#  $k$-reduced groups and Milnor invariants

Benjamin Audoux  Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille,
France<EMAIL_ADDRESS>,  Jean-Baptiste Meilhan  Univ. Grenoble
Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France  jean-
<EMAIL_ADDRESS>and  Akira Yasuhara  Faculty of
Commerce, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050,
Japan<EMAIL_ADDRESS>
###### Abstract.

We characterize, in an algebraic and in a diagrammatic way, Milnor string link
invariants indexed by sequences where any index appears at most $k$ times, for
any fixed $k\geq 1$. The algebraic characterization is given in terms of an
Artin-like action on the so-called $k$–reduced free groups; the diagrammatic
characterization uses the langage of welded knot theory. The link case is also
addressed.

## Introduction

In his seminal works [20, 21], J. Milnor introduced a family of concordance
invariants for $n$-component links, which can be seen as a wide generalization
of the linking number. Indexed by sequences $I$ of possibly repeating indices
in $\\{1,\ldots,n\\}$, these _Milnor invariants_ $\mu(I)$ are integers
extracted from longitudes within the fundamental group of the link complement,
well-defined only modulo a subtle indeterminacy. The geometric meaning of this
indeterminacy was clarified by the work of N. Habegger and X.S. Lin [12], who
showed that Milnor invariants are actually well-defined integer-valued
invariants of _string links_ , which are pure tangles without closed
components.

Recall that two (string) links are _link-homotopic_ if they are related by a
sequence of isotopies and crossing changes involving two strands of a same
component. Milnor proved in [21] that, if a sequence $I$ is without
repetition, then Milnor invariant $\overline{\mu}(I)$ is in fact a link-
homotopy invariant. He further showed how these non-repeated invariants can be
extracted from the _reduced_ fundamental group of the link complement, which
is is the ‘maximal’ quotient where each meridian commutes with any of its
conjugates. Using an Artin-like action of string links on the reduced free
group, Habegger and Lin actually showed that two string links have same Milnor
invariants indexed by sequences without repetition, if and only if they are
link-homotopic [12].

The purpose of this paper is to give a higher-order version of this
classification result of Habegger and Lin. Namely, we characterize the
information contained by Milnor invariants $\mu(I)$ of string links with
$r(I)\leq k$, where $r(I)$ denotes the maximum number of time that any index
appears in $I$. (In particular, Milnor invariants $\mu(I)$ with $r(I)=1$ are
precisely Milnor link-homotopy invariants.)
Our main result is the following; explanations of terminologies and notation
shall follow.

######  Main Theorem.

Let $L$ and $L^{\prime}$ be two $n$-component string links. The following are
equivalent.

  1. (1)

$\mu_{L}(I)=\mu_{L^{\prime}}(I)$ for any sequence $I$ with $r(I)\leq k$.

  2. (2)

$L$ and $L^{\prime}$ induce the same $k$–reduced free action:
$\varphi_{L}^{(k)}=\varphi_{L^{\prime}}^{(k)}\in\operatorname{Aut}_{c}(R_{k}F_{n})$.

  3. (3)

$L$ and $L^{\prime}$ are self $w_{k}$-concordant.

Let us first explain assertion (2). Let $G(L)$ be the fundamental group of the
complement of $L$. This group is normally generated by $n$ meridians, and for
each $i$, we denote by $N_{i}$ the normal subgroup of $G(L)$ generated by the
$i$th meridian. The _$k$ –reduced quotient_ of $G(L)$ is defined by

$\textnormal{R}_{k}G(L):=\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}N_{1}\cdots\Gamma_{k+1}N_{n}$}}},$

where $\Gamma_{q}N$ denotes the $q$th term of the lower central series of a
group $N$. This generalizes Milnor’s above-mentioned notion of reduced group
[20], which coincides with $R_{1}G(L)$. In particular, if $F_{n}$ is the free
group on $n$ generators, then $\textnormal{R}_{k}F_{n}$ is called the _$k$
–reduced free group_. Now, given an $n$-component string link $L$, we show in
Section 2.2 that, for each $k\geq 2$, there is an associated _$k$ –reduced
free action_

$\varphi^{(k)}_{L}\in\operatorname{Aut}_{c}(R_{k}F_{n}),$

where $\operatorname{Aut}_{c}(R_{k}F_{n})$ denotes the set of automorphisms of
$R_{k}F_{n}$ that act by conjugation on each generator. This can be seen as a
generalization of Habegger-Lin’s representation for the group of link-homotopy
classes of string links [12, Thm. 1.7] in the case $k=1$. We stress, however,
that our proof is very different in nature from [12].

Let us now clarify assertion (3). We use the langage of _welded knot theory_ ,
which is a diagrammatic generalization of knot theory. We stress, firstly,
that the set of classical (string) links injects into the larger set
$w\textnormal{S}L(n)$ of welded (string) links and, secondly, that Milnor
invariants extend naturally to welded objects, so that they coincide with the
classical invariants on classical objects.
In [19] a diagrammatic calculus for welded knotted objects was developped,
called _arrow calculus_ , which can be seen as a generalization of the theory
of Gauss diagrams. A $w$-tree for a welded diagram $D$, is an oriented,
unitrivalent tree, whose univalent vertices lie disjointly on $D$. Given such
a $w$-tree $T$ for $D$, there is a ‘surgery’ procedure that yields a new
welded diagram $D_{T}$, which is roughly obtained by inserting an ‘iterated
commutator of crossings’ in a neighborhood of the head of $T$. Define the
degree of a $w$-tree to be half the number of its vertices. The _self
$w_{k}$-equivalence_ is the equivalence relation on welded string links
generated by surgeries on degree $\geq k$ $w$-trees whose univalent vertices
all lie on the _same_ string link component. For $k=1$, this notion turns out
to coincide with the usual link-homotopy relation for string links [1]. On the
other hand, there is a combinatorial equivalence relation of _welded
concordance_ for welded (string) links, which is generated by birth, death and
saddle moves [5, 10] and which naturally encompasses the topological
concordance for classical (string) links. The _self $w_{k}$-concordance_ is
the equivalence relation obtained by combining the above two relations, and
our main result states that this characterizes combinatorially the information
contained in Milnor invariants $\mu(I)$ with $r(I)\leq k$.

In fact, the $k$-reduced free action involved in assertion (2) is more
generally defined for $wSL(n)$, and our main theorem follows from a more
general characterization for welded string links, see Theorems 2.8 and 3.19.
We also show that the map sending a welded string link $L$ to its associated
$k$-reduced free action $\varphi^{(k)}$, descends to a group isomorphism

$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self
$w_{k}$-concordance}$}}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{Aut}_{c}(R_{k}F_{n}),$

for any $k\geq 1$; see Proposition 3.21. The case $k=1$ was proved in [1], and
is a generalization of [12, Thm. 1.7]. This isomorphism suggests that welded
theory provides a sensible diagrammatic counterpart of the algebraic
constructions underlying Milnor invariants. Our main theorem also refines a
recent result of B. Colombari [7], who gave a diagrammatic characterization of
string links having same Milnor invariants of length $\leq q$. We note that
the geometric properties of Milnor link invariants $\overline{\mu}(I)$ with
$r(I)\leq k$ was previously investigated in [9, 27, 26], using clasper theory.

We address the (welded) link case in the final section of this paper. As
developped there, building on the proof of our main theorem for string links,
and using straightforward adaptations of the above mentioned work of Colombari
[7], we obtain in particular the following for classical links.

######  Theorem.

A link has vanishing Milnor invariants $\overline{\mu}(I)$ with $r(I)\leq k$,
if and only if it is self $w_{k}$-equivalent to the trivial link.

This follows from a more general result (Theorem 4.3) which characterizes the
so-called $k$–reduced peripheral system of _welded_ links, that is the
$k$-reduced link group endowed with peripheral elements, see Section 4.

######  Acknowledgments.

The authors thank Jacques Darné for bringing to their knowledge the result [8,
Lem. 2.37]. The first author is partially supported by the project SyTriQ
(ANR-20-CE40-0004) of the ANR, and thanks the IRL PIMS-Europe for its
hospitality during the period in which part of the work on this paper was
done. The second author is partially supported by the project AlMaRe
(ANR-19-CE40-0001-01) of the ANR. The third author is supported by the JSPS
KAKENHI grant 21K03237, and by the Waseda University grant for Special
Research Projects 2021C-120.

##  1\. Basic algebraic preliminaries

This section reviews algebraic tools that will be used in this paper, together
with basic and well-known results. Several notation used throughout the paper
will also be set in this section.

Let $n$ be a positive integer. We denote by $F_{n}$ the free group on $n$
generators $\alpha_{1},\cdots,\alpha_{n}$. For each $i\in\\{1,\cdots,n\\}$,
denote by $N_{i}$ the normal subgroup of $F_{n}$ generated by $\alpha_{i}$.

###  1.1. Commutators and the lower central series

We use the following convention for commutators and conjugates ($x,y\in
F_{n}$):

$[x,y]:=x\overline{y}\,\overline{x}y\quad\textrm{ and }\quad
x^{y}:=\overline{y}xy,$

where we write $\overline{a}$ for the inverse of an element $a$. This somewhat
nonstandard convention for commutators will be justified by the diagrammatic
counterpart of the theory, reviewed in Section 3.1. We note that with our
convention, for elements $a,b,c$ of $F_{n}$, we have the following basic
properties:

(C$0$) $\overline{[a,b]}=[\overline{b},\overline{a}];$

(C$1$) $[a,b]=\overline{b}^{\overline{a}}b=a\overline{a}^{b};$

(C$2$) $[a,b]^{c}=[a^{c},b^{c}];$

Moreover, recall the following commutator identies (compare with [18, Thm.
5.1]).

(C$3$) $[a,bc]=[a,c][a,b]^{c}\,\,\,\textrm{ and
}\,\,\,[ab,c]=[b,c]^{\overline{a}}[a,c];$ (C$4$)
$\big{[}[a,b],c\big{]}=\big{[}\overline{a},[\overline{c},\overline{b}]\big{]}^{b\overline{a}}\big{[}\overline{b},[\overline{a},\overline{c}]\big{]}^{c\overline{a}}.$

The lower central series of $F_{n}$ is the family of nested subgroups
$\\{\Gamma_{k}F_{n}\\}_{k}$ defined inductively by $\Gamma_{1}F_{n}=F_{n}$ and
$\Gamma_{k+1}F_{n}=[F_{n},\Gamma_{k}F_{n}]$. The following, less standard,
notion will also be useful.

######  Definition 1.1.

A length $k$ _linear commutator_ is a an element of $\Gamma_{k}F_{n}$ of the
form

$\left[x_{1},\left[x_{2},\left[x_{3},\cdots[x_{k-1},x_{k}]\cdots\rule{0.0pt}{8.5359pt}\right]\rule{0.0pt}{11.38092pt}\right]\rule{0.0pt}{14.22636pt}\right]$

for some elements $x_{i}\in F_{n}$.

We shall need the following basic results.

######  Lemma 1.2.

Let $C\in F_{n}$ be a length $l$ commutator, with $k$ entries in $N_{i}$ for
some $i$ ($k\leq l$). Then $C$ is a product of length $\geq l$ commutators
where each entry is an element of
$\\{\alpha_{j},\overline{\alpha}_{j}\\}_{j}$, and with at least $k$ entries
that are either $\alpha_{i}$ or its inverse.

###### Proof.

Combining (C$3$) with (C$0$) and (C$1$), gives the identities

(C$5$)
$[a,bc]=[a,c][a,b]\big{[}[\overline{b},\overline{a}],c\big{]}\,\,\,\textrm{
and }\,\,\,[ab,c]=\big{[}a,[\overline{c},\overline{b}]\big{]}[b,c][a,c].$

By assumption, $k$ entries of $C$ are products of conjugates of $\alpha_{i}$
or its inverse. Using (C$5$) on these $k$ entries, $C$ can be written as a
product of length $\geq l$ commutators, each having $\geq k$ entries that are
a single conjugate of $\alpha_{i}$ or its inverse. Since
$a^{b}=a[\overline{a},b]$ by (C$1$), we have by using (C$5$) that $C$ is
written as a product of length $\geq l$ commutators, each having at least $k$
entries that are either $\alpha_{i}$ or its inverse. Now consider one such
length $\geq l$ commutator $C^{\prime}$. One can apply (C$5$) iteratively on
all entries of $C^{\prime}$, until it is written as a product of iterated
commutators with entries in
$\big{\\{}\alpha_{j};\overline{\alpha}_{j}\big{\\}}_{j}$ and clearly, each of
these commutators necessarily contains at least $k$ entries that are either
$\alpha_{i}$ or its inverse, since $C^{\prime}$ does. ∎

######  Lemma 1.3.

Let $C\in F_{n}$ be a length $l$ commutator, with $k$ entries in $N_{i}$ for
some $i$ ($k\leq l$). Then $C\in\Gamma_{k}N_{i}$. More precisely, $C$ is a
product of length $k$ linear commutators whose entries are all conjugates of
$\alpha_{i}$ or its inverse.

###### Proof.

By assumption, $C$ is a length $l$ commutator with at least $k$ entries that
are products of conjugates of $\alpha_{i}$ or its inverse. Using (C$1$)
repeatedly, one can write $C$ as a length $k$ commutator, whose entries are
all products of conjugates of $\alpha_{i}$ or its inverse. This shows that
$C\in\Gamma_{k}N_{i}$. Now, using recursively (C$3$) and (C$2$), each such
commutator can be expressed as a product of length $k$ commutators, whose
entries are a single conjugate of $\alpha_{i}$ or its inverse. By (C$4$),
combined with (C$2$), a length $k$ commutator can be expressed as a product of
linear ones, and in our context all $k$ entries will remain a single conjugate
of $\alpha_{i}$ or its inverse. ∎

For the next two results, let $G$ be a group which is normally generated by
$n$ elements $\alpha_{1},\ldots,\alpha_{n}$. For each $i\in\\{1,\cdots,n\\}$,
denote by $N_{i}$ the normal subgroup of $G$ generated by the $i$th generator.

######  Lemma 1.4.

For any $k\in{\mathds{N}}$,

$\Gamma_{kn+1}G\subset\Gamma_{k+1}N_{1}\cdots\Gamma_{k+1}N_{n}.$

###### Proof.

Using (C$5$), any element of $\Gamma_{kn+1}G$ can be expressed as a product of
length $\geq kn+1$ commutators with entries in
$\big{\\{}\alpha_{j}^{g},\overline{\alpha}_{j}^{g}\,;\,g\in G\big{\\}}_{j}$.
For any such commutator $C$, there exists some $i$ such that at least $k+1$
entries of $C$ are elements of $N_{i}$, and Lemma 1.3 implies that
$C\in\Gamma_{k+1}N_{i}$. ∎

Define a _conjugating_ endomorphism of $G$ as an endomorphism which sends each
generator to a conjugate of itself. The following is a simple adaptation of
[8, Lem. 2.37] to our setting.

######  Lemma 1.5.

For all $k\geq 2$, any conjugating endomorphism $\varphi$ of $G$ induces an
automorphism of $\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k}G$}}}$. In particular, if $G$ is nilpotent,
then $\varphi$ is itself an automorphism of $G$.

###### Proof.

As a conjugating endomorphism, $\varphi$ induces the identity on
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{2}G$}}}$, and more generally on
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$\Gamma_{k-1}G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k}G$}}}$ for all $k\geq 2$. An induction on
$k$, using the Five Lemma on the natural exact sequence

$0\longrightarrow\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$\Gamma_{k-1}G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k}G$}}}\longrightarrow\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k}G$}}}\longrightarrow\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k-1}G$}}}\longrightarrow 0,$

then shows that $\varphi$ induces an automorphism of $\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k}G$}}}$, for any $k$.
In particular, if $G$ is nilpotent of order $N$, then $\varphi$ induces hence
an automorphism of $\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{N}G$}}}\cong G$. ∎

###  1.2. Basic commutators

We now recall the notion of basic commutators in a free group, which seems to
have first appeared in [14].

######  Definition 1.6.

A _set of basic commutators_ in the set $\\{\alpha_{1},\cdots,\alpha_{n}\\}$
is an infinite ordered set of commutators $\big{\\{}C_{i}\big{\\}}_{i}$,
defined inductively as follows. Basic commutators of length $1$ are the
$C_{i}=\alpha_{i}$ for $i=1,\cdots,n$. A basic commutator of length $m>1$ has
form $[C_{i},C_{j}]$, for some basic commutators $C_{i},C_{j}$ such that

  * •

length$(C_{i})+$length$(C_{j})=m$;

  * •

$C_{i}<C_{j}$, and $C_{j}=[C_{k},C_{l}]$ further implies that $C_{k}\leq
C_{i}$.

Basic commutators of length $m$ follow those of length $<m$, and are ordered
in an arbitrary fixed way with respect to each other.

What makes this notion significant is the following fundamental result from
[13, Thm. 11.2.4] (see also [18, Thm. 5.13.A]).

######  Theorem 1.7.

If $\big{\\{}C_{i}\big{\\}}_{i}$ is a set of basic commutators, and $k\geq 0$
is an integer, then any element $g$ in the free group $F_{n}$ can be written
in a unique way as a product

$g=C_{1}^{e_{1}}\cdots C_{N(k)}^{e_{N(k)}}h,$

with $e_{j}\in\mathbb{Z}$ and $h\in\Gamma_{k+1}F_{n}$, where $N(k)$ is the
number of basic commutators of length $\leq k$.

######  Remark 1.8.

In [13, 18], the convention $[a,b]=\overline{a}\overline{b}ab$ is used for
commutators. Although Definition 1.6 thus gives _different_ sets of basic
commutators as the ones used in [13, 18], it does share the same fundamental
properties, and in particular Theorem 1.7, as well as the coming Lemma 1.11.

###  1.3. The Magnus expansion

Denote by $\mathbb{Z}\langle\langle X_{1},\cdots,X_{n}\rangle\rangle$ the ring
of formal power series in the noncommuting variables $X_{1},\cdots,X_{n}$.

######  Definition 1.9.

The _Magnus expansion_ is the group homomorphism

$E:\leavevmode\nobreak\ F_{n}\longrightarrow\mathbb{Z}\langle\langle
X_{1},\cdots,X_{n}\rangle\rangle$

defined by sending $\alpha_{i}$ to $1+X_{i}$.

It is well-known that $E$ is in fact injective [18, Thm. 5.6], and that it is
well-behaved with respect to the lower central series, in the sense of the
following fundamental result [17, 25].

######  Theorem 1.10.

For all $k\geq 1$, we have $f\in\Gamma_{k}F_{n}$ if and only if
$E(f)=1+\textrm{(terms of degree $\geq k$)}$.

Basic commutators are well-behaved with respect to the Magnus expansion, in
the sense of the following result, which is outlined by Levine in [16, pp.
365].

######  Lemma 1.11.

Let $C$ be a basic commutator of length $k$, such that $\alpha_{i}$ occurs
$r_{i}$ times in $C$ for each $i$. We have

$E(C)=1+P+\textrm{(terms of degree $\geq k+1$)},$

where $P\neq 0$ is a sum of degree $k$ terms, each involving $r_{i}$ times the
variable $X_{i}$ ($i=1,\cdots,n$).

In the above, the sum $P$ of lowest degree (non trivial) terms in $E(C)$ is
called the _principal part_ of $E(C)$.

###### Proof.

We proceed by induction on the length $k$, where the case $k=1$ is trivial.
Let $C$ be a basic commutator of length $k$ for some $k>1$. There exists basic
commutators $C_{1}$ and $C_{2}$, of respective length $k_{1}$ and $k_{2}$,
such that $C=[C_{1},C_{2}]$ and $k_{1}+k_{2}=k$. By induction hypothesis for
$i=1,2$ we have that $E(C_{i})=1+P_{i}+\textrm{(degree $>k_{i}$ terms)}$, with
$P_{i}$ a sum of degree $k_{i}$ terms with the appropriate occurences of each
variables. Thus $E(C)=1+P_{1}P_{2}-P_{2}P_{1}+\textrm{(degree $>k$ terms)}$,
and the conclusion would follow from the fact that $P_{1}P_{2}-P_{2}P_{1}\neq
0$. In order to see that $P_{1}P_{2}-P_{2}P_{1}$ is indeed nonzero, observe
that length $k$ basic commutators form a basis for $\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$\Gamma_{k}F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}F_{n}$}}}$, as a consequence of Theorem
1.7. This in particular tells us that $C\notin\Gamma_{k+1}F_{n}$, hence by
Theorem 1.10 we have $P_{1}P_{2}-P_{2}P_{1}\neq 0$. ∎

##  2\. Milnor invariants and $k$–reduced free action

The main diagrammatic object of this paper will be the following.

######  Definition 2.1.

Consider the $2$-disk $[0,1]\times[0,1]$, equipped with fixed points
$p_{i}\times\\{\varepsilon\\}$ for $i\in\\{1,\cdots,n\\}$ and
$\varepsilon\in\\{0,1\\}$. An $n$-component _welded string link_ is the welded
equivalence class of an immersion of $n$ disjoint copies of the unit interval
into $[0,1]\times[0,1]$, such that the $i$th interval runs from
$p_{i}\times\\{0\\}$ to $p_{i}\times\\{1\\}$, and whose double points are
decorated either as a classical (as in usual knot diagrams) or a virtual
crossing (drawn as transverse double points); see Figure 1. Here, the _welded
equivalence_ is generated by the three usual Reidemeister moves involving
classical crossings, together with the OC move shown in Figure 1 and the
_Detour move_ , which replaces an arc with only virtual crossings (possibly
none) by another arc with same endpoints and only virtual crossings (possibly
none).

$\vbox{\hbox{\includegraphics[height=56.9055pt]{exsl.pdf}}}\hskip
71.13188pt\vbox{\hbox{\includegraphics[height=49.79231pt]{moves_3.pdf}}}\xleftrightarrow{\textrm{OC}}\vbox{\hbox{\includegraphics[height=49.79231pt]{moves_4.pdf}}}$
Figure 1. A $3$-component welded string link (left), and the OC move (right)

In particular, a welded string link without virtual crossing is merely a
diagram of a classical string link. A key point is that classical string links
inject in this way into welded string links: two diagrams without virtual
crossings, that are welded equivalent, represent isotopic objects, see [11,
Thm 1.B].

We denote by $\mathbf{1}$ the trivial string link diagram
$\cup_{i}p_{i}\times[0,1]$.

###  2.1. Brief review of welded Milnor invariants

Milnor invariants are classical link invariants defined by Milnor in the
fifties [20, 21], and extended to (classical) string links by Habbeger and Lin
[12]. We review here the welded extension of Milnor invariants. It was given
in [1, Sec. 6] using a topological approach, and was later reformulated in
[22] in a purely diagrammatic way.

Given a welded string link $L$, there is an associated group $G(L)$, which
coincides with the fundamental group of the complement when $L$ is classical,
see [15]. As in Wirtinger’s algorithm, a generator of $G(L)$ is associated
with each arc in a diagram of $L$, where now an arc is allowed to contain
virtual crossings, and each classical crossing gives a conjugacy relation:

$\vbox{\hbox{\includegraphics[height=42.67912pt]{Wir1.pdf}}}\ \leadsto\
\overline{a}\overline{b}a^{\prime}b\hskip
28.45274pt\vbox{\hbox{\includegraphics[height=42.67912pt]{Wir2.pdf}}}\
\leadsto\ \overline{a}ba^{\prime}\overline{b}.\\\ $

It is well-known that this is invariant under welded equivalence.

Denote by $L_{1},\ldots,L_{n}$ the components of $L$. For each $i$, label by
$a_{i,j}$ ($1\leq j\leq m_{i}+1$) the successive arcs met along $L_{i}$ when
following the orientation, where $m_{i}+1$ denotes the number of arcs of
$L_{i}$, et by $b_{i,j}$ the generator $a_{k,l}^{\pm 1}$ associated with the
overpassing arc that is met when running from $a_{i,j}$ to $a_{i,j+1}$, and
the local orientation:

$\vbox{\hbox{\includegraphics[height=49.79231pt]{Longitude.pdf}}}.$

Then the group of $L$ has a presentation of the form

(2.1) $G(L)=\big{\langle}a_{i,j},\ 1\leq i\leq n,\ 1\leq j\leq m_{i}\ \big{|}\
\overline{a}_{ij+1}\overline{b}_{i,j}a_{i,j}b_{i,j}\ 1\leq i\leq n,\ 1\leq
j\leq m_{i}-1\big{\rangle},$

######  Definition 2.2.

For $1\leq i\leq n$, the _preferred $i$th longitude_ $\lambda_{i}(L)$ is given
by $\overline{a}_{i,1}^{f_{i}}\ b_{i,1}\ b_{i,2}\ \ldots\ b_{i,m_{i}-1}$,
where $f_{i}$ is the sum of the signs of all classical crossings involving
only arcs of the $i$th component.

Let us fix some integer $q\geq 2$. It is well-known that the images
$\alpha_{1},\ldots,\alpha_{n}$ of the generators $a_{i,1}$ in the quotient
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{q}G(L)$}}}$, do generate this group [6]. In
particular, the image of the preferred $i$th longitude in
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{q}G(L)$}}}$, can be expressed as a word, which
we still denote by $\lambda_{i}(L)$, in the variables
$\alpha_{1},\ldots,\alpha_{n}$.

######  Definition 2.3.

For each sequence $I=j_{1}j_{2}\ldots j_{l}i$ of integers in
$\\{1,\cdots,n\\}$ ($l<q$), the coefficient $\mu_{L}(I)$ of $X_{j_{1}}\cdots
X_{j_{l}}$ in the Magnus expansion $E\big{(}\lambda_{i}(L)\big{)}$ is an
invariant of the welded string link $L$, called a _welded Milnor invariant_.

######  Remarks 2.4.

  1. (1)

These welded Milnor invariants naturally coincides with the classical
construction in the case of classical string links.

  2. (2)

The fact that $\lambda_{i}(L)$ represents the _preferred_ $i$th longitude
implies that $E\big{(}\lambda_{i}(L)\big{)}$ does not contain any term of the
form $X_{i}^{s}$, for $s\geq 1$, hence that $\mu_{L}(I)=0$ for any sequence of
the form $I=ii\cdots i$.

Recall from the introduction that, given a sequence $I$, we denote by $r(I)$
the maximum number of time that any index appears in $I$. For classical
(string) links, Milnor invariants $\mu(I)$ with $r(I)=1$, that is, non-
repeated Milnor invariants, are known to be link-homotopy invariants [21, 12].
Habegger and Lin actually showed that non-repeated Milnor invariants classify
string links up to link-homotopy [12], a result that was later extended to the
welded setting in [1], where non-repeated Milnor invariants are showed to
classify welded string links up to self-virtualization. Here, _self-
virtualization_ is the equivalence relation generated by the local replacement
of a classical self-crossing, by a virtual one – what turns out to coincide
with link-homotopy for classical string links (this was implicit in [1] and
formally stated in [2, Thm. 4.3]).

###  2.2. The $k$–reduced free action

As above, let $L$ be an $n$-component welded string link, with associated
group $G(L)$. By definition, $G(L)$ is normally generated by the initial arcs
$\alpha_{i}:=a_{i,1}$ of each component. As in the introduction, we can thus
consider the _$k$ –reduced quotient_ of $G(L)$

$\textnormal{R}_{k}G(L):=\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}N_{1}\cdots\Gamma_{k+1}N_{n}$}}},$

where $N_{i}$ denotes the normal subgroup generated by $\alpha_{i}$. As a
consequence of Lemma 1.4, we have that the group $\textnormal{R}_{k}G(L)$ is
nilpotent of order at most $kn+1$.

Let $F_{n}$ be the free group generated by $\alpha_{1},\cdots,\alpha_{n}$. We
have the following.

######  Lemma 2.5.

For each $k\geq 1$, we have an isomorphism

$\textnormal{R}_{k}G(L)\simeq\textnormal{R}_{k}F_{n}=\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}N_{1}\cdots\Gamma_{k+1}N_{n}$}}}.$

###### Proof.

As shown in [1, §5] (see also [19, §6.3]), for each $q\geq 1$ we have an
isomorphism

$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{q}G(L)$}}}\simeq\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{q}F_{n}$}}}.$

Hence, by Lemma 1.4, we have

$\textnormal{R}_{k}G(L)=\textnormal{R}_{k}\\!\left(\\!\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$G(L)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{kn+1}G(L)$}}}\right)\simeq\textnormal{R}_{k}\\!\left(\\!\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{kn+1}F_{n}$}}}\right)=\textnormal{R}_{k}F_{n}.\qed$

For each $i$, consider the $i$th preferred longitude of $L$ from Definition
2.2. This defines an element $\lambda^{k}_{i}(L)$ in
$\textnormal{R}_{k}G(L)\simeq\textnormal{R}_{k}F_{n}$, called the _$k$
–reduced $i$th longitude_ of $L$.

Denote by $\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$ the set of
conjugating automorphisms of $\textnormal{R}_{k}F_{n}$, which are
automorphisms sending each generator to a conjugate of itself. Since
$\textnormal{R}_{k}F_{n}$ is nilpotent, by Lemma 1.5, any conjugating
endomorphism of $R_{k}F_{n}$ is in
$\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$, for all $k\geq 1$.

######  Definition 2.6.

The _$k$ –reduced free action_ associated with $L$,

$\varphi^{(k)}_{L}\in\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n}),$

is defined by sending each generator $\alpha_{i}$ to its conjugate by
$\lambda^{k}_{i}(L)$.

This action can be seen as a generalization of Habegger-Lin’s representation
for the group of link-homotopy classes of string links [12, Thm. 1.7], in the
sense that the case $k=1$ recovers their construction.

###  2.3. Milnor invariants and $k$–reduced free action

The main purpose of this section is Theorem 2.8, which implies the equivalence
(1)$\Leftrightarrow$(2) in our main theorem.

Throughout the rest of this paper, we shall make use of the following.

######  Notation 2.7.

Recall that $N_{i}$ denotes the normal subgroup of $F_{n}$ generated by the
$i$th generator $\alpha_{i}$ ($i\in\\{1,\cdots,n\\}$). For $k\geq 1$, we set

$J^{k}:=\Gamma_{k+1}N_{1}\cdots\Gamma_{k+1}N_{n}\,\,\,\textrm{ and
}\,\,\,J_{i}^{k}:=\Gamma_{k+1}N_{1}\cdots\Gamma_{k}N_{i}\cdots\Gamma_{k+1}N_{n}.$

Denote also by $R^{k}$ the two-sided ideal of $\mathbb{Z}\langle\langle
X_{1},\cdots,X_{n}\rangle\rangle$ generated by all terms having at least $k+1$
occurences of some variable, and by $R^{k}_{i}$ the ideal generated by terms
having either at least $k+1$ occurences of some variable, or $k$ occurences of
the variable $X_{i}$.

######  Theorem 2.8.

Let $L$ and $L^{\prime}$ be two $n$-component welded string links. The
following are equivalent:

  1. (i)

$\mu_{L}(I)=\mu_{L^{\prime}}(I)$ for any sequence $I$ with $r(I)\leq k$.

  2. (ii)

the $k$–reduced $i$th longitudes $\lambda^{k}_{i}(L)$ and
$\lambda^{k}_{i}(L^{\prime})$ are congruent modulo $J_{i}^{k}$, for all $i$.

  3. (iii)

$L$ and $L^{\prime}$ induce the same $k$–reduced free action
$\varphi_{L}^{(k)}=\varphi_{L^{\prime}}^{(k)}\in\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$.

The rest of this section is devoted to the proof of Theorem 2.8, which is done
in three steps.

Firstly, we prove that (ii)$\Rightarrow$(iii). Suppose that the $k$–reduced
$i$th longitudes $\lambda^{k}_{i}(L)$ and $\lambda^{k}_{i}(L^{\prime})$ are
congruent modulo $J_{i}^{k}$, for all $i$. We have
$\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}=Xg_{i}$, for some
$g_{i}\in\Gamma_{k}N_{i}$ and some $X\in\prod_{j\neq i}\Gamma_{k+1}N_{j}$. By
(C$3$) we thus have that

$\big{[}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})},\alpha_{i}\big{]}=[g_{i},\alpha_{i}]^{\overline{X}}[X,\alpha_{i}]\in
J^{k}.$

This readily implies that
$\big{[}\alpha_{i},\lambda^{k}_{i}(L)\big{]}\equiv\big{[}\alpha_{i},\lambda^{k}_{i}(L^{\prime})\big{]}$
mod $J^{k}$, which is equivalent to saying that $L$ and $L^{\prime}$ induce
the same $k$–reduced free action:
$\varphi_{L}^{(k)}=\varphi_{L^{\prime}}^{(k)}\in\operatorname{Aut}_{c}(R_{k}F_{n})$.

Secondly, we prove that (iii)$\Rightarrow$(i). If
$\varphi_{L}^{(k)}=\varphi_{L^{\prime}}^{(k)}\in\operatorname{Aut}_{c}(R_{k}F_{n})$,
then for each $i$ we have
$\alpha_{i}^{\lambda^{k}_{i}(L)}\equiv\alpha_{i}^{\lambda^{k}_{i}(L^{\prime})}$
mod $J^{k}$, which is equivalent to

$\alpha_{i}\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}\overline{\alpha}_{i}\equiv\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\textrm{
mod $J^{k}$.}$

Taking the Magnus expansion then gives

(2.2)
$X_{i}E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}-E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}\big{(}X_{i}-X_{i}^{2}+\cdots\big{)}-X_{i}E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}\big{(}X_{i}-X_{i}^{2}+\cdots\big{)}\in
R^{k}.$

If
$E\big{(}\lambda^{k}_{i}(L)\big{)}-E\big{(}\lambda^{k}_{i}(L^{\prime})\big{)}$
lives in $R^{k}_{i}$, then $L$ and $L^{\prime}$ have same Milnor invariants
$\mu(I)$ with $r(I)\leq k$. Suppose by contradiction that

$E\big{(}\lambda^{k}_{i}(L)\big{)}-E\big{(}\lambda^{k}_{i}(L^{\prime})\big{)}\equiv
S_{q}+\left(\textrm{terms of degree $>q$}\right)\textrm{ mod $R^{k}_{i}$},$

for some $q$ and a sum $S_{q}$ of degree $q$ terms which are _not_ in
$R^{k}_{i}$. Multiplying the above by
$E\big{(}\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}$, which by definition
has constant term equal to $1$, we have

$E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}\equiv
1+S_{q}+\left(\textrm{terms of degree $>q$}\right)\textrm{ mod $R^{k}_{i}$}.$

Equation (2.2) then gives

$X_{i}E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}-E\big{(}\lambda^{k}_{i}(L)\overline{\lambda^{k}_{i}(L^{\prime})}\big{)}X_{i}\equiv
X_{i}S_{q}-S_{q}X_{i}+\left(\textrm{terms of degree $>q+1$}\right)\textrm{ mod
$R^{k}_{i}$}.$

Observe that $X_{i}S_{q}$ and $S_{q}X_{i}$ are not in $R^{k}$, since
$S_{q}\notin R_{i}^{k}$. Hence, if $X_{i}S_{q}\neq S_{q}X_{i}$, we obtain a
contradiction with (2.2). But if $X_{i}S_{q}=S_{q}X_{i}$, a simple
combinatorial argument shows that $S_{q}$ can be nothing else than the
monomial $X_{i}^{q}$. This tells us that
$E\big{(}\lambda^{k}_{i}(L)\big{)}-E\big{(}\lambda^{k}_{i}(L^{\prime})\big{)}$
contains the term $X_{i}^{q}$, which is in contradiction with the fact that
$\lambda^{k}_{i}(L)$ and $\lambda^{k}_{i}(L^{\prime})$ are images of
_preferred_ $i$th longitudes, see Remark 2.4 (ii).

Thirdly and lastly, let us prove that (i)$\Rightarrow$(ii). For this purpose,
we prove the following, which should be thought of as a ‘$k$–reduced’ version
of the injective property of the Magnus expansion (Theorem 1.10).

######  Proposition 2.9.

Let $g$ be an element of $F_{n}$. Then $g\in J_{i}^{k}$ if and only if
$E(g)\in 1+R_{i}^{k}$.

Assuming this result momentarily, we immediately deduce the desired
implication (i)$\Rightarrow$(ii) of Theorem 2.8. Indeed, consider two welded
string links $L$ and $L^{\prime}$ with same Milnor invariants $\mu(I)$ with
$r(I)\leq k$. This precisely means that, for each $i$, their respective $i$th
preferred longitudes $\lambda^{k}_{i}(L)$ and $\lambda^{k}_{i}(L^{\prime})$
satisfy

$E\big{(}\lambda^{k}_{i}(L^{\prime})\big{)}\equiv
E\big{(}\lambda^{k}_{i}(L)\big{)}\textrm{ mod $R_{i}^{k}$}.$

This rewrites as
$E\big{(}\overline{\lambda^{k}_{i}(L)}\lambda^{k}_{i}(L^{\prime})\big{)}=1+R_{i}^{k}$,
which by Proposition 2.9 gives that
$\overline{\lambda^{k}_{i}(L)}\lambda^{k}_{i}(L^{\prime})\in J_{i}^{k}$, as
desired.

###### Proof of Proposition 2.9.

The ‘only if’ part of the statement follows easily from well-known properties
of the Magnus expansion [18], so we prove here the other implication. By
Theorem 1.7, there is a set of basic commutators $\\{C_{i}\\}_{i}$ such that

$g=C_{1}^{e_{1}}\cdots C_{N(nk)}^{e_{N(nk)}}h,$

for some unique integers $e_{1},\cdots,e_{N(nk)}\in\mathbb{Z}$ and a unique
element $h\in\Gamma_{nk+1}F_{n}$. Set
$g_{j}:=\prod_{\textrm{length$(C_{j_{m}})=j$}}C_{j_{m}}^{e_{j_{m}}}$, so that
$g=g_{1}\cdots g_{kn}h$. Since $h\in J_{i}^{k}$, it remains to show that
$g_{j}\in J_{i}^{k}$ for all $j$. Suppose by contradiction that this is not
the case, and let $p$ be the smallest integer such that
$g_{p}=\prod_{m=1}^{N}C_{p_{m}}^{e_{p_{m}}}$ is not in $J_{i}^{k}$ ($N\geq
1$). This means that there is a nonempty subset $S$ of $\\{1,\cdots,N\\}$ such
that $e_{p_{l}}\neq 0$ and $C_{p_{l}}\notin J_{i}^{k}$, for all $l\in S$.
Lemma 1.11 tells us that, for each $m$, the principal part of $E(C_{p_{m}})$
is either in $R_{i}^{k}$, or is in the $\mathbb{Z}$-module generated by
monomials with $<(k-1)$ copies of $X_{i}$ and $<k$ copies of any other
variable. This implies in particular that the sum of the principal parts of
$E\big{(}\prod_{l\in S}C_{p_{l}}^{e_{p_{l}}}\big{)}$ is not in
$R_{i}^{k}\setminus\\{0\\}$. But as a property of basic commutators, we know
that these principal parts are linearly independent [18]; it follows that
$E(g_{p})-1$ is non trivial and not in $R_{i}^{k}$. By minimality of $p$, this
implies that $E(g)-1$ is not in $R_{i}^{k}$. This is a contradiction, which
concludes the proof. ∎

######  Remark 2.10.

The case $k=1$ of Proposition 2.9 was previously established in [28, Prop.
7.10]. One can actually further generalize this result as follows. Let
$\mathbf{r}=(r_{1},\cdots,r_{n})$ be an $n$-tuple of positive integers.
Consider the subgroup
$J^{\mathbf{r}}:=\Gamma_{r_{1}}N_{1}\cdots\Gamma_{r_{n}}N_{n}$ of $F_{n}$, and
the ideal $R^{\mathbf{r}}$ of $\mathbb{Z}\langle\langle
X_{1},\cdots,X_{n}\rangle\rangle$ generated by all terms where the variable
$X_{i}$ appears at least $r_{i}$ times, for $i=1,\cdots,n$. Then the above
proof generalizes in a straightforward way to show that an element $g\in
F_{n}$ is in $J^{\mathbf{r}}$ if and only if $E(g)$ is in $1+R^{\mathbf{r}}$.

##  3\. Arrow calculus and self $w_{k}$-concordance

###  3.1. Arrow calculus for welded objects

Let us briefly review arrow calculus, which is diagrammatic calculus
developped in [19] for the study of welded objects. In Section 3.2, we explain
how this diagrammatic tool is intimately related to the commutator calculus
reviewed in Section 1.

Let $L$ be an $n$-component welded string link.

######  Definition 3.1.

A _$w$ -tree_ for $L$ is a planar immersion of an oriented, connected uni-
trivalent tree $T$, such that

  * •

the set of all vertices of $T$ is embedded in the interior of
$[0,1]\times[0,1]$, such that trivalent vertices are disjoint from $L$ and
univalent vertices are in $L\setminus\\{\textrm{crossings of $L$}\\}$;

  * •

at each trivalent vertex, the three incident edges are cyclically ordered, and
there are two ingoing and one outgoing edge;

  * •

edges of $T$ may cross virtually $L$ or $T$ itself;

  * •

edges are possibly decorated by some _twists_ $\bullet$, which are disjoint
from all crossings and vertices, and which satisfy the rule .

A univalent vertex of $T$ is a _head_ if $T$ is locally oriented towards $L$,
and it is a _tail_ otherwise. For a union of $w$-trees for $L$, we allow
virtual crossings among edges, and we require all vertices to be pairwise
disjoint. See Figure 2 for an example.

We note that a $w$-tree contains a single head, due to the orientation
convention at trivalent vertices.

Figure 2. An example of $w$-tree for the trivial $3$–component string link

######  Definition 3.2.

A $w$-tree is _linear_ , if it has the following shape as an abstract tree

with possibly a number of $\bullet$ on its edges.

######  Definition 3.3.

The _degree_ of $T$ is its number of tails or, equivalently, half the total
number of vertices. A degree $k$ $w$-tree is called a _$w_{k}$ -tree_, and a
$w_{1}$-tree is also called a $w$-arrow.

Given a union of $w$-arrows $A$ for $L$, there is a notion of _surgery along
$A$_, which produces a new welded string link $L_{A}$ as follows:

.

In general, if $A$ contains some virtual crossing, this likewise introduces
pairs of virtual crossings in $L_{A}$.

Now, given a union of $w_{k}$-trees $P$ for $L$, one can define the notion of
surgery along $P$ by first replacing $P$ by a union of $w$-arrows, called the
_expansion_ of $P$ and denoted by $E(P)$, defined recursively by the local
rule illustrated below, then performing surgery on $E(P)$. The result will be
denoted by $L_{P}$.

.

######  Remark 3.4.

The above figure suggests that the union $E(P)$ of $w$-arrows obtained from
$P$ by applying (E) recursively, has the shape of an ‘iterated commutator’ of
$w$-arrows. This observation is made rigourous and further discussed in
Section 3.2.

######  Definition 3.5.

A _$w$ -tree presentation_ $(\mathbf{1},P)$ for $L$ is a union $P$ of
$w$-trees for the trivial diagram $\mathbf{1}$, such that $L=\mathbf{1}_{P}$.
Two arrow presentations $(\mathbf{1},P)$ and $(\mathbf{1},P^{\prime})$
representing equivalent welded string links are called _equivalent_.

The main point of this notion is that any welded string link admits a $w$-tree
presentation [19, Prop. 4.2]. Moreover, in [19, Thm. 5.21], a set of moves on
$w$-trees is provided, which suffice to deform any $w$-tree presentation of
$L$ into any other. These moves imply further operations, hence a full
diagrammatic calculus called _arrow calculus_ , which can be used to study
welded objects and their invariants. We do not reproduce all these moves here,
but only provide below those that will be needed in this paper :

We refer the reader to [19] for more details on arrow calculus.

We will also use the following, which refines the notion of equivalence given
in Definition 3.5.

######  Definition 3.6.

Let $(\mathbf{1},P)$ be a $w$-tree presentation for $L$, and let $S$ be a
subset of $P$. Denote by $\sigma\subset\mathbf{1}$ a neighborhood of the
endpoints of $S$, which identifies with a trivial diagram, such that $\sigma$
is disjoint from $P\setminus S$. Let $S^{\prime}$ be a union of $w$-trees for
$\sigma$. We say that $S$ and $S^{\prime}$ are _locally equivalent_ whenever
$\sigma_{S^{\prime}}$ is welded equivalent to $\sigma_{S}$.

Note that, in this setting, $(\mathbf{1},P)$ and
$\big{(}\mathbf{1},(P\setminus S)\cup S^{\prime}\big{)}$ are equivalent
$w$-tree presentations.

###  3.2. Algebraic formalism for $w$-trees

Let $(\mathbf{1},P)$ be a $w$-tree presentation for a welded diagram.

Let $A$ be a union of $w$-arrows in $P$ that are _adjacent_ , in the sense
that _all_ heads in $A$ are met consecutively on a portion $h_{A}$ of
$\mathbf{1}$, called the _support_ of $A$, without meeting any crossing or
endpoint.111We shall also say in this situation that the heads of $A$ are
_adjacent_. The _complement_ of $A$ is then defined as the $w$-tree
presentation $(\mathbf{1},P\setminus A)$ obtained from $(\mathbf{1},P)$ by
removing all $w$-arrows in $A$. The support $h_{A}$ and the arrow heads of
$P\setminus A$ then cut the strands of $\mathbf{1}$ into intervals that we
label by letters $x_{1},\ldots,x_{p}$; we set $F_{(P;A)}:=\langle
x_{1},\ldots,x_{p}\rangle$, the free group generated by these letters.

$\vbox{\hbox{\includegraphics[height=85.35826pt]{Cut1.pdf}}}\ \leadsto\
\vbox{\hbox{\includegraphics[height=85.35826pt]{Cut2.pdf}}}\ \leadsto\ \
w(A)=\overline{x}_{3}x_{5}x_{4}\overline{x}_{6}$
Figure 3. An example of word associated to a set of adjacent $w$-arrows

Assuming that all arrow heads are met to the right side when traveling along
$h_{A}$ following its orientation (this is always possible thanks to the Head
reversal move), each head of an arrow in $A$ can be labeled222It should be
noted that replacing arrows by labels corresponds actually to the _cut-
diagram_ point of view on welded objects, introduced in [4].  by the letter
$x_{i}$ or $\overline{x}_{i}$, depending on whether the arrow contains an even
or an odd number of twists, where $x_{i}$ is the label of the interval on
which the tail lies. We can then define an element $w(A)\in F_{(P;A)}$ by
reading the labels in order when running along $h_{A}$ according to its
orientation; see Figure 3 for an example. Notice that this _word_ $w(A)$
remains unchanged under a Tails exchange move and that, conversely, it
determines the union of $w$-arrows $A$ up to Tails exchange moves. More
generally, we have the following.

######  Lemma 3.7.

Suppose that two $w$-tree presentations $(\mathbf{1},P)$ and
$(\mathbf{1},P^{\prime})$ only differ by replacing a union of adjacent arrows
$A$ by another union of adjacent arrows $A^{\prime}$ with same support. Then
$F_{(P;A)}=F_{(P^{\prime};A^{\prime})}$, and $w(A)=w(A^{\prime})$ if and only
if $A$ and $A^{\prime}$ differ by a sequence of Inverse moves and Tails
exchange moves.

###### Proof.

First note that $A$ and $A^{\prime}$ have same complement, hence we indeed
have that $F_{(P;A)}=F_{(P^{\prime};A^{\prime})}$. Since this is a free group,
we have $w(A)=w(A^{\prime})$ if and only if these two words differ by
inserting or deleting copies of $x_{j}\overline{x}_{j}$ or
$\overline{x}_{j}x_{j}$ for any $j$. But this is equivalent to saying that $A$
and $A^{\prime}$ differ by a sequence of Inverse moves (with pairs of
$w$-arrows) and Tails exchange moves. ∎

Lemma 3.7 provides a one-to-one correspondence between sets of adjacent arrows
with a fixed support and complement, up to equivalence, and elements in the
associated free group. Under this correspondence, our conventions for the
commutator notation $[x,y]$ and the conjugation notation $x^{y}$, given in
Section 1.1, have natural diagrammatic counterparts.

It is easily observed that a commutator corresponds to the expansion of a
$w$–tree:

$x\overline{y}\overline{x}y=[x,y]\quad\leftrightarrow\quad\vbox{\hbox{\includegraphics[height=42.67912pt]{com1.pdf}}}\,=\,\vbox{\hbox{\includegraphics[height=42.67912pt]{com0.pdf}}}.$

In general, to a $w$-tree $T$ which is part of a $w$-tree presentation
$(\mathbf{1},P)$, corresponds a word $w(T)\in F_{\left(P;E(T)\right)}$, which
is defined as the word corresponding to its expansion; in other words, we set
$w(T)=w\big{(}E(T)\big{)}$. From the definition, the word $w(T)$ can be
directly read from $T$ using the following procedure. Label each edge of $T$
containing a tail by the generator $x_{i}$ at the tail, then label the
remaining edges of $T$ by recursively applying the local rules of Figure 4;
the label at the edge containing the head is $w(T)$; see Example 3.9 (i).

Figure 4. Procedure to compute $w(T)$ from a $w$-tree

Note that, under this correspondence, the word associated to a linear $w$-tree
(Definition 3.2), is a linear commutator with entries in
$\\{x_{i},\overline{x}_{i}\\}_{i}$, in the sense of Definition 1.1.

The conjugation notation, on the other hand, corresponds to a situation where
the Slide move can be performed:

$\overline{y}xy=x^{y}\quad\leftrightarrow\quad\vbox{\hbox{\includegraphics[height=42.67912pt]{com2.pdf}}}\,=\,\vbox{\hbox{\includegraphics[height=42.67912pt]{com3.pdf}}},$

Since two $w$-tree presentations that differ by a Slide move are locally
equivalent, the above rightmost picture can be seen as the diagrammatic
counterpart of notation $x^{y}$ in our correspondence.

####  3.2.1. Conjugated trees

Combining these two observations, we can thus extend our correspondence to a
wider range of $w$-tree presentations:

######  Definition 3.8.

A $w$-tree $T$ for $\mathbf{1}$ is called _conjugated_ if there exists a union
$U$ of _pairs of conjugating $w$-arrows_ for $T$. Here, a pair of conjugating
$w$-arrows for $T$ is a union of two parallel $w$-arrows that only differ by a
twist, and whose heads are on the same component of $\mathbf{1}$ and only
separated by a tail of $T$, and possibly other nested pairs of conjugating
$w$-arrows. See Example 3.9.

Let $T$ be a conjugated $w$-tree $T$ with union $U$ of conjugating $w$-arrows,
in a $w$-tree presentation $(\mathbf{1},P)$. One can define a corresponding
word $w^{U}(T)\in F_{(P\setminus U;E(T))}$, which is the word associated with
the union of adjacent $w$-arrows obtained by taking the expansion of $T$ and
applying Inverse moves and Slide moves333Since each tail of $T$ yields a
_union_ of $w$-arrows after expansion, one first need to use the Inverse move
several times between these tails to be able to perform the Slide moves. to
$w$-arrows in $U$. This word $w^{U}(T)$ can be directly read off $T\cup U$, by
substituting each letter $a$ in $w(T)\in F_{(P\setminus U;E(T))}$ by its
conjugate $a^{w(X)}$ whenever the corresponding tail of $T$ admits a union of
conjugating $w$-arrows $X\cup\overline{X}\subset U$, where $X$ denotes the
union of $w$-arrows met before the tail according to the orientation (and
$\overline{X}$ are the $w$-arrows met after the tail). Let us illustrate this
concretely on an example:

######  Example 3.9.

The $w_{4}$-tree $T$ shown below is a conjugated tree (here, the labels $a$,
$b$, $c$, $d$ and $e$ are not necessarily mutually distinct).

(i) Ignoring all conjugating $w$-arrow, we have that
$w(T)=\big{[}[a,b],[d,e]\big{]}$.

(ii) Denoting by $U$ the union of conjugating $w$-arrows for $T$, we have

$w^{U}(T)=\big{[}[a^{b},b^{\overline{c}}],[d^{ec},e]\big{]}.$

######  Remark 3.10.

We stress that, for a $w_{k}$–tree $T$, $w^{U}(T)$ is a length $k$ commutator
whose entries are conjugates of
$\big{\\{}x_{i},\overline{x}_{i}\big{\\}}_{i}$. Conversely, it follows from
the above that for any length $k$ commutator $w\in F_{n}$ whose entries are
conjugates of $\\{x_{i},\overline{x}_{i}\\}_{i}$, there exists a $w_{k}$-tree
$T$ and a union $U$ of conjugating arrows such that $w=w^{U}(T)$. This applies
more generaly to _products_ of such commutators, which then correspond to
_adjacent_ conjugated trees, _i.e._ conjugated trees with adjacent heads.

######  Remark 3.11.

Deleting a conjugated tree $T$, in the notation of Definition 3.8, yields the
union $U$ of conjugating $w$-arrows, which can in turn all be deleted by using
the Inverse move.

For this paper, we will need the following, which is a direct translation of
Lemma 1.3 via the above correspondence:

######  Lemma 3.12.

Let $T$ be a $w_{l}$-tree in some union of $w$-trees for $\mathbf{1}$, such
that the $i$th component of $\mathbf{1}$ contains $k$ tails of $T$ ($k\leq
l$). Then $T$ is locally equivalent to a union of adjacent conjugated linear
$w_{k}$-trees, with all tails on the $i$th component.

####  3.2.2. Relation to the welded group

Finally, let us recall how, given a $w$-tree presentation $(\mathbf{1},P)$ for
$L$, one can associate a presentation for the group $G(L)$ defined in Section
2.1, using our algebraic formalism.

Again, by the Head reversal move, one can freely assume that all heads of $P$
are attached to the right side of $\mathbf{1}$ according to the orientation.
The heads of $P$ split $\mathbf{1}$ into a union of arcs, each of which yields
a generator $x_{i}$, and we denote by $F_{P}$ the free group generated by
$\big{\\{}x_{i}\big{\\}}_{i}$.

Now, let $T$ be a single $w_{k}$-tree in $P$, and denote by $a$ and $b$ the
two generators of $F_{P}$ associated with the arcs to the left and right of
its head, respectively. Following the above, a word $w(T)$ is defined by
expanding $T$ into a union $E(T)$ of adjacent $w$-arrows, and writing the
associated word. This $w(T)$ naturally lives in $F_{P}$, since the complement
of $h_{E(T)}$ is obtained by just deleting a neighborhood of the head of $T$.

Figure 5 then illustrates how $T$ yields a conjugation relation
$R_{T}$:$\leavevmode\nobreak\ b=\overline{w(T)}aw(T)$ among the two generators
$a,b$ separated by its head.

Figure 5. Conjugating relation $R_{T}$ associated with $T$

In fact, we obtain in this way the following presentation (see [19, § 6.1.1]):

$G(L)=\langle\\{x_{i}\\}_{i}\,|\,\\{R_{T}\\}_{T\in P}\rangle.$

In particular, the $i$th preferred longitude of $L$ from Definition 2.2, can
be written as $\lambda_{i}(L)=\alpha_{i}^{s_{i}}\prod_{T\in P_{i}}w(T)$, for
some $s_{i}\in\mathbb{Z}$, where $P_{i}$ is the subset of $w$-trees in $P$
whose heads are on the $i$th component of $\mathbf{1}$, ordered according to
their occurence on the $i$th component when following the orientation.

######  Remark 3.13.

Observe that, in the notation of Figure 5, if the head of $T$ is on the $i$th
component then $a,b\in N_{i}$, and if moreover $w(T)\in\Gamma_{k}N_{i}$ for
some $k$, then we have:
$b=\overline{w(T)}aw(T)=a\big{[}\overline{a},w(T)\big{]}\equiv a\textrm{ mod
$\Gamma_{k+1}N_{i}$}$.

####  3.2.3. Self $w_{k}$-equivalence and $w^{(k+1)}$-equivalence

This section is concerned with the following families of equivalence relations
for welded objects.

######  Definition 3.14.

Let $k\geq 1$. The _$w_{k}$ -equivalence_, resp. _self $w_{k}$-equivalence_,
is the equivalence relation generated by welded equivalence and surgery along
$w$-trees, resp. self $w$-trees, of degree $\geq k$. Here, a _self $w$-tree_
is a $w$-tree whose endpoints all lie on a same component. The _$w^{(k+1)}$
-equivalence_ is the equivalence relation generated by welded equivalence and
surgery along $w$-trees having at least $k+1$ endpoints on a same component.

######  Remark 3.15.

Given a $w_{kn+1}$-tree $T$ for some $n$-component string link, there is
necessarily some index $i$ such that $T$ has at least $k+1$ ends on the $i$th
component. This elementary observation shows that the $w_{kn+1}$-equivalence
implies the $w^{(k+1)}$-equivalence.

The following will play a central role in proving our main theorem, but might
also be of independent interest in the study of arrow calculus.

######  Theorem 3.16.

Two welded string links are self $w_{k}$-equivalent, if and only if they are
$w^{(k+1)}$-equivalent.

###### Proof.

Since a self $w$-tree of degree $\geq k$ has at least $k+1$ endpoints, which
are all attached to a same component, we clearly have the ‘only if’ part of
the statement.

In order to prove the ‘if’ part, consider a $w$-tree $T^{\prime}$ for an
$n$-component welded string link $L$, with $k+1$ endpoints on the $i$th
component $L_{i}$ of $L$. We distinguish two cases. If the head of $T$ is on
$L_{i}$, then by Lemma 3.12, it is locally equivalent to a union of conjugated
self $w_{k}$-trees on the $i$th component. Hence in this case, $L_{T}$ is
clearly self $w_{k}$-equivalent to $L$ by Remark 3.11. Now, if the head of $T$
is on the $j$th component $L_{j}$ of $L$ for some $j\neq i$, then Lemma 3.12
more precisely tells us that $T$ is locally equivalent to a union of
conjugated linear $w_{k+1}$-trees, with all tails on the $i$th component and
with head on the $j$th component. Consider one such linear $w$-tree, as on the
left-hand side of the figure below:

This figure shows how applying the expansion move (E) to such a tree, followed
by the Slide move, yields a union of two $w$-arrows $A\cup A^{\prime}$ and two
self $w_{k}$-trees $B\cup B^{\prime}$ on the $i$th component. Deleting $B\cup
B^{\prime}$ up to self $w_{k}$-equivalence, then deleting $A\cup A^{\prime}$
by the Inverse move, yields the empty diagram. This observation, together with
Remark 3.11, shows that $L_{T}$ is self $w_{k}$-equivalent to $L$. ∎

######  Remark 3.17.

The argument of this proof can be used to show that the (self)
$w_{k}$-equivalence is generated by welded equivalence and surgery along
(self) $w_{k}$-trees, rather than (self) $w$-trees of degree $\geq k$.

###  3.3. Self $w_{k}$-concordance and $w^{(k+1)}$-concordance

The main purpose of this section is Theorem 3.19, which implies the
equivalence (1)$\Leftrightarrow$(3) in our main theorem. First, we introduce
the self $w_{k}$-concordance relation, along with the $w^{(k+1)}$-concordance
equivalence.

The notion of concordance for welded links was introduced and studied in [5,
10].

Two $n$-component welded string links $L$ and $L^{\prime}$ are _welded
concordant_ if one can be obtained from the other by a sequence of welded
equivalence and the birth/death and saddle moves of Figure 6, such that, for
each $i\in\\{1,\cdots,n\\}$, the number of birth/death moves used to deform
the $i$th component of $L$ into the $i$th component of $L^{\prime}$ is equal
to the number of saddle moves.

Figure 6. The birth/death and saddle moves

######  Definition 3.18.

Let $k\geq 1$. The _$w_{k}$ -concordance_, resp. _self $w_{k}$-concordance_,
is the equivalence relation generated by welded concordance and
$w_{k}$-equivalence, resp. self $w_{k}$-equivalence. The _$w^{(k+1)}$
-concordance_ is the equivalence relation generated by $w^{(k+1)}$-equivalence
and welded concordance.

The following is the main result of this section.

######  Theorem 3.19.

Let $L$ and $L^{\prime}$ be two $n$-component welded string links. The
following are equivalent:

  1. (i)

$\mu_{L}(I)=\mu_{L^{\prime}}(I)$ for any sequence $I$ with $r(I)\leq k$.

  2. (ii)

$L$ and $L^{\prime}$ are $w^{(k+1)}$-concordant.

  3. (iii)

$L$ and $L^{\prime}$ are self $w_{k}$-concordant.

The rest of this section is devoted to the proof of Theorem 3.19.

We already have directly that (ii) and (iii) are equivalent, by Theorem 3.16.
In the rest of the proof, we use the fact from Theorem 2.8 that (i) is
equivalent to saying that the $k$–reduced $i$th longitudes
$\lambda^{k}_{i}(L)$ and $\lambda^{k}_{i}(L^{\prime})$ are congruent modulo
$J_{i}^{k}=\Gamma_{k+1}N_{1}\cdots\Gamma_{k}N_{i}\cdots\Gamma_{k+1}N_{n}$ for
all $i$.

Let us prove that (iii) implies (i). It is shown in [4] that welded Milnor
invariants are invariant under welded concordance. So it suffices to show
that, if $L^{\prime}$ is obtained from $L$ by surgery along a self
$w_{k}$-tree $T$, we have
$\lambda^{k}_{i}(L^{\prime})\equiv\lambda^{k}_{i}(L)\mod J_{i}^{k}$ for all
$i$. Suppose that all endpoints of $T$ are on the $i_{0}$th component of $L$
for some $i_{0}$. Then $w(T)\in\Gamma_{k}N_{i_{0}}$, and by Remark 3.13 we
have directly that
$\lambda^{k}_{i}(L)\equiv\lambda^{k}_{i}(L)\mod\Gamma_{k+1}N_{i_{0}}(\subset
J_{i}^{k})$ for all $i\neq i_{0}$. Furthermore, there exists some words $a,b$
in $F_{n}$, such that $\lambda^{k}_{i_{0}}(L)=ab$ and, again by Remark 3.13,
$\lambda^{k}_{i_{0}}(L^{\prime})\equiv aw(T)b\mod\Gamma_{k+1}N_{i_{0}}$. This
implies that $\lambda^{k}_{i_{0}}(L^{\prime})\equiv\lambda^{k}_{i_{0}}(L)\mod
J_{i_{0}}^{k}$, as desired.

Finally, we prove that (i) implies (ii). By assumption, and using Lemmas 2.5
and 1.2, the $k$–reduced $i$th preferred longitudes
$\lambda^{k}_{i}(L^{\prime})$ and $\lambda^{k}_{i}(L)$ differ by a finite
sequence of the following operations:

  * (a)

inserting or deleting copies of $\alpha_{j}\overline{\alpha}_{j}$ or
$\overline{\alpha}_{j}\alpha_{j}$ for any $j$;

  * (b)

inserting or deleting length $\geq k+1$ commutators with entries in
$\big{\\{}\alpha_{l};\overline{\alpha}_{l}\big{\\}}_{l}$, and with at least
$k+1$ entries with some index $j\neq i$.

  * (c)

inserting or deleting length $\geq k$ commutators with entries in
$\big{\\{}\alpha_{l};\overline{\alpha}_{l}\big{\\}}_{l}$, and with at least
$k$ entries with index $i$.

We aim at showing that, under this assumption, $L$ and $L^{\prime}$ are
$w^{(k+1)}$-equivalent.

As a first step, let us assume that both $L$ and $L^{\prime}$ are given by a
so-called sorted $w$-tree presentation:

######  Definition 3.20.

A $w$-tree presentation for an $n$-component welded string link is _sorted_
444This notion was first defined in [1] under the term _ascending_ , in the
context of Gauss diagrams.  if, when running along each component following
the orientation, all tails are met before all heads.

Given a sorted presentation $(\mathbf{1},P)$, the tails of all $w_{k}$-trees
are contained in the initial arcs of $\mathbf{1}$, hence the words associated
to these $w_{k}$-trees are length $k$ commutator with entries in
$\big{\\{}\alpha_{l};\overline{\alpha}_{l}\big{\\}}_{l}$. Denote by $P_{i}$
the subset of $w$-trees in $P$ whose heads are on the $i$th component.
According to Section 3.2.2, there exists some $s_{i}\in\mathbb{Z}$ such that
the $i$th preferred longitude is equal to $\alpha_{i}^{s_{i}}\prod_{T\in
P_{i}}w(T)$. Using the Isolated move, we can introduce a union $A_{i}$ of
$w$-arrows whose endpoints are all on the $i$th component, such that
$(\mathbf{1},P\cup A_{i})$ is sorted and equivalent to $(\mathbf{1},P)$, and
such that the $i$th preferred longitude is equal to $\prod_{T\in A_{i}\cup
P_{i}}w(T)$. By this observation, we can freely assume that the words
associated with our sorted presentations, are precisely the preferred
longitudes of our string links $L$ and $L^{\prime}$. Hence by Lemma 3.7, if
$\lambda_{i}(L)$ and $\lambda_{i}(L^{\prime})$ are equal as words in
$\alpha_{1},\cdots,\alpha_{n}$, then applying (E) recursively to these sorted
presentations, yields the _same_ union of $w$-arrows.

It thus remains to realize the above three operations (a), (b) and (c) by a
$w^{(k+1)}$-equivalence among sorted tree presentations. This is done as
follows:

  * •

Operation (a) is simply achieved by Inverse moves, which insert a pair of
parallel $w$-arrows, one having a $\bullet$, running from the bottom of the
$j$th component and ending on the $i$th component.

  * •

Operation (b), resp. (c), is achieved by inserting or deleting sorted
$w$-trees, with head on the $i$th component, and with at least $k+1$ tails on
the $j$th component, resp. with at least $k$ tails on the $i$th component.

This shows that $L$ and $L^{\prime}$ are $w^{(k+1)}$-equivalent, and we are
done in the sorted case.

In order to conclude the proof, it is now enough to show that any welded
string link is $w^{(k+1)}$-concordant to a string link with a sorted
presentation, since Milnor invariants $\mu(I)$ with $r(I)\leq k$ are
invariants of $w^{(k+1)}$-concordance. This can be seen as a direct corollary
of [7, Cor. 2.5], which tells us that any $w$-tree presentation can be
deformed into a sorted one by a sequence of welded concordance and surgeries
along $w_{nk+1}$-trees. The desired result then follows by Remark 3.15.

###  3.4. Artin-like isomorphisms

The stacking product endows the set $w\textnormal{S}L(n)$ of $n$-component
welded string links with a structure of monoid, with unit the trivial diagram
$\mathbf{1}$. As seen in subsection 2.2, a conjugating automorphism
$\varphi^{(k)}_{L}\in\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$ is
associated to any welded string link $L$, which sends each generator
$\alpha_{i}$ of $R_{k}F_{n}$ to its conjugate by the $k$-reduced longitude
$\lambda^{k}_{i}(L)$. This yields, for each $k\geq 1$, a monoid homomorphism

$\varphi^{(k)}:w\textnormal{S}L(n)\longrightarrow\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n}).$

######  Proposition 3.21.

For each $k\geq 1$, the map $\varphi^{(k)}$ descends to a group isomorphism

$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self
$w_{k}$-concordance}$}}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n}).$

###### Proof.

Consider the quotient map $\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self
$w_{k}$-concordance}$}}}\longrightarrow\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$,
which will shall still denote by $\varphi^{(k)}$. The fact that this map is
injective is a consequence of Theorems 3.19 and 2.8. To prove surjectivity, it
is sufficient to observe that an element of
$\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$ is specified by an $n$-tuple
of conjugating elements in $\textnormal{R}_{k}F_{n}$, and that these
conjugating elements are easily realized as the preferred longitudes of a
sorted welded string link; see the paragraph following Def. 3.20. Since
$\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$ is a group, it follows that
$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self $w_{k}$-concordance}$}}}$ is a group
too555This fact was already known, as any welded string link is invertible up
to concordance [10, Prop. 6]., and that the isomorphism is a group
isomorphism. ∎

######  Remark 3.22.

Using the result of Colombari [7] mentioned in the introduction, the proof of
Proposition 3.21 can be adapted in a straightfoward way to show that, for all
$k\geq 1$, we have an isomorphism

$\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{$w_{k}$-concordance}$}}}\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}\operatorname{Aut}_{c}\\!\left(\hbox{\leavevmode\kern
1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}F_{n}$}}}\right),$

where $\operatorname{Aut}_{c}\\!\left(\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}F_{n}$}}}\right)$ denotes the group of
conjugating automorphisms of $\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$F_{n}$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\Gamma_{k+1}F_{n}$}}}$.

We conclude this section by a (long) remark addressing the 4-dimensional
counterpart of this study, that can be safely skipped by the reader who is not
interested in this matter.

######  Remark 3.23.

Welded theory is intimately connected to the topology of _ribbon knotted
surfaces_ in $4$–space, via the so-called _Tube map_ defined by Satoh in [23].
In our context, the Tube map is a surjective monoid homomorphism

$\textrm{Tube}:w\textnormal{S}L(n)\longrightarrow rT(n),$

where $rT(n)$ denotes the monoid of _ribbon tubes_ introduced in [1]. The
question of the injectivity of this Tube map is however still open, but it is
worth noting that our present work implies that any element in the kernel of
the Tube map is self $w_{k}$-concordant to $\mathbf{1}$ for all $k$.

Indeed, there is a $4$–dimensional analogue of arrow calculus, which was
developped (prior to [19]) by Watanabe in [24]. There, a notion of
$RC_{k}$-equivalence is defined for ribbon knotted surfaces, in terms of
surgery along degree $k$ oriented claspers, which are embedded surfaces that
realize topologically an oriented uni-trivalent tree with $2k$ vertices. A
refined notion of _self $RC_{k}$-equivalence_ is easily defined on $rT(n)$ by
further requesting that such degree $k$ oriented claspers only intersect a
single ribbon tube component. Combining furthermore this self
$RC_{k}$-equivalence with the topological concordance of ribbon tubes, one
defines a notion of _self $RC_{k}$-concordance_ for these objects. It is known
[5, Prop. 4.8] that the Tube map sends concordant welded links to concordant
ribbon tori, and it is easily verified that it sends self $w_{k}$-concordant
welded string links to self $RC_{k}$-concordant ribbon tubes. Hence, for each
$k\geq 1$, the Tube map induces a surjective homomorphism

$\textrm{Tube}_{k}:\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$w\textnormal{S}L(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self
$w_{k}$-concordance}$}}}\longrightarrow\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$rT(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self $RC_{k}$-concordance}$}}}.$

These maps are actually isomorphisms, and the proof goes as follows. Following
[1, Sec. 2.2.4], which deals with the $k=1$ case, one can use Stallings
theorem to associate an action
$\varphi^{(k)}_{T}\in\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n})$ for any
ribbon tube $T$, which actually conjugates, in
$\textnormal{R}_{k}\pi_{1}(B^{4}\setminus T)\cong\textnormal{R}_{k}F_{n}$, the
$i$th meridians of $T$ by its $i$th preferred longitude. By another use of
Stallings theorem in dimension one more, this action is invariant under
concordance, and it can be directly seen that $RC_{k}$-equivalent ribbon tubes
have $k$-reduced $i$th longitudes which are congruent modulo $J_{i}^{k}$
(borrowing notation from Thm. 2.8). As the Tube map is known to preserve the
associated groups and longitudes, this leads to a map

$\varphi^{(k)}_{r}:\hbox{\leavevmode\kern 1.00006pt\raise
1.07639pt\hbox{\sevenrm$rT(n)$}\kern-1.00006pt}\big{/}{\hbox{\kern-1.49994pt\lower
2.15277pt\hbox{\sevenrm$\textrm{self
$RC_{k}$-concordance}$}}}\longrightarrow\operatorname{Aut}_{c}(\textnormal{R}_{k}F_{n}),$

which satisfies $\varphi^{(k)}=\varphi^{(k)}_{r}\circ\textrm{Tube}_{k}$. It
follows then from the injectivity of $\varphi^{(k)}$ that $\textrm{Tube}_{k}$
is injective, hence an isomorphism. This implies that an element in the kernel
of the Tube map is trivial up to self $w_{k}$-concordance.

##  4\. The link case

By Theorems 2.8 and 3.19, welded string links are classified up to self
$w_{k}$–equivalence by their $k$–reduced longitudes. Similar phenomena occur
in the classifications up to self-virtualization [1] and $w_{k}$–concordance
[7], and these results were extended to the case of welded links in [3] and
[7] respectively, in terms of (adaptations of) the peripheral system. In this
final section, we outline how a similar extension can be derived for links up
to self $w_{k}$–concordance.

_Welded links_ are defined in the very same way as welded string links, by
replacing the copies of the unit interval with copies of the circle $S^{1}$.

Let $L$ be a welded link. Using the same Wirtinger-like procedure as in
Section 2.1, a group $G(L)$ can be associated to any welded link $L$, which
agrees with the fundamental group of the complement if $L$ is a classical
link. A choice of one generic basepoint on each component determines a set of
meridians which normally generates $G(L)$. Up to Detour moves, these
basepoints also allow to cut open $L$ into a welded string links $S_{L}$, and
$G(L)$ is isomorphic to the quotient of $G(S_{L})$ which identifies, for each
strand, the meridians associated with its two endpoints. An _$i$ th longitude_
for $L$ can then be defined as the image of the $i$th longitude for $S_{L}$ in
this quotient. Of course, the result depends on the choice of the basepoints:
moving the $i$th basepoint actually results in conjugating simultaneously the
associated $i$th meridian and longitude.666Note that, since all the meridians
defined in this way are conjugated, all these normally generating sets of
meridians lead to the same notion of $k$–reduced quotient for $G(L)$.

######  Definition 4.1.

Let $k$ be a positive integer. The _$k$ –reduced peripheral system_ of $L$ is
defined as

$\big{(}\textnormal{R}_{k}G(L),\\{x_{i}\\}_{i},\\{\lambda^{k}_{i}\Gamma_{k}N_{i}\\}_{i}\big{)}$

where, for a given choice of basepoints, $(x_{i},\lambda^{k}_{i})$ are the
images in $\textnormal{R}_{k}G(L)$ of the associated meridians and preferred
$i$th longitudes, and $\lambda^{k}_{i}.\Gamma_{k}N_{i}$ is the image in
$\textnormal{R}_{k}G(L)$ of the coset of $\lambda^{k}_{i}$ modulo
$\Gamma_{k}N_{i}$, with $N_{i}$ the normal subgroup of
$\textnormal{R}_{k}G(L)$ generated by $x_{i}$. It is well-defined up to, for
each $i$, simultaneous conjugation of $x_{i}$ and $\lambda^{k}_{i}$ by an
element of $\textnormal{R}_{k}G(L)$.

######  Remark 4.2.

The $1$–reduced peripheral system coincides with the _reduced peripheral
system_ introduced by Milnor in his foundational paper [21] on links up to
link-homotopy, see also [3].

The arrow calculus reviewed in Section 3.1 applies in the exact same way to
welded links, and leads to the notions of _self $w_{k}$-equivalence_ and _self
$w_{k}$-concordance_ for these objects, as in Definitions 3.14 and 3.18.
Theorem 3.16 also holds for welded links, since the proof is purely local.

The main result of this section is the following.

######  Theorem 4.3.

Two (welded) links have equivalent $k$–reduced peripheral systems if and only
if they are self $w_{k}$–concordant.

The rest of this section is devoted to the proof of Theorem 4.3. We stress
that all ingredients of the proof are mostly corollaries of their string link
counterparts, and straightforward adaptations of techniques of [3, 7]. Hence
we will only sketch the proof of Theorem 4.3 below, only hinting to the main
arguments and outlining the specificities of the link case.

The fact that the $k$–reduced peripheral system is invariant under self
$w_{k}$–concordance is an easy consequence of the string link case addressed
in the previous sections. Indeed, a notion of $k$–reduced peripheral system
can be defined for welded string links as in Definition 4.1, except that, in
this case, there is a natural choice of meridians, and it follows from
Theorems 3.19 and 2.8 that this is invariant under self $w_{k}$–concordance.
By fixing a set of basepoints on a welded link $L$, and considering its
$k$–reduced peripheral system as a quotient of the $k$–reduced peripheral
system of the associated welded string link $S_{L}$, we obtain the desired
invariance property.

The proof of ‘only if’ part of Theorem 4.3 goes roughly along the same lines
as the proof of (i)$\Rightarrow$(ii) of Theorem 3.19, given in Section 3.3.
The analogue of Definition 3.20 in the link case is the following, see [3]: a
$w$–tree presentation for a welded link is _sorted_ if, on each component, all
the heads are adjacent. (A $w$–tree presentation for a welded link is a union
of $w$-trees for the trivial diagram of the unlink.)

Since the $w_{kn+1}$–equivalence implies the self $w_{k}$–equivalence by
Remark 3.15 and Theorem 3.16, we have the following as a direct corollary of
[7, Prop. 3.5].

######  Proposition 4.4.

Any welded link is self $w_{k}$–equivalent to a welded link admitting a sorted
presentation.

From this proposition, and the invariance of the $k$–reduced group under self
$w_{k}$–equivalence, the exact same argument as in [3, Lem. 1.18] proves the
following.

######  Proposition 4.5.

For every welded link $L$, its $k$–reduced group admits the following
presentation

$\textnormal{R}_{k}G(L)\cong\big{\langle}x_{1},\ldots,x_{n}\ |\
\Gamma_{k+1}N_{i}\textrm{ and }[x_{i},\lambda^{k}_{i}]\textrm{ for all
$i$}\big{\rangle},$

where, for each $i$, $x_{i}$ is a meridian on the $i$th component, $N_{i}$ is
the normal subgroup generated by $x_{i}$, and $\lambda^{k}_{i}$ is a
representative word for the corresponding longitude.

The rest of the proof then follows the exact same lines as [7, Prop. 3.7]. We
start with two welded links which have equivalent $k$–reduced peripheral
systems, and consider sorted presentations using Proposition 4.4. The main
difference with the string link case of Section 3.3 is that a fourth operation
is involved on the $i$th preferred longitudes: by Proposition 4.5, these
longitudes might differ by the insertion or deletion of a commutator
$[x_{j},\lambda^{k}_{j}]$ for some $j$. This extra operation can be achieved
by a $w_{k}$-concordance on sorted presentations, with the same trick as
illustrated in the first figure of [7, Proof of Prop. 3.7]. This concludes the
proof of Theorem 4.3.

Notice that, in the above argument, the welded concordance is only needed for
the fourth operation that inserts/deletes a commutator
$[x_{j},\lambda^{k}_{j}]$ for some $j$. This is because such relators appear
in the presentation of $\textnormal{R}_{k}G(L)$ given in Proposition 4.4. In
the special case of a link $L$ with vanishing Milnor invariants
$\overline{\mu}_{L}(I)$ with $r(I)\leq k$, we have that
$\textnormal{R}_{k}G(L)\cong RF_{n}$, meaning that this extra operation
involving welded concordance is not needed for the proof. This implies that a
(welded) link has vanishing Milnor invariants $\overline{\mu}(I)$ with
$r(I)\leq k$, if and only if it is self $w_{k}$-equivalent to the unlink, as
stated in the introduction. This is shown by applying verbatim the same
argument as in [7, §3.2].777More precisely, the exact same arguments showing
that [7, Thm. 3.8] implies [7, Cor. 3.11], shows that Theorem 4.3 implies the
theorem stated at the end of the introduction.

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