\pi_{z}$. Let
$\xi_{z}^{n}\mathrel{\mathop{\mathchar 58\relax}}M_{z,p^{n}}\rightarrow S^{3}$
be the _Fox completion_ of $\pi_{z}^{n}$ (see [Mor11, section 10.2] for
further details). Note that $\xi_{z}^{n}$ is branched along $\mathcal{L}$.
Given a $\mathbb{Z}$-module $A$, set $|A|$ to denote the cardinality of $A$ if
it is finite and $0$ if $A$ is infinite.

######  Theorem 3.3.

Let $z$ be an admissible integral vector and assume that
$\widehat{\Delta}_{\mathcal{L},z}\neq 0$. Let $v=v(z)$ be the maximum of
$v_{p}(z_{i})$, where $v_{p}$ is the valuation normalized by $v_{p}(p)=1$. We
have the following:

  1. (1)

$|H_{1}(M_{z,p^{n}};\mathbb{Z})|=|H_{1}(M_{z,p^{v}};\mathbb{Z})|\times\left(\prod_{\zeta^{p^{n}}=1,\\\
\zeta^{p^{v}}\neq 1,}|\Delta_{\mathcal{L},z}(\zeta)|\right)$ for all $n\geq
v$.

  2. (2)

If $|H_{1}(M_{z,p^{n}};\mathbb{Z})|\neq 0$, then, there is an integer
$v_{\mathcal{L},z}$ such that

$|H_{1}(M_{z,p^{n}};\mathbb{Z})|=p^{\left(p^{n}\mu_{\mathcal{L},z}+n\lambda_{\mathcal{L},z}+\nu_{\mathcal{L},z}\right)}$

for $n$ large enough.

###### Proof.

The stated result is [HMM06, Theorem 2.1]. ∎

Note that when $r=1$, the singleton tuple $z$ is forced to be equal to $1$. In
the case of a knot, the Iwasawa invariants are well understood, as the
following result shows.

######  Lemma 3.4.

Let $\mathcal{K}$ be a knot. Then, the Iwasawa invariants are given by
$\mu_{\mathcal{K},1}=0$ and $\lambda_{\mathcal{K},1}=0$.

###### Proof.

The above result is [KM13, Theorem 2.2 (i)], we provide details here. As is
well known, $\Delta_{\mathcal{K},\mathbf{1}}(1)=\Delta_{\mathcal{K}}(1)=\pm
1$. Hence, $\widehat{\Delta}_{\mathcal{K},1}(0)=\pm 1$ is a unit in
$\mathbb{Z}_{p}$. Therefore, $\widehat{\Delta}_{\mathcal{K},1}(X)$ is a unit
in $\widehat{\Lambda}$, and hence, $\mu_{\mathcal{K},1}=0$ and
$\lambda_{\mathcal{K},1}=0$. ∎

##  4\. Linking numbers and calculation of Iwasawa Invariants

In this section, we introduce various methods of detecting the Iwasawa
invariants introduced in the previous section, and how they are related to
_$p$ -adic Milnor invariants_ introduced in [HMM06]. We shall describe the
precise analogy with the number field case. Let $p$ be an odd prime and $K$ be
an imaginary quadratic field in which $p$ splits, and consider the cyclotomic
$\mathbb{Z}_{p}$-extension $K_{\operatorname{cyc}}/K$. The splitting of $p$
creates a trivial zero for the $p$-adic L-function which in turn shows that
the characteristic element in $\mathbb{Z}_{p}\llbracket X\rrbracket$ is
divisible by $X$ (see the discussion on p.1 of [San93]). Therefore, in the
case where $p$ splits in $K$, we have that
$\lambda_{p}(K_{\operatorname{cyc}}/K)\geq 1$. In this setting, Gold gives a
criterion that distinguishes the case when
$\lambda_{p}(K_{\operatorname{cyc}}/K)=1$ from the case when
$\lambda_{p}(K_{\operatorname{cyc}}/K)>1$. This essentially is due a formula
due to Federer, Gross and Sinnott [FG80] for the leading (non-zero) term of
the $p$-adic L-series which is expressed in terms of the $p$-part of the class
group of $K$, and the $p$-adic regulator for the $\mathbb{Z}_{p}$-extension
$K_{\operatorname{cyc}}/K$. The result follows from work of Gold [Gol74] and
Sands [San93].

######  Theorem 4.1.

Let $p$ be an odd prime which is split in an imaginary quadratic field $K$.
Write $p$ as a product $p\mathcal{O}_{K}=\mathfrak{p}\mathfrak{p}^{*}$, where
$\mathfrak{p}$ and $\mathfrak{p}^{*}$ are the primes of $K$ above $p$. The
following conditions are equivalent:

  1. (1)

$\lambda_{p}(K_{\operatorname{cyc}}/K)>1$,

  2. (2)

either

    1. (a)

$p$ divides the class number of $K$,

    2. (b)

or, $p$ does not divide the class number of $K$. Let $N$ be a positive integer
not divisible by $p$ such that $\mathfrak{p}^{N}=\alpha\mathcal{O}_{K}$, a
principal ideal in $K$. Then, $\alpha^{p-1}\equiv
1\mod{(\mathfrak{p}^{*})^{2}}$.

###### Proof.

The result follows from [San93, Proposition 2.1]. ∎

In [Hor87], Horie shows that for any odd prime $p$, there are infinitely many
imaginary quadratic fields $K$ in which $p$ splits such that
$\lambda_{p}(K_{\operatorname{cyc}}/K)>1$. On the other hand, Jochnowitz shows
in [Joc94] that there are infinitely many $K$ in which $p$ splits such that
$\lambda_{p}(K_{\operatorname{cyc}}/K)=1$. One may formulate a more refined
question guided by arithmetic statistics. Given an imaginary quadratic field
$K$, and a real number $x>0$, let $D_{K}$ denote its discriminant and let
$\mathcal{M}_{<x}^{p}$ be the set of all imaginary quadratic fields in which
$p$ splits with $|D_{K}|<x$. Note that this is a finite set. Let
$\mathcal{M}_{<x}^{\prime}$ be the subset of $\mathcal{M}_{<x}$ consisting of
imaginary quadratic fields for which
$\lambda_{p}(K_{\operatorname{cyc}}/K)=1$. Then, the upper (resp. lower
density) of imaginary quadratic fields $K$ in which $p$ splits and
$\lambda_{p}=1$ is taken to be
$\limsup_{x\rightarrow\infty}\frac{\\#\mathcal{M}_{<x}^{\prime}}{\\#\mathcal{M}_{<x}}$
(resp.
$\liminf_{x\rightarrow\infty}\frac{\\#\mathcal{M}_{<x}^{\prime}}{\\#\mathcal{M}_{<x}}$).
If the limit
$\lim_{x\rightarrow\infty}\frac{\\#\mathcal{M}_{<x}^{\prime}}{\\#\mathcal{M}_{<x}}$
exists, then it is called the density. Let
$\bar{\mathfrak{d}}_{\lambda=1}^{p}$,
$\underline{\mathfrak{d}}_{\lambda=1}^{p}$ be the upper and lower density
respectively, and note that if
$\bar{\mathfrak{d}}_{\lambda=1}^{p}=\underline{\mathfrak{d}}_{\lambda=1}^{p}$,
the density exists, and we denote it by $\mathfrak{d}_{\lambda=1}^{p}$.

######  Question 4.2.

Let $p$ be a prime. With respect to notation above, what can be said about the
quantities $\bar{\mathfrak{d}}_{\lambda=1}^{p}$ and
$\underline{\mathfrak{d}}_{\lambda=1}^{p}$? Are they equal, and if so, what is
$\mathfrak{d}_{\lambda=1}^{p}$?

Such questions seem to be difficult in the number field setting and are
currently being pursued by the second named author. We note here that this is
a very natural question, if fact, one can ask the same question for any choice
of $\lambda\geq 1$.

In this paper, we shall study topological analogs of the above question in
section 5. In this section, we shall prove a number of analogs for Theorem 4.1
in the topological setting.

Let $\mathcal{L}$ be a link with $r$ component knots
$\mathcal{K}_{1},\dots,\mathcal{K}_{r}$. Note that in the case when $r=1$, the
$\mu$ and $\lambda$-invariants are known to vanish by Lemma 3.4, hence we
study the case when $r\geq 2$. Recall that the Alexander module is presented
by the $r\times r$ matrix $\widehat{P}_{\mathcal{L}}$. Let $\widehat{F}$ be
the free pro-$p$ group on $x_{1},\dots,x_{r}$, where we recall that $x_{i}$
corresponds to the meridian for $\mathcal{K}_{i}$, and let $y_{i}$ be the
$i$-th longitude. Let $I=(i_{1},\dots,i_{n})$ be a multi-index, the _$p$ -adic
Milnor number_ is $\hat{\mu}(I)\mathrel{\mathop{\mathchar
58\relax}}=\epsilon\left(\partial_{I^{\prime}}(y_{i_{n}})\right)$, where
$I^{\prime}=(i_{1},\dots,i_{n-1})$. Furthermore, when $n=1$, we set
$\hat{\mu}(I)=0$. Given $\ell\geq 1$, and indices $i\neq j$, we have that

(4.1) $\hat{\mu}(\underbrace{i\cdots i}_{\ell}j)={\ell_{i,j}\choose\ell},$

where we recall the $\ell_{i,j}$ is the linking number for the knots
$\mathcal{K}_{i}$ and $\mathcal{K}_{j}$. In particular,
$\hat{\mu}(ij)=\ell_{i,j}$. These Milnor invariants may also be defined in
terms of Massey products in the cohomology of $\widehat{G}_{\mathcal{L}}$, see
[HMM06, p. 127] for further details.

######  Definition 4.3.

The $p$-adic Traldi matrix $\widehat{T}_{\mathcal{L}}$ over
$\widehat{\Lambda}_{r}$ is defined by

$\widehat{T}_{\mathcal{L}}(i,j)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}-\sum_{n\geq 1}\left(\sum_{1\leq i_{1},\dots,i_{n}\leq
r,i_{n}\neq i}\hat{\mu}(i_{1}\dots i_{n}i)X_{i_{1}}\dots
X_{i_{n}}\right)&\text{ if }i=j,\\\ \hat{\mu}(ij)X_{i}+\sum_{n\geq
1}\left(\sum_{1\leq i_{1},\dots,i_{n}\leq r}\hat{\mu}(i_{1}\dots
i_{n}ji)X_{i}X_{i_{1}}\cdots X_{i_{n}}\right)&\text{ if }i\neq j.\end{cases}$

######  Theorem 4.4.

The _$p$ -adic Traldi matrix_ $\widehat{T}_{\mathcal{L}}$ coincides with the
Alexander matrix $\widehat{P}_{\mathcal{L}}$.

###### Proof.

The above result is [HMM06, Theorem 3.2.2]. ∎

Let $z=(z_{1},\dots,z_{r})$ be such that $\prod_{i}z_{i}\neq 0$ and
$\operatorname{gcd}(z_{1},\dots,z_{r})=1$. Set $\widehat{T}_{\mathcal{L},z}$
to be the matrix with entries

$\widehat{T}_{\mathcal{L},z}(i,j)=\tau_{z}\left(\widehat{T}_{\mathcal{L}}(i,j)\right).$

In particular, $\widehat{T}_{\mathcal{L},\mathbf{1}}$ is given by

$\widehat{T}_{\mathcal{L},\mathbf{1}}(i,j)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}-\sum_{n\geq 1}\left(\sum_{1\leq i_{1},\dots,i_{n}\leq
r,i_{n}\neq i}\hat{\mu}(i_{1}\dots i_{n}i)\right)X^{n}&\text{ if }i=j,\\\
\hat{\mu}(ij)X+\sum_{n\geq 1}\left(\sum_{1\leq i_{1},\dots,i_{n}\leq
r}\hat{\mu}(i_{1}\dots i_{n}ji)\right)X^{n+1}&\text{ if }i\neq j.\end{cases}$

Given $k\geq 2$, the truncated Traldi matrix
$\widehat{T}_{\mathcal{L},z}^{(k)}$ is simply the mod $X^{k}$ reduction of
$\widehat{T}_{\mathcal{L},z}$. We find that

$\widehat{T}_{\mathcal{L},z}^{(2)}\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\left(\sum_{t\neq i}-z_{t}\ell_{t,i}\right)X&\text{ if
}i=j,\\\ \ell_{i,j}z_{i}X&\text{ if }i\neq j.\end{cases}.$

Here, we have used the relation $\hat{\mu}(ij)=\ell_{i,j}$, see (4.1). The
_linking matrix_ $C_{\mathcal{L},z}$ is defined by

$C_{\mathcal{L},z}(i,j)\mathrel{\mathop{\mathchar
58\relax}}=\begin{cases}\left(\sum_{t\neq i}-z_{t}\ell_{t,i}\right)&\text{ if
}i=j,\\\ \ell_{i,j}z_{i}&\text{ if }i\neq j,\end{cases}$

we find that $\widehat{T}_{\mathcal{L},z}^{(2)}$ is represented by
$XC_{\mathcal{L},z}\mod{X^{2}}$.

Given $i,j$ such that $1\leq i,j\leq r$, let $M_{i,j}$ denote the
$(r-1)\times(r-1)$ minor of $C_{\mathcal{L},z}$ obtained by deleting the
$i$-th row and $j$-th column. Fix the absolute value $|\cdot|_{p}$ on
$\mathbb{Q}_{p}$ normalized by $|p|_{p}=p^{-1}$ and set
$|0|_{p}\mathrel{\mathop{\mathchar 58\relax}}=0$. Set $c_{\mathcal{L},z}$ to
be the maximum of the values $|\operatorname{det}M_{i,j}|_{p}$ as $i,j$ range
over all values $1\leq i,j\leq r$. Note that
$\operatorname{rank}_{\mathbb{Q}_{p}}C_{\mathcal{L},z}<r-1$ if and only if all
determinants $\operatorname{det}M_{i,j}$ vanish, if and only if
$c_{\mathcal{L},z}=0$.

######  Proposition 4.5.

Let $z=(z_{1},\dots,z_{r})$ be an admissible integral and $\mathcal{L}$ a link
with exactly $r$ components $\mathcal{K}_{1},\dots,\mathcal{K}_{r}$. Then, the
order of vanishing satisfies the lower bound

$\operatorname{ord}_{X=0}\widehat{\Delta}_{\mathcal{L},z}(X)\geq r-1.$

Write $\widehat{\Delta}_{\mathcal{L},z}(X)$ as a sum

$\widehat{\Delta}_{\mathcal{L},z}(X)=a_{r-1}X^{r-1}+a_{r}X^{r}+\dots,$

where $a_{r-1}$ is the _leading coefficient_. The following assertions hold:

  1. (1)

$\operatorname{ord}_{X=0}\widehat{\Delta}_{\mathcal{L},z}(X)>r-1$ if and only
if $\operatorname{rank}_{\mathbb{Q}_{p}}C_{\mathcal{L},z}<r-1$.

  2. (2)

Suppose that $\operatorname{rank}_{\mathbb{Q}_{p}}C_{\mathcal{L},z}=r-1$,
then, $|a_{r-1}|_{p}=c_{\mathcal{L},z}$.

###### Proof.

Note that each entry of the matrix $\widehat{T}_{\mathcal{L},z}$ is divisible
by $X$. Since $\widehat{\Delta}_{\mathcal{L},z}(X)$ is generated by the
determinants of the $(r-1)\times(r-1)$ minors of
$\widehat{T}_{\mathcal{L},z}$, it follows that
$\operatorname{ord}_{X=0}\widehat{\Delta}_{\mathcal{L},z}(X)\geq r-1$. From
the relation

$\widehat{T}_{\mathcal{L},z}\equiv C_{\mathcal{L},z}\mod{X^{2}}$

we thus find that $|a_{r-1}|_{p}$ is equal to $c_{\mathcal{L},z}$. In
particular, $\operatorname{ord}_{X=0}\widehat{\Delta}_{\mathcal{L},z}(X)>r-1$
if and only if $c_{\mathcal{L},z}=0$, if and only if
$\operatorname{rank}_{\mathbb{Q}_{p}}C_{\mathcal{L},z}<r-1$. ∎

######  Lemma 4.6.

Let $f(X)\in\widehat{\Lambda}$ be given by the power series expansion

$f(X)=a_{d}X^{d}+a_{d+1}X^{d+1}+\dots,$

where $a_{d}\neq 0$, and let $\mu$ and $\lambda$ denote the $\mu$ and
$\lambda$-invariants of $f(X)$ respectively. Then, $\lambda\geq d$ and
following conditions are equivalent

  1. (1)

the $\mu=0$ and the $\lambda=d$,

  2. (2)

$a_{d}$ is a unit in $\mathbb{Z}_{p}$.

###### Proof.

Write $f(X)=X^{d}g(X)$ with $g(X)\mathrel{\mathop{\mathchar
58\relax}}=a_{r-1}+a_{r}X+a_{r+1}X^{2}+\dots$. Suppose that $\mu=0$ and
$\lambda=d$. Then, $f(X)=P(X)u(X)$, where $P(X)$ is a distinguished polynomial
of degree $d$ and $u(X)$ is a unit in $\widehat{\Lambda}$. Note that $X^{d}$
is distinguished and divides $P(X)$ and since the degree of $P(X)$ is equal to
$d$, it follows that $P(X)=X^{d}$ and $g(X)=u(X)$ is a unit in
$\widehat{\Lambda}$. It follows that $a_{r-1}$ is a unit in $\mathbb{Z}_{p}$.

Conversely, if $a_{r-1}$ is a unit in $\mathbb{Z}_{p}$, then $g(X)$ is a unit
in $\Lambda$ and hence, $\mu=0$ and $\lambda=\operatorname{deg}X^{d}=d$. ∎

We apply the above result to the study of Iwasawa invariants. Let
$\bar{C}_{\mathcal{L},z}$ be the mod-$p$ reduction of $C_{\mathcal{L},z}$.

######  Corollary 4.7.

Let $\mathcal{L}$ be a link of $r\geq 2$ component knots and $z$ an integral
vector. Then, we have that $\lambda_{\mathcal{L},z}\geq r-1$. Furthermore, the
following conditions are equivalent

  1. (1)

$\mu_{\mathcal{L},z}=0$ and $\lambda_{\mathcal{L},z}=r-1$,

  2. (2)

$c_{\mathcal{L},z}$ is a unit in $\mathbb{Z}_{p}$,

  3. (3)

$\operatorname{rank}_{\mathbb{F}_{p}}\bar{C}_{\mathcal{L},z}=r-1$.

###### Proof.

Let $f(X)$ denote the completed Alexander polynomial
$\widehat{\Delta}_{\mathcal{L},z}(X)$. Note that according to Proposition 4.5,
$\operatorname{ord}_{X=0}f(X)\geq r-1$. In particular, this implies that
$\lambda_{\mathcal{L},z}\geq r-1$. We write
$f(X)=a_{r-1}X^{r-1}+a_{r}X^{r}+\dots$, and it follows from Proposition 4.5
that $|a_{r-1}|_{p}=c_{\mathcal{L},z}$. It follows from Lemma 4.6 that
$c_{\mathcal{L},z}$ is a unit in $\mathbb{Z}_{p}$ if and only if
$\mu_{\mathcal{L},z}=0$ and $\lambda_{\mathcal{L},z}=r-1$. It is clear that
$c_{\mathcal{L},z}$ is a unit in $\mathbb{Z}_{p}$ if and only if
$\operatorname{rank}_{\mathbb{F}_{p}}\bar{C}_{\mathcal{L},z}=r-1$. ∎

Let us explain the above result for small values of $r$.

######  Corollary 4.8.

Let $\mathcal{L}=\mathcal{K}_{1}\cup\mathcal{K}_{2}$ be a link consisting of
$2$-components, and $z=(z_{1},z_{2})$ an admissible integral vector. Then,
$\lambda_{\mathcal{L},z}\geq 1$ and the following conditions are equivalent

  1. (1)

$\mathrm{ord}_{X=0}\widehat{\Delta}_{\mathcal{L},z}(X)=1$,

  2. (2)

$\ell_{1,2}\neq 0$.

Furthermore, the following conditions are equivalent

  1. (1)

$\mu_{\mathcal{L},z}=0$ and $\lambda_{\mathcal{L},z}=1$.

  2. (2)

$p\nmid\ell_{1,2}$

###### Proof.

We have that
$C_{\mathcal{L},z}=\left({\begin{array}[]{cc}-z_{2}\ell_{1,2}&z_{1}\ell_{1,2}\\\
z_{2}\ell_{1,2}&-z_{1}\ell_{1,2}\\\ \end{array}}\right)$. According to
Proposition 4.5,

$\operatorname{ord}_{X=0}\widehat{\Delta}_{\mathcal{L},z}(X)=1$

if and only if $\operatorname{rank}_{\mathbb{Q}_{p}}C_{\mathcal{L},z}=1$. This
is equivalent to the non-vanishing of $\ell_{1,2}$.

Since it is assumed that $\operatorname{gcd}(z_{1},z_{2})=1$, it follows that
either $z_{1}$ or $z_{2}$ (or both) are not divisible by $p$. Therefore,
$\operatorname{rank}_{\mathbb{F}_{p}}\bar{C}_{\mathcal{L},z}=1$ if an only if
$p\nmid\ell_{1,2}$. ∎

######  Corollary 4.9.

Let $\mathcal{L}=\mathcal{K}_{1}\cup\mathcal{K}_{2}\cup\mathcal{K}_{3}$ be a
link consisting of $3$-components, and $z$ an admissible integral vector.
Then, $\lambda_{\mathcal{L},z}\geq 2$ and the following conditions are
equivalent

  1. (1)

$\operatorname{ord}_{X=0}\widehat{\Delta}_{\mathcal{L},z}(X)=2$,

  2. (2)

$(z_{2}\ell_{2,1}\ell_{3,2}+z_{1}\ell_{3,1}\ell_{1,2}+z_{3}\ell_{3,1}\ell_{3,2})\neq
0$, or,
$z_{2}\ell_{2,1}\ell_{3,2}+z_{1}\ell_{3,1}\ell_{1,2}+z_{3}\ell_{3,1}\ell_{3,2}\neq
0$ (or both).

Furthermore, the following conditions are equivalent:

  1. (1)

$\mu_{\mathcal{L},z}=0$ and $\lambda_{\mathcal{L},z}=2$.

  2. (2)

Both of the following conditions are satisfied

    1. (a)

$p\nmid z_{1}z_{2}z_{3}$,

    2. (b)

$p\nmid\left(z_{2}\ell_{2,1}\ell_{3,2}+z_{1}\ell_{3,1}\ell_{1,2}+z_{3}\ell_{3,1}\ell_{3,2}\right)$,
or,
$p\nmid\left(z_{2}\ell_{2,1}\ell_{3,2}+z_{1}\ell_{3,1}\ell_{1,2}+z_{3}\ell_{3,1}\ell_{3,2}\right)$
(or both).

###### Proof.

We have that
$C_{\mathcal{L},z}=\left({\begin{array}[]{ccc}-(z_{2}\ell_{2,1}+z_{3}\ell_{3,1})&z_{1}\ell_{1,2}&z_{1}\ell_{1,3}\\\
z_{2}\ell_{2,1}&-(z_{1}\ell_{1,2}+z_{3}\ell_{3,2})&z_{2}\ell_{2,3}\\\
z_{3}\ell_{3,1}&z_{3}\ell_{3,2}&-(z_{1}\ell_{1,3}+z_{2}\ell_{2,3})\\\
\end{array}}\right)$. Proposition 4.5 asserts that
$\operatorname{ord}_{X=0}\widehat{\Delta}_{\mathcal{L},z}(X)=2$ if and only if
$\operatorname{rank}_{\mathbb{Q}_{p}}C_{\mathcal{L},z}=2$. Note that the sum
of the rows of $C_{\mathcal{L},z}$ is equal to $0$, hence,
$\operatorname{rank}_{\mathbb{Q}_{p}}(C_{\mathcal{L},z})=2$ if and only if the
first two rows are linearly independent. This is the case when at least one of
the determinants

$\begin{split}&\operatorname{det}\left({\begin{array}[]{cc}-(z_{2}\ell_{2,1}+z_{3}\ell_{3,1})&z_{1}\ell_{1,2}\\\
z_{2}\ell_{2,1}&-(z_{1}\ell_{1,2}+z_{3}\ell_{3,2})\\\ \end{array}}\right)\\\
=&z_{3}\left(z_{2}\ell_{2,1}\ell_{3,2}+z_{1}\ell_{3,1}\ell_{1,2}+z_{3}\ell_{3,1}\ell_{3,2}\right),\text{
and }\\\
&\operatorname{det}\left({\begin{array}[]{cc}z_{1}\ell_{1,2}&z_{1}\ell_{1,3}\\\
-(z_{1}\ell_{1,2}+z_{3}\ell_{3,2})&z_{2}\ell_{2,3}\\\ \end{array}}\right)\\\
=&z_{1}\left(z_{2}\ell_{1,2}\ell_{2,3}+z_{1}\ell_{1,3}\ell_{1,2}+z_{3}\ell_{1,3}\ell_{3,2}\right).\end{split}$

Note that $z_{1}z_{2}z_{3}\neq 0$ and thus,
$\operatorname{rank}_{\mathbb{Q}_{p}}(C_{\mathcal{L},z})=2$ if and only if
$(z_{2}\ell_{2,1}\ell_{3,2}+z_{1}\ell_{3,1}\ell_{1,2}+z_{3}\ell_{3,1}\ell_{3,2})\neq
0$, or,
$z_{2}\ell_{2,1}\ell_{3,2}+z_{1}\ell_{3,1}\ell_{1,2}+z_{3}\ell_{3,1}\ell_{3,2}\neq
0$ (or both).

Corollary 4.7 asserts that $\mu_{\mathcal{L},z}=0$ and
$\lambda_{\mathcal{L},z}=2$ if and only if
$\operatorname{rank}_{\mathbb{F}_{p}}\bar{C}_{\mathcal{L},z}$. This is
equivalent to the above determinants not being divisible by $p$. The result
follows from this. ∎

##  5\. Statistics for Iwasawa invariants of $2$-bridge links

In this section, we study the average behavior of the Iwasawa invariants
associated to links $\mathcal{L}$ consisting of two component knots
$\mathcal{K}_{1}$ and $\mathcal{K}_{2}$. We formulate a precise problem in
arithmetic statistics. The links $\mathcal{L}$ that we consider are presented
as _$2$ -bridge links_ (see [Kaw96, p.18]).

To each oriented $2$-bridge link $\mathcal{L}$ there corresponds a relatively
prime pair of integers $(a,b)$ with $0<a<2b$, and $b$ odd when $\mathcal{L}$
is a knot and $b$ even when $\mathcal{L}$ is a link with two knot components
$\mathcal{K}_{1}$ and $\mathcal{K}_{2}$. We shall denote the link associated
to the pair $(a,b)$ by $\mathcal{L}_{a/b}$. This construction is standard
parametrization, and referred to as the _Schubert normal form_ and we refer to
[Tul81, p.542, l.-6 to l.-1] or [Kaw96, chapter 2] for an explicit description
of these links. Throughout, we shall assume that $b$ is even, thus
$\mathcal{L}_{a/b}$ has two components. Since $a$ is assumed to be coprime to
$b$, it follows that $a$ must be odd. Choose an orientation on $S^{3}$. All
links we shall consider shall also be provided with an orientation.

######  Definition 5.1.

[Kaw96, Definition 0.3.1] Two (oriented) piecewise linear links $\mathcal{L}$
and $\mathcal{L}^{\prime}$ in $S^{3}$ have the same type if there is a
piecewise linear orientation preserving homeomorphism
$h\mathrel{\mathop{\mathchar 58\relax}}S^{3}\rightarrow S^{3}$ such that
$h(\mathcal{L})=\mathcal{L}^{\prime}$ and $h_{|L}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{L}\rightarrow\mathcal{L}^{\prime}$ is orientation
preserving.

We shall consider equivalence classes of links $\mathcal{L}_{a/b}$ of the same
type. The following result explain when two pairs $\mathcal{L}_{a/b}$ and
$\mathcal{L}_{a^{\prime}/b^{\prime}}$ have the same type. Note that we are
assuming here that $b$ and $b^{\prime}$ are both even.

######  Theorem 5.2.

The $2$-component links $\mathcal{L}_{a/b}$ and
$\mathcal{L}_{a^{\prime}/b^{\prime}}$ belong to the same type if and only if

$b=b^{\prime}\text{ and }a^{\prime}a\equiv 1\mod{2b}.$

###### Proof.

We refer to [Kaw96, Theorem 2.1.3] for this result. ∎

For example, it follows from the above that the oriented links
$\mathcal{L}_{3/10}$ and $\mathcal{L}_{7/10}$ have the same type.

######  Remark 5.3.

The conventions in [Kaw96] involve a pair $(\alpha,\beta)$ that are coprime
with $-\alpha<\beta<\alpha$, and $\alpha>0$. The conventions used here are
equivalent to those of [Kaw96], however, shall be of more convenience to work
with. In order to see that the conventions indeed match up, compare the
construction of 2-bridge links in [Kaw96] with that in [Tul81]. We are using
the convention from the latter reference.

Given a rational number $a/b$ with $(a,b)=1$, and $a,b>0$, define the counting
function $\operatorname{ht}(\mathcal{L}_{a/b})\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{max}\\{a^{2},b\\}$, which we shall refer to as the
_height_ of $\mathcal{L}_{a/b}$. We do not work with the conventional notion
of height of the fraction $a/b$, and as far as we are aware, there is no
conventional notion of height that is used in such a context. In fact, our
computations have led us to believe that the result is not true for an
alternate definition of height
$\operatorname{ht}^{\prime}(a/b)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{max}\\{a,b\\}$. This proves to be an adequate
counting function to serve the purpose of formulating and resolving
statistical problem for Iwasawa invariants. We abbreviate
$\operatorname{ht}(\mathcal{L}_{a/b})$ to $\operatorname{ht}(a/b)$ in order to
simplify notation.

We shall study the average behavior of Iwasawa invariants of
$\mathcal{L}_{a/b}$, where $\operatorname{ht}(a/b)$ goes to infinity. Let
$\mathcal{N}$ be the set of all fractions $a/b$ with $(a,b)=1$, $b$ even,
$0<a<2b$. For $x>0$, set $\mathcal{N}_{<x}$ to be the subset of $\mathcal{N}$
consisting of fractions $a/b$ such that $\operatorname{ht}(a/b)<x$. It is easy
to see that $\mathcal{N}_{<x}$ is finite, we set $\\#\mathcal{N}_{<x}$ to
denote its cardinality. Given a subset $\mathcal{S}$ of $\mathcal{N}$, let
$\mathcal{S}_{<x}\mathrel{\mathop{\mathchar
58\relax}}=\mathcal{S}\cap\mathcal{N}_{<x}$. We say that $\mathcal{S}$ has
density $\mathfrak{d}(\mathcal{S})$ if the following limit exists

$\mathfrak{d}(\mathcal{S})\mathrel{\mathop{\mathchar
58\relax}}=\lim_{x\rightarrow\infty}\frac{\\#\mathcal{S}_{<x}}{\\#\mathcal{N}_{<x}}.$

Denote by $\mathcal{S}^{p,z}$ the set of all fractions $a/b$ in $\mathcal{N}$
such that $\mu_{p,\mathcal{L}_{a/b},z}=0$ and
$\lambda_{p,\mathcal{L}_{a/b},z}=1$. Our main result is as follows.

######  Theorem 5.4.

Let $p$ be a fixed odd prime. With respect to notation above, the set
$\mathcal{S}^{p,z}$ has density given by
$\mathfrak{d}(\mathcal{S}^{p,z})=1-1/p$.

######  Remark 5.5.

We thus study the proportion of $\mathcal{L}_{a/b}$ with $2$-components and
Iwasawa invariants $\mu=0$ and $\lambda=1$. Note that for simplicity we do not
count links up to equivalence, and thus the equivalence class of
$\mathcal{L}_{a,b}$ is counted twice if $a^{2}\not\equiv 1\mod{2b}$ and only
once if $a^{2}\equiv 1\mod{2b}$.

In order to ease notation, we set $\mu_{p,a/b,z}\mathrel{\mathop{\mathchar
58\relax}}=\mu_{p,\mathcal{L}_{a/b},z}$ and
$\lambda_{p,a/b,z}\mathrel{\mathop{\mathchar
58\relax}}=\lambda_{p,\mathcal{L}_{a/b},z}$. Write
$\mathcal{L}=\mathcal{K}_{1}\cup\mathcal{K}_{2}$, and recall that $\ell_{1,2}$
is the linking number of $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$. We set
$\ell_{a/b}$ to denote the linking number $\ell_{1,2}$ for the two component
knots in $\mathcal{L}_{a/b}$. Note that this only makes sense since $b$ is
assumed to be even. Corollary 4.8 asserts that $\lambda_{p,a/b,z}\geq 1$ and
the following conditions are equivalent:

  1. (1)

$\mu_{p,a/b,z}=0$ and $\lambda_{p,a/b,z}=1$,

  2. (2)

$p\nmid\ell_{a/b}$.

Note that the second condition is independent of $z$, and hence, so is the
first condition. We find that $a/b$ belongs to $\mathcal{S}^{p,z}$ if and only
if $p\nmid\ell_{a/b}$, and this condition is independent of $z$. Also observe
that if $\lambda_{p,a/b,z}=1$, then in particular, we have that
$\widehat{\Delta}_{\mathcal{L},z}\neq 0$ and
$\operatorname{ord}_{X=0}\widehat{\Delta}_{\mathcal{L},z}=1$. Before giving
the proof of Theorem 5.4, we prove a few preparatory results.

For $i\in\mathbb{Z}$, set $\epsilon_{i}\mathrel{\mathop{\mathchar
58\relax}}=(-1)^{\lfloor\frac{ia}{b}\rfloor}$. It is easy to see from the
construction of $\mathcal{L}_{a/b}$ that $\ell_{a/b}$ is given by

(5.1) $\ell_{a/b}=\sum_{i=1}^{b/2}\epsilon_{2i-1},$

we refer to [Tul81] for further details.

######  Lemma 5.6.

The following relation is satisfied

$\ell_{a/(b+2a)}=\ell_{a/b}+1.$

###### Proof.

For $0\leq j\leq(a-1)$, let $c_{j}(a/b)$ be the number of odd integers $2i-1$
such that

$\frac{jb}{a}\leq 2i-1<\frac{(j+1)b}{a}.$

Note that $2i-1$ is in the range $\frac{jb}{a}\leq 2i-1<\frac{(j+1)b}{a}$, if
and only if $\lfloor\frac{(2i-1)a}{b}\rfloor=j$. Therefore, $c_{j}(a/b)$ is
the number of odd numbers $2i-1$ such that
$\lfloor\frac{(2i-1)a}{b}\rfloor=j$. Thus the formula (5.1) yields

(5.2) $\ell_{a/b}=\sum_{j=0}^{a-1}(-1)^{j}c_{j}(a/b).$

Next, we show that $c_{j}(\frac{a}{b+2a})=c_{j}(\frac{a}{b})+1$. Note that
$c_{j}(\frac{a}{b+2a})$ is the number of odd integers $2i-1$ such that

$\frac{jb}{a}+2j\leq 2i-1<\frac{(j+1)b}{a}+2j+2.$

Let $\\{2i_{1}-1,\dots,2i_{c_{j}}-1\\}$ be the odd numbers such that

$\frac{jb}{a}\leq 2i_{k}-1<\frac{(j+1)b}{a},$

here, $c_{j}=c_{j}(a/b)$. Then, $\\{2i_{1}+2j-1,\dots,2i_{c_{j}}+2j-1\\}$ are
the odd numbers $2i-1$ such that

$\frac{jb}{a}+2j\leq 2i-1<\frac{(j+1)b}{a}+2j.$

There is exactly one odd number $2i-1$ in the range

$\frac{(j+1)b}{a}+2j\leq 2i-1<\frac{(j+1)b}{a}+2j+2,$

namely, $2i-1=2i_{c_{j}}+2j+1$. Hence, this shows that
$c_{j}(\frac{a}{b+2a})=c_{j}(\frac{a}{b})+1$. It follows from (5.2) that

$\begin{split}\ell_{a/(b+2a)}=&\sum_{j=0}^{a-1}(-1)^{j}c_{j}\left(a/(b+2a)\right)\\\
=&\sum_{j=0}^{a-1}(-1)^{j}\left(c_{j}(a/b)+1\right)\\\
=&\ell_{a/b}+\sum_{j=0}^{a-1}(-1)^{j}\\\ =&\ell_{a/b}+1.\\\ \end{split}$

The last equality holds since $a$ is odd. ∎

Since the set $\mathcal{S}^{p,z}$ consists of all fractions
$a/b\in\mathcal{N}$ with $p\nmid\ell_{a/b}$, we find that $\mathcal{S}^{p,z}$
does not depend on the choice of $z$. Thus, to simplify notation, we denote
this set by $\mathcal{S}^{p}$. Given a fixed number $a>0$, set

$\begin{split}&\mathcal{N}_{<x}(a)\mathrel{\mathop{\mathchar
58\relax}}=\\{b\mid 2b>a,a/b\in\mathcal{N}_{<x}\\}.\\\
&\mathcal{S}_{<x}^{p}(a)\mathrel{\mathop{\mathchar 58\relax}}=\\{b\mid
2b>a,a/b\in\mathcal{S}_{<x}^{p}\\}.\\\ \end{split}$

######  Lemma 5.7.

Let $a>0$, with respect to notation above

$|\frac{1}{p}\\#\mathcal{N}_{<x}(a)-\\#\mathcal{S}_{<x}^{p}(a)|<2a.$

###### Proof.

The relation $\ell_{a/(b+2a)}=\ell_{a/b}+1$ shows that among every $2ap$
consecutive numbers $b$ in $\mathcal{N}_{<x}(a)$, there are exactly $2a$ of
them that belong to $\mathcal{S}^{p}_{<x}(a)$. Therefore, we find that

$|\\#\mathcal{N}_{<x}(a)-p\\#\mathcal{S}^{p}_{<x}(a)|<2ap.$

The result follows from this. ∎

######  Lemma 5.8.

With respect to notation above,

$|\frac{1}{p}\\#\mathcal{N}_{<x}-\\#\mathcal{S}_{<x}^{p}|<x+\sqrt{x}.$

###### Proof.

Note that $\operatorname{ht}(a/b)=\operatorname{max}\\{a^{2},b\\}$, thus if
$\operatorname{ht}(a/b)<x$, we have that $a<\lfloor\sqrt{x}\rfloor$. We find
that

$\begin{split}&|\frac{1}{p}\\#\mathcal{N}_{<x}-\\#\mathcal{S}_{<x}^{p}|\\\
\leq&\sum_{a=1}^{\lfloor\sqrt{x}\rfloor}|\frac{1}{p}\\#\mathcal{N}_{<x}(a)-\\#\mathcal{S}_{<x}^{p}(a)|\\\
<&\sum_{a=1}^{\lfloor\sqrt{x}\rfloor}2a=\lfloor\sqrt{x}\rfloor(\lfloor\sqrt{x}\rfloor+1).\\\
\end{split}$

Note that in the last step, we have invoked Lemma 5.7. The result follows. ∎

######  Lemma 5.9.

With respect to notation above,

$\\#\mathcal{N}_{<x}\geq x^{3/2}-\frac{5}{4}(x+\sqrt{x}).$

###### Proof.

We find that

$\\#\mathcal{N}_{<x}=\sum_{a=1}^{\lfloor\sqrt{x}\rfloor}\\#\mathcal{N}_{<x}(a).$

Recall that $\mathcal{N}_{<x}(a)$ is the set of numbers $b$ such that $2b>a$
and $a/b\in\mathcal{N}_{<x}$. Since $a\leq\lfloor\sqrt{x}\rfloor$, the
condition $a/b\in\mathcal{N}_{<x}$ simply becomes $b\leq\lfloor x\rfloor$.
Therefore, we have that $\\#\mathcal{N}_{<x}(a)\geq x-a/2-1$, and thus,

$\begin{split}\\#\mathcal{N}_{<x}\geq&\sum_{a=1}^{\lfloor\sqrt{x}\rfloor}(x-a/2-1).\\\
=&\lfloor\sqrt{x}\rfloor
x-\frac{1}{4}\lfloor\sqrt{x}\rfloor(\lfloor\sqrt{x}\rfloor+1)-\lfloor\sqrt{x}\rfloor\\\
>&x^{3/2}-x-\frac{1}{4}\sqrt{x}(\sqrt{x}+1)-\sqrt{x}\\\
=&x^{3/2}-\frac{5}{4}(x+\sqrt{x}).\end{split}$

∎

###### Proof of Theorem 5.4.

Recall that Lemma 5.7 asserts that

$\left|\frac{1}{p}\\#\mathcal{N}_{<x}-\\#\mathcal{S}_{<x}^{p}\right|<x+\sqrt{x},$

and therefore,

$\left|\frac{1}{p}-\frac{\\#\mathcal{S}_{<x}^{p}}{\\#\mathcal{N}_{<x}}\right|<\frac{x+\sqrt{x}}{\\#\mathcal{N}_{<x}}.$

Invoking Lemma 5.9, we find that

$\left|\frac{1}{p}-\frac{\\#\mathcal{S}_{<x}^{p}}{\\#\mathcal{N}_{<x}}\right|<\frac{x+\sqrt{x}}{\left(x^{3/2}-\frac{5}{4}(x+\sqrt{x})\right)},$

which clearly goes to $0$ as $x\rightarrow\infty$. The result thus follows.

∎

######  Remark 5.10.

Note that the above proof in fact shows that the error term is bounded by
$\frac{x+\sqrt{x}}{\left(x^{3/2}-\frac{5}{4}(x+\sqrt{x})\right)}$, a quantity
which is asymptotic to ${x}^{-1/2}$.

Therefore, we have shown that the Iwasawa invariants do satisfy
$\mu_{p,a/b,z}=0$ and $\lambda_{p,a/b,z}=1$ for a density of $(1-1/p)$ of the
$2$-bridge links parametrized according to
$\operatorname{ht}(\mathcal{L}_{a/b})=\operatorname{max}\\{a^{2},b\\}$. This
does not however give much information about the links for which
$\mu_{p,a/b,z}>0$ or $\lambda_{p,a/b,z}>1$. We do believe that more answers
are obtainable for such links, for instance, it is certainly of interest to
determine if indeed $\mu=0$ holds for a density $1$ set of links parametrized
in this way for the integral vector $z=\mathbf{1}=(1,1)$. Although the authors
have not been able to answer this question, it seems that some more refined
answers may be obtainable when $a$ is chosen to a fixed odd number, and $b$ is
allowed to vary over all even integers such that $2b>a$. We illustrate this by
choosing $a=1$, thus, we are interested in the Iwasawa invariants for the
family of links $\mathcal{L}_{1,n}$ parametrized with respect to the variable
$n$. Note that $p$ is a fixed odd prime and we take $z=\mathbf{1}=(1,1)$ for
simplicity. In this setting, $\Delta_{a/b}(t)\mathrel{\mathop{\mathchar
58\relax}}=\Delta_{\mathcal{L}_{a/b},\mathbf{1}}(t)$ is given by a formula due
to Minkus, see [Min82] or [Hos20, Theorem 1]

(5.3)
$\Delta_{a/b}(t)=\sum_{k=0}^{b-1}(-1)^{k}t^{\sum_{i=0}^{k}\epsilon_{k}},$

where we recall that $\epsilon_{k}=(-1)^{\lfloor\frac{ka}{b}\rfloor}$. Note
that since for the fraction $1/n$, we have that $\epsilon_{k}=1$ since $k<n$
in the above sum. Therefore, we have that

(5.4)
$\Delta_{1/n}(t)=\sum_{k=0}^{n-1}(-1)^{k}t^{k}=\frac{(-t)^{n}-1}{-t-1}=\frac{t^{n}-1}{-t-1},$

where the last equality follows since $n$ is even.

######  Lemma 5.11.

Let $f(X)=\sum_{i}c_{i}X^{i}\in\widehat{\Lambda}$ be a formal power series
whose coefficients are not all divisible by $p$, then, the $\mu$-invariant is
$0$. Furthermore, the $\lambda$-invariant is the smallest value of $i$ such
that $p\nmid c_{i}$.

###### Proof.

We omit the proof since it is a straightforward exercise involving the
Weierstrass factorization theorem. ∎

######  Theorem 5.12.

Assume that $p$ is an odd prime. With respect to notation above, the following
assertions hold:

  1. (1)

$\mu_{p,1/n,\mathbf{1}}=0$ for all $n$.

  2. (2)

Write $n=p^{t}m$ with $p\nmid m$, then, $\lambda_{p,1/n,\mathbf{1}}=p^{t}$.

###### Proof.

From (5.4), we find that

$\widehat{\Delta}_{1/n}(X)=\Delta_{1/n}(1+X)=-\frac{(1+X)^{n}-1}{(2+X)}=-\frac{\sum_{i=1}^{n}{n\choose
i}X^{i}}{(2+X)}.$

Note that since $p>2$, the denominator $(2+X)$ is a unit in
$\widehat{\Lambda}$. Recall that $n=p^{t}m$ where $m$ is comprime to $p$. On
the other hand, it follows from elementary properties of binomial coefficients
that $n\choose i$ is divisible by $p$ for all $i<p^{t}$, and $p\nmid{n\choose
p^{t}}$. Since $p\nmid{n\choose p^{t}}$, it follows that $p$ does not divide
all coefficients of $\widehat{\Delta}_{1/n}(X)$, hence,
$\mu_{p,1/n,\mathbf{1}}=0$. Thus, from Lemma 5.11, it follows that
$\lambda_{p,1/n,\mathbf{1}}$ is the minimal value $i$ such that
$p\nmid{n\choose i}$. Thus, we have shown that
$\lambda_{p,1/n,\mathbf{1}}=p^{t}$. ∎

In light of Corollary 4.9, we expect that the results proven in this section
for the family of $2$-bridge links potentially generalize to the family of
$3$-bridge links. We do not however pursue such questions in this paper.

##  6\. Evidence for $\mu=0$ for $100\%$ of $2$-bridge links

In this final section, we study the vanishing of the $\mu$-invariant for
$2$-bridge links $\mathcal{L}=\mathcal{L}_{a/b}$, for the integral vector
$z=\mathbf{1}=(1,1)$. Note that this vector specifies a $\mathbb{Z}_{p}$-cover
of $X_{\mathcal{L}}$, whose Alexander polynomial is given by (5.4). There is a
parallel to this on the arithmetic side, namely, the cyclotomic
$\mathbb{Z}_{p}$-extension of a quadratic number field. By the celebrated
theorem of Fererro and Washington [FW79], for any prime $p$, the
$\mu$-invariant is known to vanish for the cyclotomic
$\mathbb{Z}_{p}$-extension $F_{\operatorname{cyc}}$ of any abelian number
field extension $F/\mathbb{Q}$. In analogy with the number fields case, one
may ostensibly predict that $\mu_{p,a/b,\mathbf{1}}=0$ for all $2$-bridge
links $\mathcal{L}_{a/b}$. However, this is seen to not be the case by
examining some examples. For instance, the completed Alexander polynomial for
$\mathcal{L}_{5/6}$ and $z=\mathbf{1}$ is given by

$\widehat{\Delta}_{5/6}(X)=3(1+X)(1-X)$

by Minkus’ formula (5.3), and thus, $\mu_{3,5/6,\mathbf{1}}=1$. Our
computations however indicate that the $\mu$-invariant vanishes _on average_ ,
and have led us to the following conjecture.

######  Conjecture 6.1.

Let $p$ be a fixed odd prime. Denote by $\mathcal{U}^{p}$ the set of all
fractions $a/b$ in $\mathcal{N}$ such that $\mu_{p,a/b,\mathbf{1}}=0$. Then,
the set $\mathcal{U}^{p}$ has density given by
$\mathfrak{d}(\mathcal{U}^{p})=1$, where

$\mathfrak{d}(\mathcal{U}^{p})\mathrel{\mathop{\mathchar
58\relax}}=\lim_{x\rightarrow\infty}\frac{\\#\mathcal{U}^{p}_{<x}}{\\#\mathcal{N}_{<x}}.$

To simplify the notation, set

$\mathfrak{d}_{<x}\mathrel{\mathop{\mathchar
58\relax}}=\frac{\\#\mathcal{U}^{p}_{<x}}{\\#\mathcal{N}_{<x}}.$

Our computations were performed on SageMath, see [Ste07] For our computations,
we approximate $(1+X)^{-1}$ by $\sum_{i=0}^{9}(-1)^{i}x^{i}$. and we compute
completed Alexander polynomials by the formula (5.3). Our code is available at
this link.

$p$ | $\mathfrak{d}_{<1000}$
---|---
3 | 0.99539
5 | 0.99820
7 | 0.99887
11 | 0.99955
13 | 0.99977
17 | 0.99977
19 | 0.99977
23 | 1
Table 1. Proportion $\mathfrak{d}_{<1000}$ of $2$-bridge links
$\mathcal{L}_{a/b}$ up to $\mathrm{ht}(a/b)=1000$ such that
$\mu_{p,a/b,\mathbf{1}}=0$.

The numbers on the right are soon to approach $1$ as $x$ gets larger, and thus
this gives us reason to believe that the limit is $1$ as $x\rightarrow\infty$.
The rate of convergence seems to be faster as $p$ gets larger.

######  Remark 6.2.

Computations made with various other definitions of height do not yield the
same results, at least it seems that the rate of convergence to $1$ is much
slower for with respect to the counting functions
$\operatorname{ht}^{\prime}(a/b)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{max}\\{a,b\\}$ or
$\operatorname{ht}^{\prime\prime}(a/b)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{max}\\{a,b^{2}\\}$. We have similar expectations for
all $z=(z_{1},z_{2})$ such that $p\nmid z_{1}z_{2}$.

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