# A diagrammatical characterization of Milnor invariants

Boris Colombari

###### Abstract

The goal of this paper is to give a diagrammatical characterization of the
information given by the Milnor invariants of links and string links. More
precisely, we describe when two string links have equal Milnor invariants of
length $\leq q$ and when a link has trivial Milnor invariants of lenght $\leq
q$. This will be done through the use of welded knot theory, involving the
notions of arrow calculus and $w_{q}$–concordance introduced by J-B. Meilhan
and A. Yasuhara. These results is to be compared to the classification of
links up to $C_{q}$–concordance obtained by J. Conant, R. Schneiderman and P.
Teichner.

## Introduction

In 1954, J. Milnor defined the now well-known Milnor invariants of a link in
[19], encoding the information given by the longitudes of the link in the
coefficients of some formal power series. These longitudes are part of the
peripheral system of the link, which is composed of its fundamental group,
meridians and longitudes. Although topological in nature, the peripheral
system can be described in a combinatorial way using the Wirtinger
presentation of the group for a diagram representing the link. This is the
approach that will be used in this paper, as we will deal with objects that
are defined diagrammatically.

The goal of this paper is to give some new insights on the type of information
the Milnor invariants are able to detect, which is related to the study of
knotted objects in $4$ dimensions. Indeed the characterization will be made
through the use of welded objects, which can be seen as a diagrammatical
intermediary between classical knots and (the ribbon subclass of) knotted
surfaces in $\mathbb{R}^{4}$. In [1], B. Audoux, P. Bellingeri, J-B. Meilhan
and E. Wagner gave a diagrammatical characterization of the non-repeating
Milnor invariants in terms of link-homotopy on welded string links, which
consists in allowing strands to cross themselves. They proved that two string
links have equal non-repeating Milnor invariants if and only if they are link-
homotopic as welded string links. In this paper we obtain similar
diagrammatical characterizations of the general Milnor invariants of length
$\leq q$, using the notion of $w_{q}$–concordance on welded string links. In
[2], B. Audoux and J-B. Meilhan also gave a characterization of the reduced
peripheral system of links in terms of link-homotopy, and we obtain a similar
result for the $q$–nilpotent peripheral system of links in terms of
$w_{q}$–concordance, leading to a diagrammatical characterization of links
with trivial Milnor invariants of length $\leq q$.

The main tool in this paper is the arrow calculus developped by J-B. Meilhan
and A. Yasuhara in [17], which describes surgeries on virtual diagrams by
unitrivalent trees, giving an analogue on welded objects of the notion of
clasper calculus introduced on classical links by K. Habiro in [15]. The
notion of $w_{q}$–equivalence consists in allowing surgeries along oriented
trees of degree $\geq q$, defining a finer equivalence relation on welded
objects as $q$ increases. When combined with welded concordance, we obtain the
notion of $w_{q}$–concordance, which we will use to characterize the Milnor
invariants of length $\leq q$ on welded objects. More precisely, we obtain the
following result:

######  Theorem 1.

The Milnor invariants of length $\leq q$ are complete invariants of
$w_{q}$–concordance on welded string links.

As classical (string) links inject into welded ones (see Theorem 1.B of [13]
for the case of virtual knots, which extends to welded objects), this gives a
diagrammatical characterization for classical string links with equal Milnor
invariants:

######  Corollary 1.

Two classical string links have equal Milnor invariants of length $\leq q$ if
and only if they are $w_{q}$–concordant when considered as welded string
links.

As proven by A. Casejuane and J-B. Meilhan in the upcoming paper [8] following
[7], for $q\leq 3$ welded string links are classified up to
$w_{q}$–equivalence by their finite type invariants of order $<q$, and it is
conjectured to still be true for $q\geq 4$. Theorem 1 shows that among these
finite type invariants, the Milnor invariants are able to distinguish welded
string links up to $w_{q}$–concordance. The others invariants, namely the
closure invariants which are extracted from the normalized Alexander
polynomials of the welded long knots obtained by putting some of the
components end to end, are needed to distinguish between two welded string
links which are concordant but not $w_{q}$–equivalent.

Moreover, combining Theorem 1 with results from [17] and [7] we have the
following corollary on the finite type concordance invariants of welded string
links:

######  Corollary 2.

For two welded string links $L$ and $L^{\prime}$, the following statements are
equivalent:

  1. 1.

$L$ and $L^{\prime}$ are $w_{q}$–concordant,

  2. 2.

$L$ and $L^{\prime}$ have the same Milnor invariants of length $\leq q$,

  3. 3.

$L$ and $L^{\prime}$ the same finite type concordance invariants of degree
$<q$.

This can be seen as a welded analogue of the result obtained by N. Habegger
and G. Masbaum in [14], stating that all rational finite type concordance
invariants of string links are given by the Milnor invariants.

We also obtain a classification result similar to Theorem 1 on welded links,
but in terms of $q$–nilpotent peripheral systems instead of Milnor invariants.
More precisely, we use the equivalence class of $q$–nilpotent peripheral
systems under a natural equivalence relation which removes the ambiguity
coming from the choice of meridians (see Section 3.1).

######  Theorem 2.

The equivalence class of $q$–nilpotent peripheral system is a complete
invariant of $w_{q}$–concordance on welded links.

The Milnor invariants can still be used to detect the unlink modulo
$w_{q}$–equivalence:

######  Corollary 3.

A welded link is $w_{q}$–concordant to the unlink if and only if it has
vanishing Milnor invariants of length $\leq q$.

As in the case of string links, we obtain a diagrammatical charaterization for
classical links with trivial Milnor invariants:

######  Corollary 4.

A classical link has vanishing Milnor invariants of length $\leq q$ if and
only if it is $w_{q}$–concordant to the unlink when considered as a welded
link.

This final result is to be compared to the classification of classical links
up to $C_{q}$–concordance in terms of Milnor invariants and higher order Sato-
Levine and Arf invariants obtained by J. Conant, R. Schneiderman and P.
Teichner in [11]. The notion of $C_{q}$–concordance is obtained by combining
$C_{q}$–equivalence, which comes from clasper surgeries described by K. Habiro
in [15], and concordance. This construction inspired that of
$w_{q}$–concordance through arrow calculus on welded objects. To illustrate
the role of these additional invariants, we give an example at the end of the
paper of a link which is $w_{3}$–concordant to the unlink as a welded link,
but not $C_{3}$–concordant to it.

This paper is divided in three main sections. Section 1 is dedicated to the
introduction of the objects which will be used throughout the paper, starting
with some algebraic properties of commutators in Section 1.1. In Section 1.2,
we define welded links and string links, give their representation by virtual
diagrams and then by arrow and tree presentations coming from arrow calculus.
In Section 1.3, we define the notion of concordance and $w_{q}$–concordance
for welded objects. Section 2 is dedicated to the classification of welded
string links up to $w_{q}$–concordance. It starts with the introduction of the
Milnor invariants, then we prove that any tree presentation of a welded string
link is $w_{q}$–concordant to an ascending presentation, which is a
prerequisite to the proof of Theorem 1. Section 3 is dedicated to the
classification of welded links up to $w_{q}$–concordance, starting with the
definition and properties of $q$–nilpotent peripheral systems in Section 3.1,
then proving that any tree presentation of a welded link is $w_{q}$–equivalent
to a sorted presentation and giving a classification of these presentations in
Section 3.2.

Acknowledgements. This article summarizes some results obtained during my PhD
thesis. I would like to thank my thesis tutor B. Audoux for his guidance and
helpful advice in the writing of this paper, as well as J-B. Meilhan, A.
Yasuhara and M. Chrisman for their insightful comments on a previous version
of this paper. I am also grateful to l’Institut de Mathématiques de Marseille
for hosting me during the redaction of my PhD thesis, which was supported by
l’École Normale Supérieure de Cachan.

##  1 Notations and settings

###  1.1 Algebraic notions

For elements $a,b$ in a group $G$, we denote by $a^{b}:=b^{-1}ab$ the
_conjugate of $a$ by $b$_, and $[a,b]:=a^{-1}b^{-1}ab$ the _commutator of $a$
and $b$_. For subsets $A,B$ of $G$, we denote by $[A,B]$ the normal subgroup
of $G$ generated by elements $[a,b]$ for $a\in A$ and $b\in B$.

The _lower central series of $G$_, denoted by $(\Gamma_{q}G)_{q\geq 1}$, is
defined by induction on $q$ by $\Gamma_{1}G:=G$ and
$\Gamma_{q+1}G:=[G,\Gamma_{q}G]$ for $q\geq 1$. The $\Gamma_{q}G$ are normal
subgroups of $G$, and we denote by $N_{q}G:=G/\Gamma_{q}G$ the _nilpotent
quotients of $G$_. The subgroup $\Gamma_{q}G$ contains all iterated
commutators of weight $\geq q$, where the weight corresponds to the number of
elements. For example, $[[g_{1},g_{2}],g_{3}]$ is a commutator of weight $3$,
and $[[g_{1},g_{2}],[g_{3},g_{4}]]$ and $[g_{1},[[g_{2},g_{3}],g_{4}]]$ are
commutators of weight $4$. The classification of ascending and sorted
presentations in Sections 2 and 3 will require the following description of
elements of $\Gamma_{q}G$, which follows from well known identites given in
[16]:

######  Proposition 1.1.

If a group $G$ is generated by elements $g_{1},\ldots,g_{n}$, any element of
$\Gamma_{q}G$ can be written as a product of commutators of weight $\geq q$ on
$g_{1}^{\pm 1},\ldots,g_{n}^{\pm 1}$.

For $n\geq 1$, let $F_{n}$ denote the free group on $n$ generators
$m_{1},\ldots,m_{n}$.
We denote by $\mathcal{S}_{n}:=\mathbb{Z}\langle\langle
X_{1},\ldots,X_{n}\rangle\rangle$ the $\mathbb{Z}$–algebra of formal power
series with $\mathbb{Z}$ coefficients on $n$ non-commutating variables. We
denote by $U_{1}(\mathcal{S}_{n})$ the set of power series with constant term
equal to $1$, which is a subgroup of the multiplicative group of units of
$\mathcal{S}_{n}$.
We denote by $\mathcal{X}$ the two-sided ideal of $\mathcal{S}_{n}$ of power
series whose constant term is zero. For $q\geq 1$, $\mathcal{X}^{q}$ is the
two-sided ideal of power series whose monomials are of degree greater than or
equal to $q$. We denote by
$\mathcal{S}_{n}^{(q)}:=\mathcal{S}_{n}/\mathcal{X}^{q}$ the
$\mathbb{Z}$–algebra obtained from $\mathcal{S}_{n}$ where all monomials of
degree at least $q$ are made equal to zero. We denote by
$U_{1}(\mathcal{S}_{n}^{(q)})$ the subgroup of units of
$\mathcal{S}_{n}^{(q)}$ with contant term equal to $1$. Note that we have a
canonical isomorphism $U_{1}(\mathcal{S}_{n}^{(q)})\simeq
U_{1}(\mathcal{S}_{n})/(1+\mathcal{X}^{q})$.

We will consider the _Magnus expansion_ $E:F_{n}\to U_{1}(\mathcal{S}_{n})$,
defined as the group homomorphism sending $m_{i}$ to $1+X_{i}$. It is easily
checked (see Section 5.5 of [16]) that $E$ is injective, so
$E:F_{n}\to\text{Im}(E)$ is an isomorphism. Moreover, it follows from results
in Section 5.7 of [16] that
$E(\Gamma_{q}F_{n})=\Gamma_{q}\text{Im}(E)=\text{Im}(E)\cap(1+\mathcal{X}^{q})$.
As a result, we have:

######  Proposition 1.2.

For $q\geq 1$, the Magnus expansion induces a homomorphism
$E_{q}:N_{q}F_{n}\to U_{1}(\mathcal{S}_{n}^{(q)})$, which is injective.

We will use this injective homomorphism $E_{q}$ to extract the Milnor
invariants from the longitudes of welded objects, and conversely to recover
the longitudes from the Milnor invariants.

###  1.2 Representations of welded links and welded string links

####  1.2.1 Virtual diagrams

In what follows, we fix a positive integer $n$, and $n$ points
$p_{1},\ldots,p_{n}$ in $(0,1)$ such that $<p_{1}<\ldots<p_{n}$.

######  Definition 1.3.

An _$n$ –component virtual link diagram_ is an immersion
$f:\\{1,\ldots,n\\}\times S^{1}\to\mathbb{R}^{2}$ whose singularities are
transverse double points labeled with an element of $\\{+,-,v\\}$. An _$n$
–component virtual string link diagram_ is a proper immersion
$f:\\{1,\ldots,n\\}\times[0,1]\to[0,1]\times[0,1]$ such that
$f(i,\varepsilon)=(p_{i},\varepsilon)$ for $1\leq i\leq n$ and
$\varepsilon=0,1$, and whose singularities are transverse double points
labeled with an element of $\\{+,-,v\\}$.

We denote by $D$ such a diagram, and $D_{i}$ its $i^{th}$ component, given by
$f(i,\cdot)$. This component is given the canonical orientation from $S^{1}$
for a link, or from $[0,1]$ for a string link. A double point is called a
positive (resp. negative, resp. virtual) crossing if it is labeled with a $+$
(resp. $-$, resp. $v$). A positive or negative crossing is called classical,
and is represented by erasing part of one of the strands (depending on the
sign) at the crossing, while a virtual crossing is represented by drawing a
circle around it :

positive crossingnegative crossingvirtual crossing

At a classical crossing, the strand that has been partly erased is called the
_lower strand_ , and the other one is called the _upper strand_. The preimage
of the double point on the lower strand is called an _undercrossing_ , and the
one on the upper strand is called an _overcrossing_. We call _arcs of $D$_ the
portions of strands delimited by the undercrossings.

######  Definition 1.4.

An _$n$ –component welded link_ (resp. _welded string link_) is an equivalence
class of $n$–component virtual link (resp. string link) diagrams up to the
local moves, and their left-right mirror images, displayed in Figure 1.

R1R2R3vR1vR2vR3MixedOC Figure 1: Welded Reidemeister moves.

######  Definition 1.5.

We associate a group $G_{D}$ to a virtual diagram $D$, defined by a
presentation similar to the Wirtinger presentation of the fundamental group
for classical links. More precisely, we associate one generator to each arc of
$D$, and one relation $c=b^{a^{\varepsilon}}$ for each classical crossing,
where $a$ is the generator associated to the upper strand, $b$ (resp. $c$) is
the generator associated to the lower strand preceding (resp. following) the
crossing, and $\varepsilon$ is the sign of the crossing.

As we will often need to consider the product of the conjugating elements
involved in such relations along a subinterval of a component, we introduce
the following notation:

######  Definition 1.6.

For a subinterval $J$ of a component of a virtual diagram $D$, we denote by
$w_{J}$ the word on the generators of $G_{D}$ and their inverse obtained by
taking the product of the generators associated to the overstrands at the
undercrossings on $J$, in the order given by the orientation of the diagram.

We will make use of such words throughout this paper, sometimes as elements of
$G_{D}$ or $N_{q}G_{D}$, and sometimes strictly as words on the generators and
their inverse, with no relations between them. For example, on the diagram
illustrated below we have $w_{J}=ab^{-1}bc$.

$J$$a$$b$$b$$c$

For a virtual diagram $D$, we denote by $a_{ij}$ the generators of $G_{D}$
associated to the arcs of $D_{i}$. Since the relations in the definition of
$G_{D}$ are conjugating relations between generators associated to the same
component, the homomorphisms $a_{i}^{\ast}:\langle a_{ij}\rangle\to\mathbb{Z}$
defined by $a_{i}^{\ast}(a_{kj})=\delta_{ik}$ induce homomorphisms
$a_{i}^{\ast}:G_{D}\to\mathbb{Z}$.

Let $D$ be a virtual diagram of a welded link. For a given point $p_{i}$ on an
arc of $D_{i}$ between two undercrossings, we denote by $m_{i}$ the generator
of $G_{D}$ associated to this arc, and call it a _meridian of $D_{i}$_. Let
$D_{i}^{p_{i}}$ denote the subinterval $D_{i}\setminus\\{p_{i}\\}$ of $D_{i}$.
We call _longitude of $D_{i}$ associated to $m_{i}$_ the element
$\ell_{i}:=m_{i}^{-k}w_{D_{i}^{p_{i}}}\in G_{D}$, where
$k:=a_{i}^{\ast}(w_{D_{i}^{p_{i}}})$. These notions of meridian and longitude
come from the case of a classical link $L$ in $\mathbb{R}^{3}$, where a
meridian of a component $L_{i}$ is the boundary of a small disk which is
positively transverse to $L_{i}$ in one point, whose corresponding element in
$G_{L}:=\pi_{1}(L)$ is the generator associated to that point, and a longitude
of $L_{i}$ is a loop which runs parallel to it and has linking number with
$L_{i}$ equal to $0$. The correcting term $m_{i}^{-k}$ in $\ell_{i}$
corresponds to the linking number constraint of the classical case. A
peripheral system of $D$ is given by its group $G_{D}$ and a choice of
meridians and longitudes. Up to isomorphisms which conjugate each meridian and
associated longitude by the same element, it is an invariant of the welded
link represented by $D$.

We can also consider the notion of peripheral system for a virtual diagram $D$
of a welded string link. In this case, there is a natural choice for the
meridians: $m_{i}$ is the generator associated to the bottom arc of $D_{i}$,
and the “longitude” $\ell_{i}$ is defined as $\ell_{i}:=m_{i}^{-k}w_{D_{i}}$,
with $k:=a_{i}^{\ast}(w_{D_{i}})$. With these notation, the generator
associated to the top arc of $D_{i}$ is equal to $m_{i}^{\ell_{i}}$ in
$G_{D}$.

####  1.2.2 Arrow calculus

In [17], J-B. Meilhan and A. Yasuhara developed the notion of _arrow calculus_
, inspired by clasper calculus on classical objects, which consists in adding
arrows, called w-arrows, on virtual diagrams representing some surgeries as
indicated below (note that in [17], the virtual crossings are represented
without the small circle):

$=$$=$

Note that the relative position of the classical and virtual crossings
involved in the surgery depend on the orientation of the strand containing the
tail of the w-arrow. Moreover, the side of the strand from which the w-arrow
is attached is also relevant. A w-arrow may cross other strands of the virtual
diagram or other w-arrows, in which case virtual crossings are created when
performing the surgery.

A w-arrow may also contain a dot, representing a twist, which modifies the
surgery as indicated below. A w-arrow may contain multiple dots (inducing as
many twists), but since the twists can by cancelled pairwise by using $vR2$
moves, we may assume that each w-arrow contains at most one dot.

$=$

######  Definition 1.7.

An _arrow presentation_ of a virtual diagram $D$ is a pair $(V,\mathcal{A})$,
where $V$ is a virtual diagram with no classical crossing and $\mathcal{A}$ is
a family of w-arrows on $V$ such that the diagram obtained from $V$ by
performing the surgeries corresponding to $\mathcal{A}$ is $D$. Two arrow
presentations are called _equivalent_ if they give equivalent virtual
diagrams.

As noted in [17], any classical crossing on a virtual diagram can be replaced
by a virtual crossing and a w-arrow next to it:

$vR2$surgery

As a result, any virtual diagram admits an arrow presentation. In Section 4 of
[17], Meilhan and Yasuhara give a list of local moves on w-arrows which result
in equivalent arrow presentations. First, the extremities of w-arrows can pass
through virtual crossings, but in general not through classical crossings or
other w-arrow extremities. A twist can be moved along a w-arrow, passing
through the strands or w-arrows that it intersects.

$=$$=$$\underbrace{}$strands or w-arrows$=$

Changing the side of the strand from which the extremity of a w-arrow is
attached has the following effect:

$=$head reversal$=$tail reversal

Finally, the four local moves illustrated on Figure 2 correspond to the
classical Reidemeister moves and the $OC$ move, where a white dot indicates
that the w-arrow may or may not contain a twist.

$=$Isolated arrow$=$Inverse$=$Tail exchange$=$Slide$=$ Figure 2: Local moves
on w-arrows.

As stated in Theorem 4.5 of [17], two arrow presentations represent equivalent
diagrams if and only if they are related by these arrow moves. As a result, we
can study welded objects through their arrow presentations up to these local
moves.

##### Remark:

Using the virtual Reidemeister moves $vR1$, $vR2$ and $vR3$, and the fact that
w-arrow extremities can pass through virtual crossings, we can assume that the
underlying virtual diagram $V$ in an arrow presentation is trivial, i.e. a
disjoint union of intervals for a string link and a disjoint union of circles
for a link. As a result, when working on arrow presentations with a set number
of components, we will only focus on the w-arrows and consider that the
underlying virtual diagram is trivial.

The use of arrow calculus in this paper revolves around the notion of w-tree,
which represent “stacked commutators” of w-arrows. A w-tree is a unitrivalent
tree whose degree one vertices are positioned on the diagram. One of these
vertices is the head of the w-tree, and the other ones are tails. There is a
cyclic order on the three edges at each trivalent vertex, given by the
orientation of the plane in which the diagram is represented. Each edge of the
tree can contain a twist. The degree of a w-tree $T$ is defined as the number
of tails of $T$. See Figure 3 for an example of a w-tree of degree $3$.

Figure 3: A w-tree of degree $3$.

A w-tree represents a collection of w-arrows, obtained by expanding the w-tree
iteratively as indicated on Figure 4, where $T_{1}$ and $T_{2}$ are two
unitrivalent trees, and the $T_{i}$’s obtained after expanding are parallel
copies of $T_{i}$. The surgery along a w-tree is then defined as the surgery
along its expansion.

$T_{1}$$T_{2}$$=$$T_{1}$$T_{1}$$T_{2}$$T_{2}$$T_{1}$$T_{2}$$=$$T_{1}$$T_{1}$$T_{2}$$T_{2}$
Figure 4: Expanding a w-tree.

######  Definition 1.8.

A _tree presentation_ of a virtual diagram $D$ is a pair $(V,\mathcal{T})$,
where $V$ is a virtual diagram with no classical crossing and $\mathcal{T}$ is
a family of w-trees on $V$ such that the diagram obtained from $V$ by
performing the surgeries corresponding to $\mathcal{T}$ is $D$. Two tree
presentations are called _equivalent_ if they give equivalent virtual
diagrams.

The presentation of the group $G_{D}$ can be adapted to take into account
w-trees as follows: the arcs are now delimited not only by undercrossings but
also by the heads of the w-trees. Moreover, there is a relation associated to
the head of a w-tree, obtained by propagating the generators along the edges
from the tails to the head, keeping track of the twists and forming a
commutator at each trivalent vertex as illustrated in Figure 5 (note that in
[17], $[a,b]$ is defined as $ab^{-1}a^{-1}b$ to match more closely with the
expansion of w-trees).

$a$$a$$a$$a^{-1}$$a$$b$$[a^{-1},b]$$b$$a$$c=b^{a}$ Figure 5: Propagating
generators along w-trees.

As shown in Lemma 5.17 in [17], we can exchange the sides of the incoming
edges at a trivalent vertex by adding a twist on each of the three edges at
this vertex (see Figure 6). This can be used to extrapolate some local moves
on w-trees by changing the orientation of certain strands in order to get to a
configuration where an established local move can be performed.

$T_{1}$$T_{2}$$=$$T_{1}$$T_{2}$ Figure 6: Changing the cyclic order at a
trivalent vertex.

Local moves on w-arrows induce local moves on w-trees, and such moves are
detailed in Section 5.2 of [17]. We will focus here on the moves that will be
needed in the following sections of this paper. First, the inverse move
generalizes to w-trees: two parallel copies of the same w-tree which only
differ by a twist on the last edge cancel each other out (see Figure 7).

$T$$T$$=$$=$$T$$T$ Figure 7: Inverse move for w-trees

The tail exchange move directly generalizes to w-trees: we can permute the
positions of two adjacent tails on a strand, whether they belong to the same
w-tree or to two distinct ones. The permutation of two adjacent heads can be
performed by adding a w-tree whose degree is the sum of the degrees of the two
w-trees, as illustrated in Figure 8.

$T_{1}$$T_{2}$$=$$T_{1}$$T_{1}$$T_{2}$$T_{2}$$T_{1}$$T_{2}$$=$$T_{1}$$T_{1}$$T_{2}$$T_{2}$
Figure 8: Permuting two adjacent heads.

Finally, as detailed in the proof of Lemma 7.10 of [17], we can permute the
head of a degree $k$ w-tree $T_{1}$ with a tail of a degree $k^{\prime}$
w-tree $T_{2}\neq T_{1}$ by adding a degree $k+k^{\prime}$ w-tree, and a
collection $C$ of w-trees of degree $>k+k^{\prime}$ whose heads are positioned
next to the head of $T_{2}$, as indicated in Figure 9.

$T_{1}$$T_{2}$$=$$T_{1}$$T_{2}$$T_{1}$$T_{2}$$\vdots$$C$ Figure 9: Permuting a
head and a tail from two different w-trees.

##### Remark:

We will also need to be able to permute the head of a w-tree with one of its
tails. We will see in Section 2 that, up to the addition of w-trees of higher
degree, it can be done using the preceding moves up to $w_{q}$–equivalence
(see Definition 1.9) and welded concordance, which is defined in Section 1.3.

One important aspect of the moves described above is that the heads of the
additional w-trees of higher degree are positioned next to some pre-existing
heads.

The notion of $w_{q}$–equivalence is defined in Section 7.1 of [17] as
follows:

######  Definition 1.9.

Two virtual diagrams are said to be _$w_{q}$ –equivalent_ if one can be
obtained from the other by a finite sequence of welded Reidemeister moves and
surgeries along w-trees of degree $\geq q$.

An important step in the classification of welded string links (resp. welded
links) will be showing that any virtual diagram of welded string link (resp.
welded link) is equivalent up to $w_{q}$–equivalence and welded concordance to
a diagram admitting an ascending tree presentation (resp. is
$w_{q}$–equivalent to a diagram admitting a sorted tree presentation), defined
as follows:

######  Definition 1.10.

A tree presentation of a welded string link is called _ascending_ if on each
component, the heads of the w-trees are positioned above the tails (the
strands being oriented upwards). A tree presentation of a welded link is
called _sorted_ if on each component, the heads and the tails of the w-trees
are positioned in two disjoint intervals.

The corresponding notion for diagrams of tangles was introduced by D. Bar-
Natan, Z. Dansco and R. van der Veen in [4], where it is called _over-then-
under tangles_ , as all the overcrossings appear before the undercrossings on
each strand. The closely related notions of ascending and sorted Gauss
diagrams were introduced in [1] and [2] respectively.

###  1.3 Welded concordance

In this section, we introduce the notion of welded concordance between two
welded links or two welded string links. This notion was defined on virtual
objects by J. Carter and S. Kamada in [6], and adapted to welded objects by
Gaudreau in [12] and by H. Boden and M. Chrisman in [5].

A _cobordism of welded links_ is a finite sequence of virtual diagrams of
welded links (with potentially differing number of components), where two
consecutive diagrams differ by one of the following moves:

  * •

a welded Reidemeister move;

  * •

a _birth_ or a _death_ , i.e. the creation or deletion of a closed component
with no crossings;

  * •

a _saddle point_ , i.e. the fusion of two components or separation of a
component in two as illustrated below.

In order to define cobordisms of welded string links, we need to introduce the
notion of mixed virtual diagrams, which is defined in the same way as a
virtual diagram of a welded string link, but with the addition of some
(possibly none) oriented closed components. A _cobordism of $n$–component
welded string links_ is a finite sequence of mixed virtual diagrams, each
containing $n$ interval components, where two consecutive diagrams differ by
one of the moves listed above.

Given a cobordism of welded links or welded string links, we can join the
consecutive diagrams to form an immersed surface, capped by disks at births
and deaths, and with saddles at saddle points. The surface $S$ contains a
certain number of connected components $S_{i}$, with Euler characteristic
$\chi(S_{i})$.

######  Definition 1.11.

A _welded concordance_ is a cobordism of welded links (resp. welded string
links) between two virtual diagrams of $n$–component welded links (resp.
welded string links), satisfying the following conditions:

  * •

the underlying surface $S$ has $n$ connected components, connecting the
$i^{th}$ component of the first diagram to the $i^{th}$ component of the last
diagram;

  * •

each connected component of $S$ is homeomorphic to an annulus $S^{1}\times I$
(resp. a disk $I\times I$).

The definition of a saddle point guaranties that the underlying surface of a
cobordism of welded links (resp. welded string links) is orientable. Since the
boundary of each connected component $S_{i}$ is a disjoint union of two
circles (resp. a circle), its homeomorphic class is determined by its Euler
characteristic, which must be $0$ (resp. $1$) for it to be an annulus (resp. a
disk). We can compute $\chi(S_{i})$ by counting the number of births, deaths
and saddle points involving the connected component $S_{i}$. More precisely,
for a welded link (resp. welded string link), the Euler characteristic of
$S_{i}$ starts at $0$ (resp. $1$), increases by $1$ for each birth and each
death, and decreases by $1$ for each saddle point. As a result, $S_{i}$ is
homeomorphic to an annulus (resp. a disk) if and only if it involves as many
births and deaths as saddle points.

######  Definition 1.12.

Two virtual diagrams are said to be _$w_{q}$ –concordant_ if one can be
obtained from the other by a finite sequence of $w_{q}$–equivalences and
welded concordance.

##  2 Classification of welded string links

In this section, we introduce the Milnor invariants of welded string links,
which give invariants of $w_{q}$–concordance. The main goal is to prove that
the Milnor invariants of length $\leq q$ are complete invariants of
$w_{q}$–concordance. In order to do that, we will show that any tree
presentation of a welded string link is $w_{q}$–concordant to an ascending
presentation, as the classification result on ascending presentations follows
from the fact that longitudes can be read directly on diagrams. This will in
turn give a diagrammatical characterization of classical string links with
equal Milnor invariants.

Since this section is dedicated to welded string links and will not involve
welded links, we refer to virtual diagrams of $n$–component welded string
links as simply virtual diagrams.

At the end of Section 1.2.1, we defined the peripheral system of a welded
string link $L$ represented by a virtual diagram $D$. We will now use it to
obtain the Milnor invariants of $L$. More precisely, we will consider the _$q$
–nilpotent peripheral system_ of $L$, which consists of the triplet
$(N_{q}G_{D},\overline{m},\overline{\ell})$, where
$\overline{m}=(\overline{m}_{1},\ldots,\overline{m}_{n})\in(N_{q}G_{D})^{n}$
is the image of the meridians in the $q$–niloptent quotient of $G_{D}$, and
$\overline{\ell}=(\overline{\ell}_{1},\ldots,\overline{\ell}_{n})\in(N_{q}G_{D})^{n}$
is the image of the longitudes.

It is known that the $q$–nilpotent quotient $N_{q}G_{D}$ is generated by the
$\overline{m}_{i}$’s, so each longitude $\overline{\ell}_{j}$ can be expresed
as a word on the $\overline{m}_{i}$’s. The power series
$E_{q}(\overline{\ell}_{j})$ is then well defined modulo monomials of degree
$\geq q$ (see [20] for more details). For a sequence of indices
$I=(i_{1},\ldots,i_{k},j)\in[\\![1,n]\\!]^{k+1}$ with $k<q$, we denote by
$\mu_{I}(D)\in\mathbb{Z}$ the coefficient of the monomial $X_{i_{1}}\cdots
X_{i_{k}}$ in $E_{q}(\overline{\ell}_{j})$ the _Milnor invariant of index $I$
of $D$_. The length of a Milnor invariant $\mu_{I}(D)$ is defined as the
length of the sequence of indices $I$.

Since the effect of a surgery along a w-tree of degree $k$ is to add a
commutator of weight $k$ in one of the longitudes $\ell_{j}$, it does not
modify $\overline{\ell}_{j}\in N_{q}G_{D}$ if $k\geq q$, and as such does not
modify the Milnor invariants of length $\leq q$. As a result, we have the
following, as noted in Remark 7.6 of [17]:

######  Proposition 2.1.

The Milnor invariants of length $\leq q$ are invariants of $w_{q}$–equivalence
on welded string links.

######  Proposition 2.2.

[3, Cor. 5.6] The Milnor invariants are welded concordance invariants.

This can be established using the terminology of cut-diagrams introduced in
[3], which generalizes Gauss diagrams in higher dimensions. It can easily be
checked that our definition of welded concordance (Definition 1.11) between
two welded string links represented by virtual diagrams implies the existence
of a cut-diagram of a surface between the two, where the lines of double
points correspond to the preimages of the undercrossings. As a result, the two
welded string links are cut-concordant (see Definition 5.1 in [3]), and by
Corollary 5.6 of [3] cut-concordance preserves the Milnor invariants, which
proves Proposition 2.2.

##### Remark:

A topological proof of Proposition 2.2 was given in [9] by M. Chrisman using
the Tube map, which associates a knotted surface in $\mathbb{R}^{4}$ to a
virtual (or welded) object. This knotted surface has the same Milnor
invariants as the diagrammatical object, and the invariance under topological
concordance of surfaces follows from a theorem of Stallings.

We will now prove that the Milnor invariants of length $\leq q$ are complete
invariants of $w_{q}$–concordance on welded string links. The main step in the
proof is showing that any tree presentation is $w_{q}$–concordant to an
ascending presentation, for which the classification result is relatively
straightforward.

Using the moves described in Section 1.2, we have the following result:

######  Proposition 2.3.

Let $T_{1}$ and $T_{2}$ be two different w-trees of respective degree $k$ and
$k^{\prime}$ in a tree presentation $(V,\mathcal{T})$. Then $(V,\mathcal{T})$
is equivalent to a tree presentation $(V,\mathcal{T}^{\prime})$ which only
differs from $(V,\mathcal{T})$ by the permutation of two adjacent ends of
$T_{1}$ and $T_{2}$ and the addition of w-trees of degree $\geq k+k^{\prime}$
whose heads are positioned next to the heads of w-trees in $(V,\mathcal{T})$.
In particular, the degrees of the additional w-trees are greater than the
degrees of $T_{1}$ and $T_{2}$.

Since the tail exchange move generalizes to w-trees, we can permute two
adjacent tails without adding any new w-tree, whether they come from the same
w-tree or not. We now show how to perform the last move needed to get to an
ascending presentation.

######  Proposition 2.4.

Let $q\geq 1$, and let $T$ be a w-tree of degree $k$ in a tree presentation
$(V,\mathcal{T})$, such that $T$ has a tail adjacent to its head. Then
$(V,\mathcal{T})$ is $w_{q}$–concordant to a tree presentation
$(V,\mathcal{T}^{\prime})$ which only differs from $(V,\mathcal{T})$ by the
permutation of the head of $T$ with its tail and the addition of w-trees of
degree $>k$. Moreover, the heads of the additional w-trees are positioned in a
small neighborhood of the head of $T$.

###### Proof.

If $k\geq q$, we can simply delete $T$ by $w_{q}$–equivalence and recreate it
with the positions of its head and adjacent tail swapped on the strand. We
illustrate the proof for $k<q$ on Figure 10, where we do not represent the
additional w-trees of degree $>k$ created by the moves described in
Proposition 2.3. The heads of these w-trees all end up on the portion of
strand represented in the figure.

To go from picture (a) to picture (b), we use a birth to create an empty
closed component. To go from (b) to (c), we use the inverse move to introduce
two copies of $T$ parallel to each other which only differ by a twist on the
last edge, but whose head and adjacent tail are positioned on the closed
component. To go from (c) to (d), we use Proposition 2.3 to permute the head
of the upper copy of $T$ with the tail of the lower one. This creates w-trees
of degree $\geq 2k$ whose heads are positioned next to the heads of these two
copies of $T$. To go from (d) to (e), we slide the tail of the upper copy of
$T$ along the closed component before performing a saddle move on picture (f).
Picture (g) depicts the same configuration as picture (f) after straightening
the portion of strand. To go from (g) to (h), we use Proposition 2.3 to bring
the tail of the upper copy of $T$ up to its head, and the head of the middle
copy of $T$ next to the head of the lower copy of $T$. There might be heads
and tails of higher degree w-trees in the way, but since we can delete g-trees
of degree $\geq q$ by $w_{q}$–equivalence, we can get to the configuration
depicted in picture (h) in a finite number of steps. Finally, to go from (h)
to (i) we use the inverse move to cancel out the two lower copies of $T$. We
are left with a copy of $T$ whose head and adjacent tail have been permuted,
and a collection of additional w-trees of degree $>k$ whose heads are
positioned on the portion of strand displayed in picture (i). ∎

$T$(a)$T$(b)$T$$T$$T$(c)$T$$T$$T$(d)$T$$T$$T$(e)$T$$T$$T$(f)$T$$T$$T$(g)$T$$T$$T$(h)$T$(i)

Figure 10: Permuting the head and a tail of the same w-tree.

The additional w-trees of higher degree are not represented.

######  Corollary 2.5.

Let $(V,\mathcal{T})$ be a tree presentation of a welded string link. For any
$q\geq 1$, $(V,\mathcal{T})$ is $w_{q}$–concordant to an ascending
presentation.

###### Proof.

First, note that by $w_{q}$–equivalence all w-trees of degree $\geq q$ can be
deleted, and this will be done implicitly in all the steps of this proof.
Using the moves $vR1$, $vR2$ and $vR3$ on the underlying virtual diagram $V$,
we can assume that it is made up of vertical strands without any virtual
crossing. We start by sliding the extremities of the w-trees of $\mathcal{T}$
upward (without permuting any) in order to get to a configuration where all
the heads are above the midway point on each strand. The goal is then to get
to a configuration where all the tails of w-trees are below the midway point,
since such a presentation would be ascending.

Let $t=(t_{1},\ldots,t_{q-1})\in\mathbb{N}_{0}^{q-1}$ be the $(q-1)$–tuple
indicating the number $t_{i}$ of tails belonging to w-trees of degree $i$
which are above the midway point. We will proceed by induction on $c$, the set
$\mathbb{N}_{0}^{q-1}$ being given the lexicographic order (which is a well-
order, so the principle of induction applies). By Propositions 2.3 and 2.4, we
can permute two adjacent ends of w-trees which are above the midway point by
adding w-trees of higher degree, whose heads are positioned above the midway
point as they appear next to pre-existing heads, which are all above it. If
$t=(0,\ldots,0)$, we are done. If $t>(0,\ldots,0)$, let $T$ be a w-tree with a
tail $\tau$ above the midway point, and whose degree is minimal among such
w-trees. Let $h=(h_{1},\ldots,h_{q-1})\in\mathbb{N}_{0}^{q-1}$ the
$(q-1)$–tuple indicating the number $h_{i}$ of heads belonging to w-trees of
degree $i$ which are positionned between the tail $\tau$ and the midway point.
We will bring the tail $\tau$ under the midway point by induction on $h$.

If $h=(0,\ldots,0)$, there are no heads between $\tau$ and the midway point,
so we can slide $\tau$ under it. If $h>(0,\ldots,0)$, we slide $\tau$ down to
the first head below it. Let $j$ be the degree of the w-tree of this head. We
can permute $\tau$ with this head by adding w-trees of degree greater than
$\deg(T)$ and $j$, whose heads are above the midway point. We have replaced
$h=(h_{1},\ldots,h_{j},h_{j+1},\ldots,h_{q-1})$ by
$h^{\prime}=(h_{1},\ldots,h_{j}-1,h_{j+1}^{\prime},\ldots,h_{q-1}^{\prime})<h$.
By the induction hypothesis, we can then bring $\tau$ below the midway point.

By doing this, we created new w-trees of degree $>i:=\deg(T)$ and whose heads
are above the midway point, so we replaced
$t=(0,\ldots,0,t_{i},t_{i+1},\ldots,t_{q-1})$ by
$t^{\prime}=(0,\ldots,0,t_{i}-1,t_{i+1}^{\prime},\ldots,t_{q-1}^{\prime})<t$.
This concludes the proof by induction on $t$. ∎

######  Lemma 2.6.

Two ascending presentations which have equal Milnor invariants of length $\leq
q$ are $w_{q}$–equivalent.

###### Proof.

Let $(V,\mathcal{T})$ be an ascending presentation. After expanding the
w-trees, we can assume that we have an arrow presentation $(V,\mathcal{A})$.
The conjugating elements associated to the w-arrows are then meridians
$m_{j}^{\pm}$, i.e. the generators associated to the bottom arcs of the
virtual diagram. As a result, the longitudes can be directly read as words on
the meridians, and there is a one-to-one correspondance between words
$\ell_{1},\ldots,\ell_{n}$ in the $m_{j}^{\pm 1}$’s and ascending arrow
presentations, each $m_{j}^{\pm 1}$ in $\ell_{i}$ corresponding to a
appropriately twisted w-arrow from the $j^{th}$ component to the $i^{th}$ one.
Using isolated arrow moves at the bottom of $i^{th}$ component, we can replace
$\ell_{i}$ by $m_{i}^{k}\ell_{i}$, so we can assume that
$a_{i}^{\ast}(\ell_{i})=0$ for $1\leq i\leq n$, so that the $\ell_{i}$’s are
the longitudes of the diagram. We will show that an ascending tree
presentation $(V,\mathcal{T}^{\prime})$ with Milnor invariants of length $\leq
q$ equal to those of $(V,\mathcal{A})$ can be modified using
$w_{q}$–equivalence into an ascending arrow presentation
$(V,\mathcal{A}^{\prime\prime})$ which has the same $\ell_{i}$’s as
$(V,\mathcal{A})$, and thus represent equivalent virtual diagrams.

Let us write the longitudes $\ell_{i}^{\prime}$ of $(V,\mathcal{T}^{\prime})$
as words on the generators $m_{j}$’s instead of the $m_{j}^{\prime}$’s. By the
equality of the Milnor invariants of legnth $\leq q$, we have
$E(\ell_{i})\equiv E(\ell_{i}^{\prime})\\!\mod\mathcal{X}^{q}$. By Proposition
1.2, we have $\ell_{i}\equiv\ell_{i}^{\prime}\\!\mod\Gamma_{q}F_{n}$. By
Proposition 1.1, $\ell_{i}$ and $\ell_{i}^{\prime}$ differ by a finite
sequence of insertion/deletion of $m_{j}m_{j}^{-1}$ and $m_{j}^{-1}m_{j}$,
which can be performed on arrow presentations by inverse moves, and
insertion/deletion of commutators of weight $\geq q$ on the $m_{j}^{\pm 1}$’s,
which can be performed on tree presentations by insertion/deletion of suitable
w-trees of degree $\geq q$ by $w_{q}$–equivalence. As a result,
$(V,\mathcal{T}^{\prime})$ is $w_{q}$–equivalent to an ascending tree
presentation $(V,\mathcal{T}^{\prime\prime})$ such that
$\ell_{i}^{\prime\prime}=\ell_{i}$ as words on the $m_{j}^{\pm 1}$’s. After
expanding the w-trees to obtain an arrow presentation
$(V,\mathcal{A}^{\prime\prime})$, we thus have
$(V,\mathcal{A}^{\prime\prime})=(V,\mathcal{A})$, so
$(V,\mathcal{A}^{\prime\prime})$ and $(V,\mathcal{T}^{\prime})$ are
$w_{q}$–equivalent. ∎

Using Propositions 2.2 and 2.1, Corollary 2.5 and Lemma 2.6, we obtain the
main result of the section:

######  Theorem 2.7.

The Milnor invariants of length $\leq q$ are complete invariants of
$w_{q}$–concordance on welded string links.

As a result, we obtain the following diagrammatical characterization on
classical string links:

######  Corollary 2.8.

Two classical string links have equal Milnor invariants of length $\leq q$ if
and only if they are $w_{q}$–concordant as welded string links.

We also have the following corollary on the finite type concordance invariants
of welded string links:

######  Corollary 2.9.

For two welded string links $L$ and $L^{\prime}$, the following statements are
equivalent:

  1. 1.

$L$ and $L^{\prime}$ are $w_{q}$–concordant,

  2. 2.

$L$ and $L^{\prime}$ have the same Milnor invariants of length $\leq q$,

  3. 3.

$L$ and $L^{\prime}$ the same finite type concordance invariants of degree
$<q$.

###### Proof.

We know that $1\Rightarrow 3$ by Proposition 7.5 of [17], and $3\Rightarrow 2$
was proved in the appendix of [7] using arguments similar to those of D. Bar-
Natan in the classical case. Finally, Theorem 2.7 above gives
$2\Leftrightarrow 1$. ∎

This shows that on welded string links, the Milnor invariants are universal
among $\mathbb{Z}$-valued finite type concordance invariants. This is to be
compared to the result of N. Habegger and G. Masbaum (Corollary 6.4 in [14])
giving the universality of the Milnor invariants among the rational finite
type concordance invariants on classical string links.

Since the Milnor invariants of any length are welded concordance invariants,
and the restriction to invariants of length $\leq q$ for the classification of
$w_{q}$–concordance comes from the addition of $w_{q}$–equivalence, a natural
question is to ask whether the Milnor invariants are complete invariants of
welded concordance on welded string links, or if other invariants are
required. This is still an open question.

##  3 Classification of welded links

The goal of this section is to prove that welded links are classified by their
$q$–nilpotent peripheral systems up to $w_{q}$–concordance. We will use the
fact that any tree presentation of a welded link is $w_{q}$–equivalent to a
sorted presentation, and that two sorted presentations with equivalent
$q$–nilpotent peripheral systems (see Definition 3.1) are $w_{q}$–concordant.
We will then use this classification to characterize welded links, and in turn
classical links, with vanishing Milnor invariants of length $\leq q$.

Since this section is dedicated to welded links, we refer to virtual diagrams
of $n$–component welded links as simply virtual diagrams.

###  3.1 The $q$–nilpotent peripheral system of a welded link

We defined the peripheral system of a virtual diagram $D$ at the end of
Section 1.2.1 as the group $G_{D}$ together with meridians $m_{i}$ and
longitudes $\ell_{i}$. Similarly to welded string links, we consider the _$q$
–nilpotent peripheral system of $D$_
$(N_{q}G_{D},\overline{m},\overline{\ell})$. In order to remove the
indeterminacy coming from the choice of the meridians, we introduce the
following equivalence relation on $q$–nilpotent peripheral systems:

######  Definition 3.1.

A $q$–nilpotent peripheral system $(N_{q}G_{D},\overline{m},\overline{\ell})$
of a virtual diagram $D$ is said to be _equivalent_ to a $q$–nilpotent
peripheral system
$(N_{q}G_{D^{\prime}},\overline{m}^{\prime},\overline{\ell}^{\prime})$ of a
virtual diagram $D^{\prime}$ if there exists an isomorphism
$\varphi:N_{q}G_{D}\to N_{q}G_{D^{\prime}}$ and elements
$g_{1},\ldots,g_{n}\in N_{q}G_{D}$ such that
$\varphi\big{(}\overline{m}_{i}^{g_{i}}\big{)}=\overline{m}_{i}^{\prime}$ and
$\varphi\big{(}\overline{\ell}_{i}^{g_{i}}\big{)}=\overline{\ell}_{i}^{\prime}$
for $1\leq i\leq n$.

######  Proposition 3.2.

Two $q$–nilpotent peripheral systems of a virtual diagram $D$ are equivalent.

###### Proof.

Let $(N_{q}G_{D},\overline{m},\overline{\ell})$ and
$(N_{q}G_{D},\overline{m}^{\prime},\overline{\ell}^{\prime})$ be two
$q$–nilpotent peripheral systems of a virtual diagram $D$. Let $p_{i}$ (resp.
$p_{i}^{\prime}$) denote the point on $D_{i}$ associated to $m_{i}$ (resp.
$m_{i}^{\prime}$), let $J_{i}$ denote the subinterval $]p_{i},p_{i}^{\prime}[$
of $D_{i}$, and let $g_{i}:=w_{J_{i}}\in N_{q}G_{D}$. Due to the Wirtinger
relations associated to the undercrossings on $J_{i}$, we have
$\overline{m}_{i}^{\prime}=\overline{m}_{i}^{g_{i}}$.

$D_{i}$$p_{i}$$\overline{m}_{i}$$p_{i}^{\prime}$$\overline{m}_{i}^{\prime}=\overline{m}_{i}^{g_{i}}$$J_{i}$$\dots$$\overbrace{{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>
Let $D_{i}^{p_{i}}$ (resp. $D_{i}^{p_{i}^{\prime}}$) be the subinterval
$D_{i}\setminus\\{p_{i}\\}$ (resp. $D_{i}\setminus\\{p_{i}^{\prime}\\}$) of
$D_{i}$, and $k\in\mathbb{Z}$ such that
$\overline{\ell}_{i}=\overline{m}_{i}^{-k}w_{D_{i}^{p_{i}}}$. Then
$w\\!_{D_{i}^{p_{i}^{\prime}}}=(w_{D_{i}^{p_{i}}})^{g_{i}}$ and
$a_{i}^{\ast}(w\\!_{D_{i}^{p_{i}^{\prime}}})=a_{i}^{\ast}(w_{D_{i}^{p_{i}}})=k$,
so
$\overline{\ell}_{i}^{g_{i}}=(\overline{m}_{i}^{-k})^{g_{i}}(w_{D_{i}^{p_{i}}})^{g_{i}}=\overline{m}_{i}^{\prime-k}w\\!_{D_{i}^{p_{i}^{\prime}}}=\overline{\ell}_{i}^{\prime}$.
As a result, the $q$–nilpotent peripheral systems
$(N_{q}G_{D},\overline{m},\overline{\ell})$ and
$(N_{q}G_{D},\overline{m}^{\prime},\overline{\ell}^{\prime})$ are equivalent
by the isomorphism $\varphi=\text{id}_{N_{q}G_{D}}$, with conjugating elements
$g_{i}=w_{J_{i}}$. ∎

The following proposition shows that the equivalence class of $q$–nilpotent
peripheral system is an invariant of welded links up to $w_{q}$–equivalence.

######  Proposition 3.3.

If two virtual diagrams differ by a Reidemeister move or by a surgery along a
w-tree of degree $\geq q$, then they have equivalent $q$–nilpotent peripheral
systems. Moreover, if the points on the diagrams associated to the meridians
stay fixed and outside of the local move, then the isomorphism between the
$q$–nilpotent peripheral systems is such that $g_{i}=1$ for $1\leq i\leq n$.

###### Proof.

If two virtual diagrams $D$ and $D^{\prime}$ differ by a Reidemeister move,
there is an isomorphism $\widetilde{\varphi}:G_{D}\to G_{D^{\prime}}$ sending
each generator associated to an arc of $D$ outside the local move to the
corresponding one in $D^{\prime}$, inducing an isomorphism
$\varphi:G=N_{q}G_{D}\to G^{\prime}=N_{q}G_{D^{\prime}}$. If the points
associated to the meridians stay fixed outside of the local move, then we have
$\varphi(\overline{m}_{i})=\overline{m}_{i}^{\prime}$, and it is easily
checked for each Reidemeister move that
$\varphi(\overline{\ell}_{i})=\overline{\ell}_{i}^{\prime}$. If some points
associated to the meridians are inside the local move, we change their
positions before and after the move to get to the previous case, which does
not change the equivalence class of the $q$–nilpotent peripheral systems by
Proposition 3.2.

If $D^{\prime}$ is obtained from $D$ by a surgery along a w-tree of degree
$k\geq q$, the presentation of $G_{D^{\prime}}$ differs from the presentation
of $G_{D}$ by duplicating the generator associated to the arc of $D$
containing the head of the w-tree, and adding a Wirtinger relation between
them, whose conjugating element is the commutator of weight $k$ obtained by
propagating generators along the w-tree. Since $k\geq q$, this conjugating
element is trivial in $N_{q}G_{D^{\prime}}$, so the added Wirtinger relation
identifies the duplicated generators. As a result, the map
$\varphi:N_{q}G_{D}\to N_{q}G_{D^{\prime}}$ sending each generator of $G_{D}$
to the corresponding one in $G_{D^{\prime}}$ is an isomorphism, which gives an
equivalence of $q$–nilpotent peripheral systems for which $g_{i}=1$ for $1\leq
i\leq n$. ∎

Since the $q$–nilpotent peripheral system of a welded string link is
equivalent to its Milnor invariants of length $\leq q$, by Proposition 2.2 it
is an invariant of welded concordance. The same is true for the (equivalence
class of) $q$–nilpotent peripheral system on welded links. This follows from
Theorem 5.5 in [3], as it is easily checked that the existence of a welded
concordance between two welded links implies that of a cut-concordance, as was
the case for welded string links.

######  Proposition 3.4.

[3, Thm. 5.5] If two virtual diagrams differ by a welded concordance, then
they have equivalent $q$–nilpotent peripheral systems for all $q\geq 1$.

As a result, the equivalence class of $q$–nilpotent peripheral system is an
invariant of $w_{q}$–concordance on welded links.

###  3.2 Classification of welded links up to $w_{q}$–concordance

In this section, we will prove that the equivalence class of $q$-nilpotent
peripheral systems is a complete invariant of $w_{q}$–concordance on welded
links. The first step is to show that any tree presentation is
$w_{q}$–equivalent to a sorted one (see Definition 1.10), whose associated
group admits a presentation involving only meridians and longitudes. As a by-
product, we obtain the Chen–Milnor presentation of $N_{q}G_{D}$ for any
virtual diagram $D$. We then prove that two sorted tree presentations with
equivalent $q$–nilpotent peripheral systems are $w_{q}$–concordant. This in
turn gives a characterization of links with trivial Milnor invariants of
length $\leq q$.

######  Proposition 3.5.

For $q\geq 1$, any tree presentation is $w_{q}$–equivalent to a sorted
presentation.

###### Proof.

The proof is similar to that of Corollary 2.5. The portion of strand $P_{i}$
which will contain the tails on the $i^{th}$ component of the sorted tree
presentation can be chosen arbitrarily, starting as a small neighborhood of a
point containing no head. We need to push the tails on this component toward
$P_{i}$, with the notable difference from the case of string links being that
we can choose the direction in which we push each tail. As a result, we can
avoid having to permute a tail with the head of its w-tree, simply by pushing
it in the direction opposite to its head if they are on the same component.

$P_{i}$$P_{i}$

Since the permutation of a tail with its head was the only move requiring a
welded concordance in the proof of Corollary 2.5, we only need moves which can
be achieved by $w_{q}$–equivalence in order to get to a sorted presentation.
This can be done using the same induction process as in the aforementioned
proof, first on the number of tails outside $P_{i}$, and then for each tail on
the number of heads in between that tail and $P_{i}$ in the direction which we
choose to push the tail. ∎

An important detail to note in the proof of Proposition 3.5 is that at no
point in the process does any head enter the portion of strand $P_{i}$. By
Proposition 3.3, if we choose the meridian $m_{i}$ in $P_{i}$, then the
isomorphism relating the $q$–nilpotent peripheral system of the original
diagram to that of the sorted diagram is such that $g_{i}=1$.

If $(V,\mathcal{T})$ is a sorted presentation of a virtual diagram $D$ with
meridians on the arcs containing the tails, every generator can be expressed
in terms of the meridians, and the only relations left are the commutation of
each meridian with its longitude, so $G_{D}\simeq\langle
m_{1},\ldots,m_{n}\,|\,[m_{1},\ell_{1}],\ldots,[m_{n},\ell_{n}]\rangle$. By
Proposition 1.1, we obtain a presentation of $N_{q}G_{D}$ by adding the
commutators of weight $\geq q$ on generators $m_{i}^{\pm 1}$ in the relations.
Since any tree presentation is $w_{q}$–equivalent to a sorted one, and
$w_{q}$–equivalent diagrams have isomorphic $q$–nilpotent groups, we obtain
the following Chen–Milnor presentation, which was established by J. Milnor for
classical links in [18]:

######  Proposition 3.6.

For any virtual diagram $D$ and $q\geq 1$, we have:

$N_{q}G_{D}\simeq\langle
m_{1},\ldots,m_{n}\,|\,C_{q},[m_{1},\overline{\ell}_{1}],\ldots,[m_{n},\overline{\ell}_{n}]\rangle,$

where the $m_{i}$’s are meridians of $D$, the $\overline{\ell}_{i}$’s are the
associated longitudes expressed in terms of the meridians and $C_{q}$ is the
set of commutators of weight $\geq q$ on the generators $m_{i}^{\pm 1}$.

Similar Chen–Milnor type theorems on welded string links and welded links were
already established by M. Chrisman in [9], and by B. Audoux, J-B. Meilhan and
A. Yasuhara in [3] using the notion of cut diagrams.

######  Proposition 3.7.

Two sorted tree presentations which have equivalent $q$–nilpotent peripheral
systems are $w_{q}$–concordant.

We now prove that the equivalence class of $q$–nilpotent peripheral systems
classifies welded links up to $w_{q}$–concordance, starting with the special
case of sorted presentations.

###### Proof.

By expanding the w-trees, we can assume that the sorted presentations are
arrow presentations, denoted $(V,\mathcal{A})$ and $(V,\mathcal{A}^{\prime})$
respectively. Let $D$ and $D^{\prime}$ be the virtual diagrams obtained from
these presentations. By Proposition 3.5, we can assume the points $p_{i}$
(resp. $p_{i}^{\prime}$) associated to the meridians of the presentation
$(V,\mathcal{A})$ (resp. $(V,\mathcal{A}^{\prime})$) are placed on the arcs
containing the tails. We first deal with the case of an isomorphism
$\varphi:N_{q}G_{D}\to N_{q}G_{D^{\prime}}$ such that $g_{i}=1$ for $1\leq
i\leq n$. Using the same notation as in the previous section, and performing
some isolated arrow moves if necessary, we can assume that the words
$l_{i}:=w_{D_{i}^{p_{i}}}$ (resp. $l_{i}^{\prime}:=w\\!_{D_{i}^{\prime
p_{i}^{\prime}}}$) satisfy $a_{i}^{\ast}(l_{i})=0$ (resp.
$a_{i}^{\prime\ast}(l_{i})=0$). Here the $l_{i}$’s (resp. $l_{i}^{\prime}$’s)
are considered as words on the $m_{j}^{\pm 1}$ (resp. $m_{j}^{\prime\pm 1}$’s)
without any relation between the generators. In order to compare $l_{i}$ and
$l_{i}^{\prime}$, we introduce the word $\tilde{l}_{i}^{\prime}$ on the
$m_{j}^{\pm 1}$ obtained by replacing $m_{j}^{\prime}$ by $m_{j}$ in
$l_{i}^{\prime}$. Then the words $l_{i}$ and $\tilde{l}_{i}^{\prime}$ both
represent the longitude
$\overline{\ell}_{i}=\varphi^{-1}(\overline{\ell}_{i}^{\prime})\in
N_{q}G_{D}$. As a result, by the presentation of $N_{q}G_{D}$ given above, we
can go from $l_{i}$ to $\tilde{l}_{i}^{\prime}$ by adding/deleting occurences
of $m_{j}m_{j}^{-1}$, $m_{j}^{-1}m_{j}$, commutators of weight $\geq q$ on the
$m_{j}$’s, and $[m_{j},\overline{\ell}_{j}]^{\pm 1}$.

For each $1\leq i\leq n$ and each addition/deletion of
$[m_{j},\overline{\ell}_{j}]^{\pm 1}$ in $\ell_{i}$, we preemptively create
one closed component, which will be attached to the $i^{th}$ component of
$(V,\mathcal{A})$. By an inverse move, we add a pair of cancelling w-arrows
from the arc of $V_{j}$ containing the meridian $m_{j}$ to the new closed
component, then we slide the tail of the non-twisted arrow along $V_{j}$ with
the reverse orientation, using inverse and slide moves to cross the heads (see
Figure 11). This has the effect of creating the word
$[m_{j},\overline{\ell}_{j}]$ on the new component. To create the word
$[m_{j},\overline{\ell}_{j}]^{-1}=[\overline{\ell}_{j},m_{j}]$, we slide the
tail of the twisted arrow instead of the non-twisted one.

$V_{i}$$V_{j}$$m_{j}$$V_{i}$$V_{j}$$m_{j}$$V_{i}$$V_{j}$$m_{j}$$\vdots$$\Big{\\}}$$\overline{\ell}_{j}$$\vdots$$\Big{\\}}$$\overline{\ell}_{j}^{-1}$

We need to create these extra closed components in order to make sure that the
words $\ell_{i}$ representing the longitudes are not modified before all the
required words $[m_{j},\overline{\ell}_{j}]^{\pm 1}$ have been created.
Moreover, trying to insert the word $[m_{i},\overline{\ell}_{i}]^{\pm 1}$
directly into the $i^{th}$ component without creating an extra one would
result in a neverending loop as new w-arrows would constantly be created when
trying to slide the tail of the w-arrow illustrated above through its head.

Once these extra components have been added, we can start to implement the
necessary modifications on $(V,\mathcal{A})$ to go from $l_{i}$ to
$\tilde{l}_{i}^{\prime}$: an addition/deletion of $m_{j}m_{j}^{-1}$ or
$m_{j}^{-1}m_{j}$ can be achieved by an inverse move, the addition/deletion of
a commutator of weight $\geq q$ on the $m_{j}^{\pm 1}$ can be achieved by
$w_{q}$–equivalence, and the addition of $[m_{j},\overline{\ell}_{j}]^{\pm 1}$
in the word $l_{i}$ can be achieved by attaching one of the extra components
containing the word $[m_{j},\overline{\ell}_{j}]^{\pm 1}$ to $V_{i}$ on the
appropriate arc by a saddle move. In order to delete such a commutator, we
insert its inverse by the same process, then cancel out the product by inverse
moves. As a result, we can modify $(V,\mathcal{A})$ into a $w_{q}$–concordant
tree presentation $(V,\mathcal{T}^{\prime\prime})$ whose longitudes are
represented by the same words as those of $(V,\mathcal{A}^{\prime})$. After
expanding the w-trees, we hence obtain an arrow presentation equal to
$(V,\mathcal{A}^{\prime})$, and as a result $(V,\mathcal{A})$ and
$(V,\mathcal{A}^{\prime})$ are $w_{q}$–concordant.

inverseslide Figure 11: Crossing a head using inverse and slide moves.

We now deal with the general case of an isomorphism $\varphi$ with potentially
non trivial conjugating elements $g_{i}\in N_{q}G_{D}$. We will modify
$(V,\mathcal{A})$ in order to get to the configuration above. Using inverse
moves, we add w-arrow heads representing the word $g_{1}$ followed by the word
$g_{1}^{-1}$ next to the arc containing the tails on $V_{1}$. As in the proof
of Proposition 3.5, we can slide the tails on $D_{1}$ to the portion of strand
between the words $g_{1}$ and $g_{1}^{-1}$ by using $w_{q}$–equivalence.
Moreover, we can preserve the sorted structure of the other components, and
keep all the heads outside of the portion of strand between the words $g_{1}$
and $g_{1}^{-1}$. We obtain a sorted presentation
$(V,\mathcal{T}^{\prime\prime})$ of a virtual diagram $D^{\prime\prime}$ which
is $w_{q}$–equivalent to $D$, with a $q$–nilpotent peripheral system
$(N_{q}G_{D^{\prime\prime}},m^{\prime\prime},\overline{\ell}^{\prime\prime})$
such that each meridian $m_{i}^{\prime\prime}$ is placed on the arc containing
the tails. By the proof of Proposition 3.2 and by Proposition 3.3, the
$q$–nilpotent peripheral system
$(N_{q}G_{D^{\prime\prime}},m^{\prime\prime},\overline{\ell}^{\prime\prime})$
is equivalent to $(N_{q}G_{D},m,\overline{\ell})$ by an isomorphism
$\psi:N_{q}G_{D^{\prime\prime}}\to N_{q}G_{D}$ such that
$\psi(m_{1}^{\prime\prime})=m_{1}^{g_{1}}$,
$\psi(\overline{\ell}_{1}^{\prime\prime})=\overline{\ell}_{1}^{g_{1}}$ and
$\psi(m_{i}^{\prime\prime})=m_{i}$,
$\psi(\overline{\ell}_{i}^{\prime\prime})=\overline{\ell}_{i}$ for $i\geq 2$.
The isomorphism $\varphi\circ\psi:N_{q}G_{D^{\prime\prime}}\to
N_{q}G_{D^{\prime}}$ sends $m_{1}^{\prime\prime}$ to $m_{1}^{\prime}$,
$\overline{\ell}_{1}^{\prime\prime}$ to $\overline{\ell}_{1}^{\prime}$, and
$(m_{i}^{\prime\prime})^{\psi^{-1}(g_{i})}$ to $m_{i}^{\prime}$,
$(\overline{\ell}_{i}^{\prime\prime})^{\psi^{-1}(g_{i})}$ to
$\overline{\ell}_{i}^{\prime}$ for $i\geq 2$. By iterating this process for
each component, we obtain a sorted tree presentation which is
$w_{q}$–equivalent to $(V,\mathcal{A})$ and whose $q$–nilpotent peripheral
system is equivalent to that of $(V,\mathcal{A}^{\prime})$ by an isomorphism
which sends meridians to meridians and longitudes to longitudes. By the
special case above, this concludes the proof of the proposition. ∎

Note that contrary to the case of welded string links, the notion of welded
concordance is not used to obtain a sorted presentation, but is necessary to
relate two sorted presentations with the same invariants. Using Propositions
3.3, 3.4 and 3.7, we obtain the main result of the section:

######  Theorem 3.8.

The equivalence class of $q$–nilpotent peripheral systems is a complete
invariant of $w_{q}$–concordance on welded links.

The Milnor invariants of a welded link can also be obtained as the
coefficients of the power series $E_{q}(\overline{\ell}_{i})$. However, as in
the case of classical links, because of the remaining relations in
$N_{q}G_{D}$ the Milnor invariants of length $q$ are only well defined modulo
some of the Milnor invariants of length $<q$, and only the first non-vanishing
Milnor invariants of the link are well-defined as integers. We will use
Theorem 3.8 to give a characterization of welded links with Milnor invariants
of length $\leq q$ equal to zero. This will require a more precise description
of the Chen–Milnor presentation, using the Chen maps defined below.

For a virtual diagram $D$, let $\langle a_{ij}\rangle$ be the free group with
one generator $a_{ij}$ for each arc of $D_{i}$. Once the points $p_{i}$
associated to the meridians $m_{i}=a_{i0}$ are chosen, we denote by $w_{ij}$
the word $w_{J}$, where $J$ is the subinterval of $D_{i}$ from $p_{i}$ to a
point on the arc associated to the generator $a_{ij}$. The group $G_{D}$ is
then equal to $\langle a_{ij}\,|\,a_{ij}=m_{i}^{w_{ij}}\rangle$.

######  Definition 3.9.

The Chen maps $\eta_{q}:\langle a_{ij}\rangle\to F_{n}=\langle m_{i}\rangle$
associated to a virtual diagram $D$ are defined by induction on $q\geq 1$ by
setting $\eta_{1}(a_{ij}):=m_{i}$ for $1\leq i\leq n$, and
$\eta_{q+1}(a_{ij}):=\eta_{q}\big{(}m_{i}^{w_{ij}}\big{)}$ for $q\geq 1$ and
$1\leq i\leq n$.

It is easily checked that $\eta_{q+1}\equiv\eta_{q}\\!\mod\Gamma_{q}G_{D}$ for
$q\geq 1$. Moreover, the Chen maps induce homomorphisms
$\overline{\eta}_{q}:N_{q}G_{D}\to N_{q}\langle
m_{i}\,|\,[m_{i},\eta_{q}(\ell_{i})]\rangle$, and these are isomorphisms (see
[9]). The Milnor invariants of length $\leq q$ of $D$ can be obtained from the
power series $E_{q}(\overline{\eta}_{q}(\overline{\ell}_{i}))$, as the Chen
maps give a way to express the longitudes in terms of the meridians.

######  Corollary 3.10.

A welded link is $w_{q}$–concordant to the unlink if and only if it has
vanishing Milnor invariants of length $\leq q$.

###### Proof.

Let $L$ be a welded link represented by a virtual diagram $D$. If $L$ is
$w_{q}$–concordant to the unlink, by Theorem 3.8 its $q$–nilpotent peripheral
system is equivalent to the $q$–nilpotent peripheral system of the unlink, and
in particular the longitudes $\overline{\ell}_{i}$ are trivial in
$N_{q}G_{D}$. As a result, $E_{q}(\overline{\eta}_{q}(\overline{\ell}_{i}))=1$
for $1\leq i\leq n$, and the Milnor invariants of length $\leq q$ of $L$ are
all zero.

Conversely, suppose that the Milnor invariants of length $\leq q$ of $L$ are
all zero. We prove by induction on $1\leq k\leq q$ that
$\eta_{k}(\ell_{i})\in\Gamma_{k}F_{n}$ for $1\leq i\leq n$. For $k=1$ it is
trivially true since $\Gamma_{1}F_{n}=F_{n}$. If it is true for $k-1$, then
$\eta_{k-1}(\ell_{i})\in\Gamma_{k-1}F_{n}$, and since
$\eta_{k}\equiv\eta_{k-1}\\!\mod\Gamma_{k-1}F_{n}$, we have:

$[m_{i},\eta_{k}(\ell_{i})]\equiv[m_{i},\eta_{k-1}(\ell_{i})]\equiv
1\\!\mod\Gamma_{k}F_{n},$

hence $[m_{i},\eta_{k}(\ell_{i})]\in\Gamma_{k}F_{n}$. As a result, the induced
Chen map $\overline{\eta}_{k}$ is:

$\overline{\eta}_{k}:N_{k}G_{D}\to N_{k}\langle
m_{i}\,|\,[m_{i},\eta_{k}(\ell_{i})]\rangle\simeq N_{k}F_{n}.$

The Milnor invariants of length $k$ of $L$ are then the coefficients of the
monomials of degree $k-1$ of the power series
$E_{k}(\overline{\eta}_{k}(\overline{\ell}_{i}))\in
U_{1}(\mathcal{S}_{n})/(1+\mathcal{X}^{k})$. Since these are zero by our
hypothesis on $L$, $E_{k}(\overline{\eta}_{k}(\overline{\ell}_{i}))=1$, and by
Proposition 1.2 we have $\overline{\eta}_{k}(\overline{\ell}_{i})=1\in
N_{k}F_{n}$ for $1\leq i\leq n$. As a result
$\eta_{k}(\ell_{i})\in\Gamma_{k}F_{n}$, which concludes the induction on $k$.
The $q$–nilpotent peripheral system of $L$ is then equivalent to that of the
unlink by the induced Chen map $\overline{\eta}_{q}:N_{q}G_{D}\to N_{q}F_{n}$,
which sends the meridians of $L$ to the generators of $N_{q}F_{n}$ and the
longitudes of $L$ to $1$. By Theorem 3.8, $L$ is $w_{q}$–concordant to the
unlink. ∎

As a result, we have the following diagrammatical characterization of
classical links with vanishing Milnor invariants of length $\leq q$:

######  Corollary 3.11.

A classical link has vanishing Milnor invariants of length $\leq q$ if and
only if it is $w_{q}$–concordant to the unlink as a welded link.

This is to be compared to the result of J. Conant, R. Schneiderman and P.
Teichner on the classification of $C_{q}$–concordance on classical links [11,
Cor. 3]. As this classification involves the additional higher-order Sato-
Levine and Arf invariants, it is to be expected that some classical links
might be $w_{q}$–concordant to the unlink as a welded link but not
$C_{q}$–concordant to it as a classical link. Note that by Theorem 8.1 of
[17], welded knots are already trivial up to $w_{q}$–equivalence (and hence up
to $w_{q}$–concordance), and by Theorem 5 of [12], welded long knots (i.e.
welded string links with one component) are trivial up to welded concordance,
so the case of one component is always trivial up to $w_{q}$–concordance. We
give an example below illustrating the difference between $C_{q}$–concordance
and $w_{q}$–concordance on a link with $2$ components.

##### Example:

Illustrated in Figure 12 is a link which is $w_{3}$–concordant to the unlink,
as it has trivial Milnor invariants of length $\leq 3$, but not
$C_{3}$–concordant to the unlink, as it has non-trivial Sato-Levine invariants
of order 2 (see [10] Section 5.1 for indications on how to compute these
invariants). This link is obtained, modulo one $R1$ move, by performing a
clasper surgery on the unlink along the tree of order 1 (or degree 2)
represented on the left. Due to the self-annihilation relation $[X,X]=0$ in a
Lie algebra, the Milnor invariants of length $3$, the first potentially non-
trivial ones, vanish. The Sato-Levine invariants are designed to detect such
trees, as they take their values in a quasi-Lie algebra where the self-
annihilation relation is replaced by the antisymmetry $[Y,X]=-[X,Y]$.

$1$$2$$=$$1$$2$ Figure 12: A link which is $w_{3}$–concordant but not
$C_{3}$–concordant to the unlink

## References

  * [1] Benjamin Audoux, Paolo Bellingeri, Jean-Baptiste Meilhan, and Emmanuel Wagner.  Homotopy classification of ribbon tubes and welded string links.  Annali della Scuola Normale Superiore di Pisa Classe di Scienza, 17(2):713–761, 2017.
  * [2] Benjamin Audoux and Jean-Baptiste Meilhan.  Characterization of the reduced peripheral system of links.  arXiv:1904.04763v4, 2020.
  * [3] Benjamin Audoux, Jean-Baptiste Meilhan, and Akira Yasuhara.  Milnor concordance invariant for knotted surfaces and beyond.  arXiv:2109.14578, 2021.
  * [4] Dror Bar-Natan, Zsuzsanna Dancso, and Roland van der Veen.  Over then under tangles.  arXiv:2007.09828, 2020.
  * [5] Hans U. Boden and Micah Chrisman.  Virtual concordance and the generalized Alexander polynomial.  J. Knot Theory Ramifications, 30(5):Paper No. 2150030, 35, 2021\.
  * [6] J. Scott Carter, Seiichi Kamada, and Masahico Saito.  Stable equivalence of knots on surfaces and virtual knot cobordisms.  J. Knot Theory Ramifications, 11(3):311–322, 2002.
  * [7] Adrien Casejuane.  Formules combinatoires pour les invariants d’objets noués.  PhD thesis, Institut Fourier, 2021.
  * [8] Adrien Casejuane and Jean-Baptiste Meilhan.  On finite type invariants of welded string links and ribbon tubes.  2021\.
  * [9] Micah Chrisman.  Milnor’s concordance invariants for knots on surfaces.  arXiv:2002.01505v2, 2021.
  * [10] James Conant, Rob Schneiderman, and Peter Teichner.  Whitney tower concordance of classical links.  Geom. Topol., 16(3):1419–1479, 2012.
  * [11] James Conant, Rob Schneiderman, and Peter Teichner.  Clasper concordance, Whitney towers and repeating Milnor invariants.  arXiv:2005.05381, 2020.
  * [12] Robin Gaudreau.  Classification of virtual string links up to cobordism.  Ars Math. Contemp., 19(1):37–49, 2020.
  * [13] Mikhail Goussarov, Michael Polyak, and Oleg Viro.  Finite-type invariants of classical and virtual knots.  Topology, 39(5):1045–1068, 2000.
  * [14] Nathan Habegger and Gregor Masbaum.  The Kontsevich integral and Milnor’s invariants.  Topology, 39(6):1253–1289, 2000.
  * [15] Kazuo Habiro.  Claspers and finite type invariants of links.  Geom. Topol., 4:1–83, 2000.
  * [16] Wilhelm Magnus, Abraham Karrass, and Donald Solitar.  Combinatorial group theory. Presentations of groups in terms of generators and relations.  Dover Publications, 1976.
  * [17] Jean-Baptiste Meilhan and Akira Yasuhara.  Arrow calculus for welded and classical links.  Algebraic and Geometric Topology, 19:397–456, 2019.
  * [18] John Milnor.  Isotopy of links.  In Algebraic geometry and topology, A Symposium in honor of S. Lefschetz, pages 280–306. Princeton Mathematical Series, No. 12. Princeton, New Jersey: Princeton University Press VIII, 1957.
  * [19] John W. Milnor.  Link groups.  Ann. Math. (2), 59:177–195, 1954.
  * [20] Hakuro A. Miyazawa, Kodai Wada, and Akira Yasuhara.  Combinatorial approach to milnor invariants of welded links.  arXiv:2003.13273v2, 2020.