Valeriy G Bardakov

Paolo Bellingeri
email: paolo.bellingeri@unicaen.fr

GROUPS OF VIRTUAL AND WELDED LINKS

Keywords: 1991 Mathematics Subject Classification. Primary 20F36 Braid groups, virtual braid, welded braids, virtual knots, welded knots, group of knot

published or not. The documents may come

introduction

Virtual knot theory has been introduced by Kauffman [21] as a generalization of classical knot theory. Virtual knots (and links) are represented as generic immersions of circles in the plane (virtual link diagrams) where double points can be classical (with the usual information on overpasses and underpasses) or virtual. Virtual link diagram are equivalent under ambient isotopy and some types of local moves (generalized Reidemeister moves): classical Reidemeister moves (Figure 1), virtual Reidemeister moves and mixed Reidemeister moves (Figures 2 and3). [START_REF] Goussarov | Finite-type invariants of classical and virtual knots[END_REF]Theorem 1B] states that if two classical knot diagrams are equivalent under generalized Reidemeister moves, then they are equivalent under the classical Reidemeister moves. In this sense virtual link theory is a nontrivial extension of classical theory. This Theorem is a straightforward consequence of the fact that the knot group (more precisely the group system of a knot, see for instance [START_REF] Conway | A group to classify knots[END_REF][START_REF] Joyce | A Classifying Invariant of Knots, the Knot Quandle[END_REF]) is a complete knot invariant which can be naturally extended in the realm of virtual links. Nevertheless this notion of invariant does not appear satisfactory for virtual objects (see Section 4): the main goal of this paper is to explore new invariants for virtual links using braids and their virtual generalizations.

In fact, using virtual generalized Reidemeister moves we can introduce a notion of "virtual" braids (see for instance [START_REF] Kauffman | Virtual knot theory[END_REF][START_REF] Vershinin | On homology of virtual braids and Burau representation[END_REF]). Virtual braids on n strands form a group, usually denoted by V B n . The relations between virtual braids and virtual knots (and links) are completely determined by a generalization of Alexander and Markov Theorems [START_REF] Kamada | Invariants of virtual braids and a remark on left stabilisations and virtual exchange moves[END_REF].

To the generalized Reidemeister moves on virtual diagrams one could add the following local moves, called forbidden moves of type F 1 and F 2 (Figure 4):

We can include one or both of them to obtain a "quotient" theory of the theory of virtual links. If we allow the move F 1, then we obtain the theory of Welded links whose interest is growing up recently, in particular because of the fact that the welded braid counterpart can be defined in several equivalent ways (for instance in terms of configuration spaces, mapping classes and automorphisms of free groups). The theory with both forbidden moves added is called the theory of Fused links but this theory is trivial, at least at the level of knots, since any knot is equivalent to the trivial knot [START_REF] Kanenobu | Forbidden moves unknot a virtual knot[END_REF][START_REF] Nelson | Unknotting virtual knots with Gauss diagram forbidden moves[END_REF].

The paper is organized as follows: in Sections 2 and 3 we recall some definitions and classical results and we construct a representation of V B n into Aut F n+1 ; using this representation we define (Section 4) a new notion of group of a virtual knot (or link) and we compare our invariant to other known invariants. In the case of welded objects (Section 5) our construction gives an invariant which is a straightforward generalization to welded knots of Kauffman notion of group of virtual knots. We conclude with some observations on the analogous of Wada groups in the realm of welded links.

Acknowledgements. The research of the first author was partially supported by RFBR-10-01-00642. The research of the second author was partially supported by French grant ANR-11-JS01-002-01.

This work started during the staying of the second author at the University of Caen, december 2010, in the framework of the French -Russian grant 10-01-91056. The first author would like to thank the members of the Laboratory of Mathematics of the University of Caen for their kind hospitality.

Braids vs virtual and welded braids

During last twenty years several generalizations of braid groups were defined and studied, according to their definition as "diagrams" in the plane: in particular singular braids [START_REF] Baez | Link invariants of finite type and perturbation theory[END_REF], virtual braids [START_REF] Kauffman | Virtual knot theory[END_REF][START_REF] Vershinin | On homology of virtual braids and Burau representation[END_REF] and welded braids [START_REF] Fenn | The braid-permutation group[END_REF].

It is worth to mention that for all of these generalizations it exists an Alexander-like theorem, stating that any singular (respectively virtual or welded) link can be represented as the closure of a singular (respectively virtual or welded) braid. Moreover, there are generalizations of classical Markov's theorem for braids giving a characterization for two singular (respectively virtual or welded) braids whose closures represent the same singular (respectively virtual or welded) link [START_REF] Gemein | Singular braids and Markov's theorem[END_REF][START_REF] Kamada | Invariants of virtual braids and a remark on left stabilisations and virtual exchange moves[END_REF].

In the following we introduce virtual and welded braid groups as quotients of free product of braid groups and corresponding symmetric groups.

Virtual and welded braid groups have several other definitions, more intrinsic, see for instance [START_REF] Bar Natan | Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne[END_REF][START_REF] Kamada | Invariants of virtual braids and a remark on left stabilisations and virtual exchange moves[END_REF] for the virtual case and [START_REF] Brendle | Configuration spaces of rings and wickets, to appear in[END_REF][START_REF] Fenn | The braid-permutation group[END_REF][START_REF] Kamada | Invariants of virtual braids and a remark on left stabilisations and virtual exchange moves[END_REF] for the welded one.

The braid group B n , n ≥ 2, on n strings can be defined as the group generated by σ 1 , σ 2 , . . . , σ n-1 , with the defining relations:

σ i σ i+1 σ i = σ i+1 σ i σ i+1 , i = 1, 2, . . . , n -2, σ i σ j = σ j σ i , |i -j| ≥ 2.
The virtual braid group V B n can be defined as the group generated by the elements σ i , ρ i , i = 1, 2, . . . , n -1 with the defining relations:

σ i σ i+1 σ i = σ i+1 σ i σ i+1 , i = 1, 2, . . . , n -2, σ i σ j = σ j σ i , |i -j| ≥ 2. ρ i ρ i+1 ρ i = ρ i+1 ρ i ρ i+1 , i = 1, 2, . . . , n -2, ρ i ρ j = ρ j ρ i , |i -j| ≥ 2. ρ 2 i = 1, i = 1, 2, . . . , n -1; σ i ρ j = ρ j σ i , |i -j| ≥ 2, ρ i ρ i+1 σ i = σ i+1 ρ i ρ i+1 , i = 1, 2, . . . , n -2.
It is easy to verify that ρ i 's generate the symmetric group S n and that the σ i 's generate the braid group B n (see Remark 1).

In [START_REF] Goussarov | Finite-type invariants of classical and virtual knots[END_REF] it was proved that the relations

ρ i σ i+1 σ i = σ i+1 σ i ρ i+1 , ρ i+1 σ i σ i+1 = σ i σ i+1 ρ i
corresponding to the forbidden moves F1 and F2 for virtual link diagram, are not fulfilled in V B n .

According to [START_REF] Fenn | The braid-permutation group[END_REF] the welded braid group W B n is generated by σ i , α i , i = 1, 2, . . . , n -1. Elements σ i generate the braid group B n and elements α i generate the symmetric group S n , and the following mixed relations hold

α i σ j = σ j α i , |i -j| ≥ 2, α i+1 α i σ i+1 = σ i α i+1 α i , i = 1, 2, . . . , n -2, α i σ i+1 σ i = σ i+1 σ i α i+1 , i = 1, 2, . . . , n -2.
Comparing the defining relations of V B n and W B n , we see that the group presentation of W B n can be obtained from the group presentation of V B n replacing ρ i by α i and adding relations of type α i σ i+1 σ i = σ i+1 σ i α i+1 , i = 1, 2, . . . , n -2 which are related to F 1 moves.

Notice that if we add to relations of V B n the relations related to F 2 moves:

ρ i+1 σ i σ i+1 = σ i σ i+1 ρ i , i = 1, 2, . . . , n -2,
we get a group, W B n , which is isomorphic to W B n : this isomorphism is given by the map ι n : W B n → W B n that sends ρ i in α i and σ i in σ -1 i .

Braids as automorphisms of free groups and generalizations

As remarked by Artin, the braid group B n may be represented as a subgroup of Aut(F n ) by associating to any generator σ i , for i = 1, 2, . . . , n -1, of B n the following automorphism of F n :

σ i :    x i -→ x i x i+1 x -1 i , x i+1 -→ x i , x l -→ x l , l = i, i + 1.
Artin proved a stronger result (see for instance [START_REF] Hansen | Braids and coverings[END_REF]Theorem 5.1]), by giving a characterization of braids as automorphisms of free groups. He proved that any automorphism β of Aut(F n )1 corresponds to an element of B n if and only if β satisfies the following conditions:

i) β(x i ) = a -1 i x π(i) a i , 1 ≤ i ≤ n, ii) β(x 1 x 2 . . . x n ) = x 1 x 2 . . . x n ,
where π ∈ S n and a i ∈ F n .

The group of conjugating automorphisms C n consists of automorphisms satisfying the first condition. In [START_REF] Fenn | The braid-permutation group[END_REF] it was proved that W B n is isomorphic to C n and therefore the group W B n can be also considered as a subgroup of Aut(F n ). More precisely the generators σ 1 , . . . , σ n-1 of W B n correspond to previous automorphisms of F n while any generator α i , for i = 1, 2, . . . , n -1 is associated to the following automorphism of F n :

α i :    x i -→ x i+1 x i+1 -→ x i , x l -→ x l , l = i, i + 1.
Remark 1. As noticed by Kamada [START_REF] Kauffman | Virtual knot theory[END_REF], the above representation of W B n as conjugating automorphisms and the fact W B n is a quotient of V B n imply that the σ i 's generate the braid group B n in V B n .

On the other hand the construction of an embedding of V B n into Aut(F m ) for some m remains an open problem.

Theorem 2. [2]

There is a representation ψ of V B n in Aut(F n+1 ), F n+1 = x 1 , x 2 , . . . , x n , y which is defined by the following actions on the generators of V B n :

ψ(σ i ) :        x i -→ x i x i+1 x -1 i , x i+1 -→ x i , x l -→ x l , l = i, i + 1; y -→ y, ψ(ρ i ) :        x i -→ y x i+1 y -1 , x i+1 -→ y -1 x i y, x l -→ x l , l = i, i + 1, y -→ y, for all i = 1, 2, . . . , n -1.
This representation was independently considered in [START_REF] Manturov | On the recognition of virtual braids[END_REF]. Remark that the group W B n can be considered as a quotient of ψ(V B n ): in fact a straightforward verification shows that: Proposition 3. Let q n : V B n → W B n be the projection defined by q n (σ i ) = σ i and q n (ρ i ) = α i for i = 1, . . . , n. Let F n+1 = x 1 , x 2 , . . . , x n , y and F n = x 1 , x 2 , . . . , x n . The projection

p n : F n+1 → F n+1 / y F n induces a map p # n : ψ(V B n ) → W B n such that p # n • ψ = q n .
The faithfulness of the representation given in Theorem 2 is evident for n = 2 since in this case V B 2 C 2 Z * Z 2 . In fact, from the defining relations it follows that V B 2 C 2 and if we consider composition of ψ with p 2 :

F 3 = x 1 , x 2 , y → F 2 = x 1 , x 2 we get the representation of C 2 by automorphisms of F 2 .
For n > 2 we do not know if above representation is faithful: since ψ(V B n ) ⊆ W B n+1 the faithfulness of ψ for any n would imply that virtual braid groups can be considered as subgroup of welded groups, whose structure and applications in finite type invariants theory is much more advanced (see for instance [START_REF] Bar Natan | Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne[END_REF][START_REF] Berceanu | Universal representations of braid and braid-permutation groups[END_REF]).

We remark also that a quite tedious computation shows that image by ψ of the Kishino braid :

Kb = σ 2 σ 1 ρ 2 σ -1 1 σ -1 2 ρ 1 σ -1 2 σ -1 1 ρ 2 σ 1 σ 2 σ 2 σ 1 ρ 2 σ -1 1 σ -1 2 ρ 1 σ -1 2 σ -1 1 ρ 2 σ 1 σ 2 is non trivial while its Alexander invariant is trivial [7]. Notice that Kb = σ -1 2 b 2 1 σ 2 where b 1 = σ 2 2 σ 1 ρ 2 σ -1 1 σ -1 2 ρ 1 σ -1 2 σ -1 1 ρ 2 σ 1 .

Groups of virtual links

In the classical case the group of a link L is defined as the fundamental group π 1 (S 3 \N (L)) where N (L) is a tubular neighborhood of the link in S 3 . To find a group presentation of this group we can use Wirtinger method as follows.

One can consider the oriented diagram of the link as the union of oriented arcs in the plane. Define a base point for π 1 (S 3 \ N (L)) and associate to any arc a loop starting from the base point, which goes straight to the chosen arc, encircles it with linking number +1 and returns straight to the base point. Let us consider the loops a i , a j , a k around three arcs in a crossing of the diagram as in Figure 5: One can easily verify that in the first case the loop a j is homotopic to a k a i a -1 k and in the second case the loop a j is homotopic to a -1 k a i a k . The group π 1 (S 3 \ N (L)) admits the following presentation: Let D be an oriented diagram of a a link L and A 1 , . . . , A n be the arcs determined by D. The group π 1 (S 3 \ N (L)), admits the following group presentation: Generators: {a 1 , . . . a n }, where a j is the loop associated to the arc A j . Relations: To each crossing corresponds a relation as follows

a j a k = a k a i if a i , a j , a k meet in a crossing like in case a) of figure 5; a k a j = a i a k if a i , a j , a k meet in a crossing like in case b) of figure 5.
We recall that this presentation is usually called upper Wirtinger presentation while the lower Wirtinger presentation is obtained applying Wirtinger method to the diagram where all crossings are reversed; these presentations are generally different but the corresponding groups are evidently isomorphic because of their geometrical meaning.

Another way to obtain a group presentation for π 1 (S 3 \N (L)) is to consider a braid β ∈ B n such that its Alexander closure is isotopic to L; therefore the group π 1 (S 3 \ N (L)) admits the presentation:

π 1 (S 3 \ N (L)) = x 1 , x 2 , . . . , x n || x i = β(x i ), i = 1, . . . , n ,
where we consider β as an automorphism of F n (this a consequence of van Kampen's Theorem, see for instance [START_REF] Wada | Group invariants of links[END_REF]).

Given a virtual link vL, according to [START_REF] Kauffman | Virtual knot theory[END_REF] the group of the virtual knot vL, denoted with G K,v (vL), is the group obtained extending the Wirtinger method to virtual diagrams, forgetting all virtual crossings. This notion of group of a virtual knot (or link) does not seem satisfactory: for instance if vT is the virtual trefoil knot with two classical crossings and one virtual crossing and U is unknot then G K,v (vT ) G K,v (U ) Z, although that vT is not equivalent to U . In addition, as noted Goussarov-Polyak-Viro [START_REF] Goussarov | Finite-type invariants of classical and virtual knots[END_REF], the upper Wirtinger group of a virtual knot is not necessary isomorphic to the corresponding lower Wirtinger group.

We introduce another notion of group G v (vL) of a virtual link vL. Let vL = β v be a closed virtual braid, where

β v ∈ V B n 2 . Define G v (vL) = x 1 , x 2 , . . . , x n , y || x i = ψ(β v )(x i ), i = 1, . . . , n .
We will consider the action from left to right and for simplicity of notation we will write

x i β v instead of ψ(β v )(x i ).
Notation. In the following we use the notations

[a, b] = a -1 b -1 ab and a b = b -1 ab.
The following Theorem was announced in [START_REF] Bardakov | Virtual and welded links and their invariants[END_REF].

Theorem 4. The group G v (vL) is an invariant of the virtual link vL.

Proof. According to [START_REF] Kamada | Invariants of virtual braids and a remark on left stabilisations and virtual exchange moves[END_REF] two virtual braids have equivalent closures as virtual links if and only if they are related by a finite sequence of the following moves: 1) a braid move (which is a move corresponding to a defining relation of the virtual braid group),

2) a conjugation in the virtual braid group,

3) a right stabilization of positive, negative or virtual type, and its inverse operation, 4) a right/left virtual exchange move.

Here a right stabilization of positive, negative or virtual type is a replacement of b ∈ V B n by bσ n , bσ -1 n or bρ n ∈ V B n+1 , respectively, a right virtual exchange move is a replacement

b 1 σ -1 n b 2 σ n ←→ b 1 ρ n b 2 ρ n ∈ V B n+1 and a left virtual exchange move is a replacement s(b 1 )σ -1 1 s(b 2 )σ 1 ←→ s(b 1 )ρ 1 s(b 2 )ρ 1 ∈ V B n+1 , where b 1 , b 2 ∈ V B n and s : V B n -→ V B n+1 is the shift map i.e. s(σ i ) = σ i+1 .
We have to check that under all moves 1) -4), the group

G v (vL) does not change. Let vL = β v where β v ∈ V B n . 1) If β v ∈ V B n is another braid such that β v = β v in V B n , then ψ(β v ) = ψ(β v ) since ψ is a homomorphism. Hence G( β v ) = G( β v
) and the first move does not change the group of virtual link.

2) Evidently it is enough to consider only conjugations by the generators of V B n . Let

G 1 = G( β v ) = x 1 , x 2 , . . . , x n , y || x i = x i β v , i = 1, 2, . . . , n and G 2 = G( σ k β v σ -1 k ) = x 1 , x 2 , . . . , x n , y || x i = x i (σ k β v σ -1 k ), i = 1, 2, . . . , n ,
where k ∈ {1, 2, . . . , n -1}. To prove that G 2 G 1 we rewrite the defining relations of G 2 in the form

x i σ k = x i (σ k β v ), i = 1, 2, . . . , n. If i = k, k -1 then this relation is equivalent to x i = x i β v since x i σ k = x i . But it is a relation in G 1 .
Hence, we have to consider only two relations:

x k σ k = x k (σ k β v ), x k+1 σ k = x k+1 (σ k β v ).
By the definition of ψ these relations are equivalent to

x k x k+1 x -1 k = (x k x k+1 x -1 k )β v , x k = x k β v . The second relation is a relation from G 1 .
Rewrite the first relation:

x k x k+1 x -1 k = (x k β v )(x k+1 β v )(x -1 k β v
) and using the second relation we get

x k+1 = x k+1 β v ,
which is a relation from G 1 . Hence we proved that any relation from G 1 is true in G 2 and analogously one can prove that any relation from G 2 is true in G 1 .

Consider the conjugation by element ρ k . In this case we have

G 2 = G( ρ k β v ρ k ) = x 1 , x 2 , . . . , x n , y || x i = x i (ρ k β v ρ k ), i = 1, 2, . . . , n .
Rewrite the relations of G 2 in the form

x i ρ k = x i (ρ k β v ), i = 1, 2, . . . , n. If i = k, k -1 then we have x i = x i β v , since x i ρ k = x i But it is a relation in G 1 .
Hence, we have to consider only two relations:

x k ρ k = x k (ρ k β v ), x k+1 ρ k = x k+1 (ρ k β v ).
By the definition of ψ these relations are equivalent to

yx k+1 y -1 = (yx k+1 y -1 )β v , y -1 x k y = (y -1 x k y)β v or yx k+1 y -1 = y(x k+1 β v )y -1 , y -1 x k y = y -1 (x k β v )y.
These relations are equivalent to

x k+1 = x k+1 β v , x k = x k β v
which are relations from G 1 . Hence we proved that the set of relations from G 2 is equivalent to the set of relations from G 1 .

3) Consider the move from a braid

β v = β v (σ 1 , σ 2 , . . . , σ n-1 , ρ 1 , ρ 2 , . . . , ρ n-1 ) ∈ V B n to the braid β v σ -1 n ∈ V B n+1 .
We have two groups:

G 1 = G( β v ) = x 1 , x 2 , . . . , x n , y || x i = x i β v , i = 1, 2, . . . , n and
G 2 = G( β v σ -1 n ) = x 1 , x 2 , . . . , x n , x n+1 , y || x i = x i (β v σ -1 n ), i = 1, 2
, . . . , n + 1 and we need to prove that they are isomorphic.

Rewrite the relations of G 2 in the form

x i σ n = x i β v , i = 1, 2, . . . , n + 1.
If i = 1, 2, . . . , n -1 then we have

x i = x i β v , which is a relation in G 1 .
Hence, we have to consider only two relations:

x n σ n = x n β v , x n+1 σ n = x n+1 β v ,
these relations are equivalent to

x n x n+1 x -1 n = x n β v , x n = x n+1 .
Using the second relation rewrite the first in the form

x n = x n β v ,
which is a relation from G 1 . Also we can remove x n+1 from the set of generators of G 2 . Hence we proved that the set of relations from G 2 is equivalent to the set of relations from G 1 .

The move from a braid β v ∈ V B n to the braid β v σ n is similar.

Consider the move from a braid β v ∈ V B n to the braid β v ρ n ∈ V B n+1 . We have two groups:

G 1 = G( β v ) = x 1 , x 2 , . . . , x n , y || x i = x i β v , i = 1, 2, . . . , n and
G 2 = G( β v ρ n ) = x 1 , x 2 , . . . , x n , x n+1 , y || x i = x i (β v ρ n ), i = 1, 2, . . . , n + 1 .
For i = n and i = n + 1 we have the following relations in G 2 :

x n ρ n = x n β v , x n+1 ρ n = x n+1 β v ,
which are equivalent to

yx n+1 y -1 = x n β v , y -1 x n y = x n+1 .
Rewrite the second relation in the form x n = yx n+1 y -1 and substituting in the first relation we have

x n = x n β v ,
which is a relation from G 1 . Also we can remove x n+1 from the set of generators of G 2 . Hence we proved that the set of relations from G 2 is equivalent to the set of relations from G 1 .

4) Finally, consider the exchange move

3 b 1 σ -1 n b -1 2 σ n ←→ b 1 ρ n b -1 2 ρ n , b 1 , b 2 ∈ V B n .
We have two groups:

G 1 = G( b 1 σ -1 n b -1 2 σ n ) = x 1 , x 2 , . . . , x n+1 , y || x i = x i (b 1 σ -1 n b -1 2 σ n ), i = 1, 2, . . . , n + 1
and

G 2 = G( b 1 ρ n b -1 2 ρ n ) = x 1 , x 2 , . . . , x n+1 , y || x i = x i (b 1 ρ n b -1 2 ρ n ), i = 1, 2, . . . , n + 1
and we need to prove that they are isomorphic.

Rewrite the defining relations from G 1 in the form

x i (σ -1 n b 2 ) = x i (b 1 σ -1 n ), i = 1, 2, . . . , n + 1,
and defining relations from G 2 in the form

x i (ρ n b 2 ) = x i (b 1 ρ n ), i = 1, 2, . . . , n + 1.
Let b 1 and b 2 are the following automorphisms

b 1 :                                    x 1 -→ x a 1 π(1) , x 2 -→ x a 2 π(2) , ................... x n -→ x an π(n) , x n+1 -→ x n+1 , y -→ y, b 2 :                                    x 1 -→ x c 1 τ (1) , x 2 -→ x c 2 τ (2) , ................... x n -→ x cn τ (n) , x n+1 -→ x n+1 , y -→ y,
, where π, τ ∈ S n and a i , c i ∈ x 1 , x 2 , . . . , x n , y . Then the automorphism σ -1 n b 2 has the form

σ -1 n b 2 :                                            x 1 -→ x c 1 τ (1) , x 2 -→ x c 2 τ (2) , ................... x n-1 -→ x c n-1 τ (n-1) , x n -→ x n+1 , x n+1 -→ x -1 n+1 x cn τ (n) x n+1 , y -→ y,
and the automorphism b 1 σ -1 n has the form

b 1 σ -1 n :                                            x 1 -→ (x π(1) σ -1 n ) a 1 σ -1 n , x 2 -→ (x π(2) σ -1 n ) a 2 σ -1 n ,
................................

x n-1 -→ (x π(n-1) σ -1 n ) a n-1 σ -1 n , x n -→ (x π(n) σ -1 n ) anσ -1 n , x n+1 -→ x -1 n+1 x n x n+1 , y -→ y.
Hence the first group has the following presentation

G 1 = x 1 , x 2 , . . . , x n+1 , y || x c 1 τ (1) = (x π(1) σ -1 n ) a 1 σ -1 n , x c 2 τ (2) = (x π(2) σ -1 n ) a 2 σ -1 n , . . . , x c n-1 τ (n-1) = (x π(n-1) σ -1 n ) a n-1 σ -1 n , x n+1 = (x π(n) σ -1 n ) anσ -1 n , x cn τ (n) = x n .
Analogously, construct the presentation for G 2 . Calculate the automorphism

ρ n b 2 :                                            x 1 -→ x c 1 τ (1) , x 2 -→ x c 2 τ (2) , .................... x n-1 -→ x c n-1 τ (n-1) , x n -→ yx n+1 y -1 , x n+1 -→ y -1 x cn τ (n) y, y -→ y,
and the automorphism b 1 ρ n has the form

b 1 ρ n :                                          x 1 -→ (x π(1) ρ n ) a 1 ρn , x 2 -→ (x π(2) ρ n ) a
x n-1 -→ (x π(n-1) ρ n ) a n-1 ρn , x n -→ (x π(n) ρ n ) anρn , x n+1 -→ y -1 x n y, y -→ y.
Hence the second group has the following presentation

G 2 = x 1 , x 2 , . . . , x n+1 , y || x c 1 τ (1) = (x π(1) ρ n ) a 1 ρn , x c 2 τ (2) = (x π(2) ρ n ) a 2 ρn , . . . , x c n-1 τ (n-1) = (x π(n-1) ρ n ) a n-1 ρn , yx n+1 y -1 = (x π(n) ρ n ) anρn , x cn τ (n) = x n . Compare G 1 and G 2 : since a i = a i (x 1 , x 2 , . . . , x n , y), let us denote a i = a i σ -1 n = a i (x 1 , x 2 , . . . , x n-1 , x n+1 , y) and a i = a i ρ n = a i (x 1 , x 2 , . . . , x n-1 , yx n+1 y -1 , y). Then G 1 = x 1 , x 2 , . . . , x n+1 , y || x c 1 τ (1) = (x π(1) σ -1 n ) a 1 , x c 2 τ (2) = (x π(2) σ -1 n ) a 2 , . . . , x c n-1 τ (n-1) = (x π(n-1) σ -1 n ) a n-1 , x n+1 = (x π(n) σ -1 n ) a n , x cn τ (n) = x n . and G 2 = x 1 , x 2 , . . . , x n+1 , y || x c 1 τ (1) = (x π(1) ρ n ) a 1 , x c 2 τ (2) = (x π(2) ρ n ) a 2 , . . . , x c n-1 τ (n-1) = (x π(n-1) ρ n ) a n-1 , yx n+1 y -1 = (x π(n) ρ n ) a n , x cn τ (n) = x n .
Denote by z n+1 = yx n+1 y -1 ; the group G 2 has therefore the following presentation

G 2 = x 1 , x 2 , . . . , x n , z n+1 , y || x c 1 τ (1) = (x π(1) ρ n ) a 1 , x c 2 τ (2) = (x π(2) ρ n ) a 2 , . . . , x c n-1 τ (n-1) = (x π(n-1) ρ n ) a n-1 , z n+1 = (x π(n) ρ n ) a n , x cn τ (n) = x n , where a i = a i σ -1 n = a i (x 1 , x 2 , .
. . , x n-1 , z n+1 , y). We will assume that n = π(1) (other cases consider analogously). Then our groups have presentations:

G 1 = x 1 , x 2 , . . . , x n+1 , y || x c 1 τ (1) = x a 1 n+1 , x c 2 τ (2) = (x π(2) ) a 2 , . . . , x c n-1 τ (n-1) = (x π(n-1) ) a n-1 , x n+1 = (x π(n) ) a n , x cn τ (n) = x n , G 2 = x 1 , x 2 , . . . , x n , z n+1 , y || x c 1 τ (1) = (z n+1 ) a 1 , x c 2 τ (2) = (x π(2) ) a 2 , . . . , x c n-1 τ (n-1) = (x π(n-1) ) a n-1 , z n+1 = (x π(n) ) a n , x cn τ (n) =
x n . and therefore they are isomorphic.

Example 0. For the unknot U we have

G v (U ) = x, y F 2 .
Example 1. For the virtual trefoil vT we have that vT = σ 2 1 ρ 1 and then

G v (vT ) = x, y || x (y x y -2 x y) = (y x y -2 x y) x F 2 .
Hence, we obtain a new proof of the fact that vT a non-trivial virtual knot.

Example 2. The group G v (K) is not a complete invariant for virtual knots. Let c = ρ 1 σ 1 σ 2 σ 1 ρ 1 σ -1 1 σ -1 2 σ -1 1 ∈ V B 3
; the closure c is equivalent to the Kishino knot (see Figure 6). The Kishino knot is a non trivial virtual knot [START_REF] Elhamdadi | Virtual knot invariants from group biquandles and their cocycles[END_REF] with trivial Jones polynomial and trivial fundamental group (G K,v ( c) = Z). For this knot we have that G v ( c) = F 2 . In fact, it is not difficult to prove that c defines the following automorphism of F 4 ψ(c) :

                   x 1 -→ y 2 x -1 3 x 2 x 3 y -2 x 3 y 2 x -1 3 x -1 2 x 3 y -2 , x 2 -→ x -1 3 x 2 x 3 y -2 x 3 yx -1 3 x -1 2 x 1 x 2 x 3 y -1 x -1 3 y 2 x -1 3 x -1 2 x 3 , x 3 -→ yx -1 3 x 2 x 3 y -1 , y -→ y,
and therefore the image of c as automorphism of F 4 is non trivial: nevertheless we have that

G v ( c) = x 1 , x 2 , x 3 , y || x 1 = y 2 x -1 3 x 2 x 3 y -2 x 3 y 2 x -1 3 x -1 2 x 3 y -2 , x 2 = x -1 3 x 2 x 3 y -2 x 3 yx -1 3 x -1 2 x 1 x 2 x 3 y -1 x -1 3 y 2 x -1 3 x -1 2 x 3 , x 3 = yx -1 3 x 2 x 3 y -1 .
Using the first relation we can remove x 1 and using the third relation we can remove x 2 . We get

G v ( c) = x, y || xy -1 xyx -1 = xy -1 xyx -1 F 2 , where x = x 3 Example 3. Let b 1 = b -1 2 = σ 1 ρ 1 σ 1 and b = b 1 ρ 2 b 2 ρ 2 .
It is easy to check that for this virtual braid its group

G v ( b) = x 1 , x 2 , y || yx 1 y -1 = x 2
is an HNN-extension of the free group x 1 , x 2 with cyclic associated subgroups. Therefore the closure of b is a non trivial virtual link.

The following Propositions establish the relations between the different notions of groups of (virtual) knots. Proposition 5. Let K be a classical knot then

G v (K) = Z * π 1 (S 3 \ N (K)), Z = y .
Proof. As previously recalled, if β ∈ B n is a braid such that its Alexander closure is isotopic to K, the group π 1 (S 3 \ N (K)) admits the presentation x 1 , x 2 , . . . , x n || x i = β(x i ), i = 1, . . . , n .

The claim follows therefore from the remark that the representation ψ of V B n in Aut(F n+1 ) restraint to B n coincides with the usual Artin representation composed with the natural inclusion ι :

Aut(F n ) → Aut(F n+1 ).
The Proposition below will be proved at the end of Section 5. Proposition 6. Let vK be a virtual knot.

(

) The group G v (vK)/ y is isomorphic to G K,v (vK); (2) The abelianization of G v (vK) is isomorphic to Z 2 . 1
We recall that a group G is residually finite if for any nontrivial element g ∈ G there exists a finite-index subgroup of G which does not contain g.

According to [START_REF] Hempel | Residual Finiteness for 3-manifolds[END_REF] every knot group of a classical knot is residually finite, while the virtual knot groups defined by Kauffman does not need to have this property (see [START_REF] Silver | Virtual knot groups, Knots in Hellas '98[END_REF]). Proposition 7. Let K be a classical knot then G v (K) is residually finite.

Proof. The free product of two residually finite groups is residually finite [START_REF] Gruenberg | Residual properties of infinite soluble groups[END_REF].

We do not know whether G v (vK) is residually finite for any virtual knot vK. Remark 8. We can modify Wirtinger method to virtual links using previous representation of V B n . Let us now consider each virtual crossing as the common endpoint of four different arcs and mark the obtained arcs with labels x 1 , . . . , x m and add an element y of this set. Now consider the group G y (vK) generated by elements x 1 , . . . , x m , y under the usual Wirtinger relations for classical crossings plus the following relations for virtual crossing: the labeling of arcs x k , x l , x i , x j meeting in a virtual crossing as in Figure 7 respect the relations x i = y -1 x l y and x j = yx k y -1 . One can easily check that G y (vK) is actually invariant under virtual and mixed Reidemeister moves and hence is an invariant for virtual knots. Arguments in Proposition 11 can be therefore easily adapted to this case to prove that G y (vK) is isomorphic G v (vK). Remark 9. Using the "Wirtinger like" labeling proposed in previous Remark it is also possible to extend the notion of group system to G v (vK). We recall that the group system of a classical knot K is given by the knot group, a meridian and its corresponding longitude: in the case of G v (vK) we can call meridian the generator corresponding to any arc. The longitude corresponding to this meridian is defined as follows: we go along the diagram starting from this arc (say a i ) and we write a -1 k when passing under a k as in Figure 5 a) and a k when passing under a k as in Figure 5 b). On the other hand if we encounter a virtual crossing according Figure 7 we write y when we pass from x l to x i and we write y -1 when we pass from x k to x j . Finally we write a -m i where m is the length (the sum of exponents) of the word that we wrote following the diagram. It is easy to verify that such an element belongs to the commutator subgroup of G v (vK) and that meridian and longitude are well defined under generalized Reidemeister moves.

Groups of welded links

As in the case of virtual links, Wirtinger method can be naturally adapted also to welded links: it suffices to check that also the forbidden relation F 1 is preserved by Wirtinger labeling. Given a welded link wL we can therefore define the group of the welded link wL, G K,w (wL), as the group obtained extending the Wirtinger method to welded diagrams, forgetting all welded crossings. This fact has been already remarked: see for instance [START_REF] Winter | The classification of spun torus knots[END_REF], where the notion of group system is extended to welded knot diagrams.

Notice that the forbidden move F 2 is not preserved by above method and then that the lower Wirtinger presentation does not extend to welded link diagrams.

On the other hand, considering welded links as closure of welded braids we can deduce as in the virtual case another possible definition of group of a welded link. Let wL = β w be the closure of the welded braid

β w ∈ W B n . Define G w (wL) = x 1 , x 2 , . . . , x n || x i = β w (x i ), i = 1, . . . , n .
Theorem 10. The group G w (wL) is an invariant of the virtual link wL.

Proof. We recall that two welded braids have equivalent closures as welded links if and only if they are related by a finite sequence of the following moves [START_REF] Kamada | Invariants of virtual braids and a remark on left stabilisations and virtual exchange moves[END_REF]:

1) a braid move (which is a move corresponding to a defining relation of the welded braid group),

2) a conjugation in the welded braid group, 3) a right stabilization of positive, negative or welded type, and its inverse operation, We have to check that under all moves 1) -3), the group G w (wL) does not change: the move 1) is evident and for the moves 2) and 3) we can repeat verbatim the proof of Theorem 4.

Proposition 11. Let wK be a welded knot. The groups G w (wK) and G K,w (wK) are isomorphic.

Proof. Any generator of W B n acts trivially on generators of F n except a pair of generators: more precisely we have respectively that

x i • σ i = x i x i+1 x -1 i := u + (x i , x i+1 ) , x i+1 • σ i = x i := v + (x i , x i+1 ) , x j • σ i = x j j = i, i + 1 , x i • σ -1 i = x i+1 := u -(x i , x i+1 ) , x i+1 • σ -1 i = x -1 i+1 x i x i+1 := v -(x i , x i+1 ) , x j • σ -1 i = x j j = i, i + 1 , x i • α i = x i+1 := u • (x i , x i+1 ) , x i+1 • α i = x i := v • (x i , x i+1 ) , x j • α i = x j j = i, i + 1 .
Let β w a welded braid with closure equivalent to wK. Let us regard to the diagram representing the closure of β w , β w , as a directed graph and denote the edges of β w , the generators of G K,w (wK) by labels x 1 , . . . , x m : according to Wirtinger method recalled in Section 4, for each crossing of β w (see Figure 7) we have the following relations:

(1) If the crossing is positive

x i = x l and x j = x -1 l x k x l ; (2) If the crossing is negative x i = x k x l x -1
k and x j = x k where the word u + (x k , x l ) and v + (x k , x l ) are the words defined above;

(3) If the crossing is welded x i = x l and x j = x k are the words defined above. Now let us remark that we have that

x l = u -(x k , x l ) , x -1 l x k x l = v -(x k , x l ), x k x l x -1 k = u + (x k , x l ), x k = v + (x k , x l ), x l = u • (x k , x l ) and x k = v • (x k , x l
) and let us recall that the action of welded braids is from left to right (β 1 β 2 (x i ) denotes ((x i )β 1 )β 2 ). Therefore if we label x i (for i = 1, . . . , n) the arcs on the top of the braid β w , the arcs on the bottom will be labelled by β -1 w (x i ) (for i = 1, . . . , n). Since we are considering the closure of β w , we identify labels on the bottom with corresponding labels on the top and we deduce that a possible presentation of G K,w (wK) is : • with the usual overpasses/underpasses information;

G K,w (wK) = x 1 , x 2 , . . . , x n || x i = β -1 w (x i ), i =
• as "singular" points. Clearly D will represent a virtual knot diagram if we allow virtual local moves or respectively a welded knot diagram if we allow welded local moves.

In the following we will write G v (D) and G K,v (D) when we consider D as a virtual knot diagram, while we will set G w (D) and G K,w (D) when we see D as a welded knot diagram. The following Proposition is a consequence of the fact that Wirtinger labeling is preserved by F 1 moves.

Proposition 12. Let D be a diagram as above. Then G K,v (D) = G K,w (D).
Remark also that if D 1 and D 2 are equivalent as welded diagrams then G K,v (D 1 ) = G K,v (D 2 ) even if D 1 and D 2 are not equivalent as virtual diagrams.

Proof of Proposition 6. Let β v be a virtual braid with closure equivalent to vK. Let F n+1 = x 1 , x 2 , . . . , x n , y and F n = x 1 , x 2 , . . . , x n . As in recalled in Proposition 3, the projection

p n : F n+1 → F n+1 / y F n induces a surjective map p # n : ψ(V B n ) → W B n and therefore that G v (vK)/ y is isomorphic to G w (wK) = x 1 , x 2 , . . . , x n || x i = β w (x i ), i = 1, . . . , n , where β w = p # (ψ(β v )).
The first claim is therefore a straightforward consequence of Proposition 12.

The second claim of the Proposition follows easily from the fact that any generator of type x i is a conjugated of x π(i) where π ∈ S n is of order n. Therefore in the abelianization all generators of type x i have the same image.

From Propositions 6 and 11 therefore it follows that: Proposition 13. Let D be a diagram as above. Then G v (D)/ y = G w (D).

Wada groups for virtual and welded links

In [START_REF] Wada | Group invariants of links[END_REF] Wada found several representations of B n in Aut(F n ) which, by the usual braid closure, provide group invariants of links. These representation are of the following special form: any generator (and therefore its inverse) of B n acts trivially on generators of F n except a pair of generators:

x i • σ i = u + (x i , x i+1 ) , x i+1 • σ i = v + (x i , x i+1 ) , x j • σ i = x j j = i, i + 1 . x i • σ -1 i = u -(x i , x i+1 ) , x i+1 • σ -1 i = v -(x i , x i+1 ) , x j • σ -1 i = x j j = i, i + 1 .
where u and v are now words in the generators a, b, with a, b F 2 . In [START_REF] Wada | Group invariants of links[END_REF] Wada found four families of representations providing group invariants of links (they are types 4 -7 in Wada's paper):

• Type 1:

u + (x i , x i+1 ) = x h i x i+1 x -h i and v + (x i , x i+1 ) = x i ; • Type 2: u + (x i , x i+1 ) = x i x -1 i+1 x i and v + (x i , x i+1 ) = x i ; • Type 3: u + (x i , x i+1 ) = x i x i+1 x i and v + (x i , x i+1 ) = x -1 i ; • Type 4: u + (x i , x i+1 ) = x 2
i x i+1 and v + (x i , x i+1 ) = x -1 i+1 x -1 i x i+1 . As in the case of Artin representation we can ask if these representations extend to welded braids providing group invariants for welded links. More precisely, let χ k : G W Bn → Aut(F n ) (for k = 1, . . . , 4) be the set map from the set of generators G W Bn := {σ 1 , . . . , σ n-1 , α 1 , . . . , α n-1 } of W B n to Aut(F n ) which associates to any generators σ i the Wada representation of type k and to any generators α i the usual automorphism

x i • α i = x i+1 := u • (x i , x i+1 ) x i+1 • α i = x i := v • (x i , x i+1 )
x j • α i = x j j = i, i + 1 . Proof. For k = 1, 2 the proof is a straightforward verification that relations of W B n hold in Aut(F n ): for k = 3, 4 it suffices to remark that relation of type α i σ i+1 σ i = σ i+1 σ i α i+1 is not preserved.

When k = 1, 2 we can therefore define for any β w ∈ W B n the group W k (β w ) = x 1 , x 2 , . . . , x n || x i = (χ k (β w ))(x i ), i = 1, . . . , n that we will call Wada group of type k for β w ∈ W B n .

Theorem 15. The Wada group W k is a link invariant for k = 1, 2.

Proof. One can repeat almost verbatim the arguments from [START_REF] Wada | Group invariants of links[END_REF] for classical braids. Notice that W k (β w ) is the group of co-invariants of β w , i.e. the maximal quotient of F n on which χ k (β w ) acts trivially. Since the group of co-invariants is invariant up to conjugation by an automorphism, we deduce that conjugated welded braids have isomorphic Wada groups. To prove the statement it is therefore sufficient to verify that W k (β w ) = x 1 , x 2 , . . . , x n || x i = (χ k (β w ))(x i ), i = 1, . . . , n and W stab k (β w ) = x 1 , x 2 , . . . , x n x n+1 || x i = (χ k (β w σ n ))(x i ), i = 1, . . . , n + 1 are isomorphic: this a straightforward computation similar to the case 3) in Theorem 4, the key point being that (χ k (β w σ n ))(x n+1 ) = x n for k = 1, 2 (see also Section 2 of [START_REF] Wada | Group invariants of links[END_REF]).

In [START_REF] Elhamdadi | Virtual knot invariants from group biquandles and their cocycles[END_REF] Wada representations of type 2, 3, 4 have been extended to group invariants for virtual links using a Wirtinger like presentation of virtual link diagrams: contrarily to the classical case these groups are not necessarily isomorphic.

Analogously it would be interesting to understand the geometrical meaning of Wada groups of welded links. In this perspective, Proposition 16 shows that Wada representations of type 1 and 2 are not equivalent.

We will say that two representations ω 1 : W B n → Aut(F n ) and ω 2 : W B n → Aut(F n ) are equivalent, if there exist automorphisms φ ∈ Aut(F n ) and µ : W B n → W B n such that φ -1 ω 1 (β w ) φ = ω 2 (µ(β w )), for any β w ∈ G W Bn . Proposition 16. Wada representations of type 1 and 2 are not equivalent.

Proof. The proof is the same as in Proposition A.1 of [START_REF] Crisp | Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups[END_REF]: the claim follows from considering the induced action on H 1 (F n ). Under Wada representations of type 1 a welded braid has evidently finite order as automorphism of H 1 (F n ) while we have that χ 2 (σ t i )[x i ] = (t+1)[x i ]t[x 2 ] for all t ∈ N, where [u] denote the equivalence class in H 1 (F n ) of an element u ∈ F n .

Figure 1 .

 1 Figure 1. Classical Reidemeister moves

Figure 2 .Figure 3 .

 23 Figure 2. Virtual Reidemeister moves

Figure 4 .

 4 Figure 4. Forbidden moves of type F 1 (on the left) and type F 2 (on the right)

Figure 5 .

 5 Figure 5. Arcs around two types of crossings

2

 2 ρn , ................................

Figure 6 .

 6 Figure 6. The Kishino knot: usual diagram and as the closure of a virtual braid.

  1, . . . , n . and therefore G K,w (wK) is clearly isomorphic to G w (wK).

Figure 7 .

 7 Figure 7. Wirtinger-like labelling

Proposition 14 .

 14 The set map χ k :G W Bn → Aut(F n ) (for k = 1, . . . , 4) induces a homomorphism χ k : W B n → Aut(F n ) if and only if k = 1, 2.

In the following we will consider the action of (classical, virtual or welded) braids from left to right and β 1 β

(x i ) will denote ((x i )β 1 )β 2 .

The indices v and w are given to precise when we are considering virtual or welded braids

In this formula we take b -1 2 instead b 2 for convenience.