Matthieu Calvez

DUAL GARSIDE STRUCTURE AND REDUCIBILITY OF BRAIDS

HAL is

Introduction

Braid groups are both mapping class groups and Garside groups. The links between these two features of braid groups seem to be very deep and their investigation is currently the goal of several works. In particular the Garside-theoretic approach to the problem of deciding algorithmically the Nielsen-Thurston type of a given braid turns out to be very fruitful [START_REF] Benardete | A combinatorial approach to reducibility of mapping classes[END_REF], [START_REF] Lee | A Garside theoretic approach to the reducibility problem in braid groups[END_REF], [START_REF] González-Meneses | Reducible braids and Garside theory[END_REF], [START_REF] González-Meneses | On reduction curves and Garside properties of braids[END_REF], [START_REF] Calvez | Fast algorithmic Nielsen-Thurston classification of four-strand braids[END_REF].

The n-strand braid group is naturally identified with the mapping class group of the n-times punctured disk D n . Braids induce a (right) action on the set of isotopy classes of simple closed curves in D n : considering the isotopy class [C] of a simple closed curve C and an n-braid x, the isotopy class of simple closed curves resulting from the action of x on [C] will be denoted by [C] x . The simple closed curves we shall consider in the present paper will be nondegenerate, that is surrounding more than one and less than n punctures.

Let us assume for the moment that D n is parametrized as the disk with diameter [0, n + 1] in C and points 1, 2, . . . , n removed. In this setting, a curve C is said to be standard, or round, if it is isotopic to a geometric circle in D n . That is, if the punctures enclosed by C are consecutive. An isotopy class [C] is said to be standard, or round, if some (hence every) representative is round. Round curves are particularly useful for decomposing a reducible braid into its corresponding components [START_REF] González-Meneses | On reduction curves and Garside properties of braids[END_REF]. As every reducible braid has a conjugate which preserves a family of round curves [START_REF] Benardete | A combinatorial approach to reducibility of mapping classes[END_REF], searching for such a conjugate becomes a possible strategy both for determining whether a braid is reducible and for finding its geometric components. Benardete, Gutierrez and Nitecki [START_REF] Benardete | A combinatorial approach to reducibility of mapping classes[END_REF] explain how to determine whether such a conjugate exists, and also how to find it, thanks to the classical Garside structure of braid groups.

Braid groups are the main examples of Garside groups [START_REF] Dehornoy | Gaussian groups and Garside groups, two generalisations of Artin groups[END_REF]. This means that they admit a lattice structure, and a special element denoted ∆, satifying some properties first discovered by Garside in [START_REF] Garside | The braid group and other groups[END_REF]. We will refer to this as the classical Garside structure of the braid group. Using this structure, one can define the left normal form of a braid x, which is a unique decomposition of the form x = ∆ p x 1 • • • x r (see [START_REF] Epstein | Word processing in groups[END_REF], [START_REF] Elrifai | Algorithms for positive braids[END_REF]) in which the factors belong to the set of the so-called simple elements.

2010 Mathematics Subject Classification. 20F36. This work was done while the author was staying at the University of Seville. The visit was partially supported by a grant for international mobility by the Université Européenne de Bretagne and by the spanish research group FQM-218 (Junta de Andalucía).

The result by Benardete, Gutierrez and Nitecki, which relates round curves and left normal forms, is the following: Theorem 1 ( [START_REF] Benardete | A combinatorial approach to reducibility of mapping classes[END_REF], [START_REF] Lee | A Garside theoretic approach to the reducibility problem in braid groups[END_REF]). Let C be a standard curve in D n . Let x = ∆ p x 1 • • • x r be a braid in (classical) left normal form. If [C] x is standard, then [C] ∆ p x1•••xm is standard for m = 1, . . . , r.

Thanks to this result, and applying special conjugations called cyclings and decyclings, it is possible to determine whether a braid has a conjugate which preserves a family of round curves, hence it is possible to know whether a braid is reducible, with the aid of the Garside structure of the braid group [START_REF] Benardete | A combinatorial approach to reducibility of mapping classes[END_REF].

The method mentioned in the previous paragraph needs to compute a big subset of the conjugacy class of a braid. This has been improved in [START_REF] González-Meneses | Reducible braids and Garside theory[END_REF], avoiding the computation of such a big subset, at the cost of enlarging the set of standard curves to include round and also almost-round curves. This raises the question whether the notion of round curves, and the use of the classical Garside structure, are the best choices for this kind of techniques.

There is another well-known Garside structure of the braid group, discovered by Birman, Ko and Lee [START_REF] Birman | A new approach to the word problem in the braid groups[END_REF], which is known as the dual Garside structure. With respect to this structure, the left normal form of a braid is a unique decomposition x = δ p x 1 • • • x r , where δ is the special (Garside) element, and the factors are simple, with respect to the dual structure.

In this paper we prove the analogue of Theorem 1 in the dual setting. We remark that the proofs of Theorem 1 given in [START_REF] Benardete | A combinatorial approach to reducibility of mapping classes[END_REF] and [START_REF] Lee | A Garside theoretic approach to the reducibility problem in braid groups[END_REF] cannot be adapted in a natural way to the dual Garside structure of the braid group. For that reason we give a new proof of Theorem 1, in the classical setting, which can be naturally adapted to the dual setting. To this purpose we also introduce a natural notion of standard curve related to the dual Garside structure of the braid group. Namely, as round curves determine standard parabolic subgroups of the braid group, with the Artin strucure, the standard curves in the dual setting will be those determining standard parabolic subgroups, with the dual structure.

It is important to mention that a generalization of Theorem 1 to Artin-Tits groups of spherical type, with the classical Garside structure, is given in [START_REF] Godelle | Parabolic subgroups of Garside groups II: ribbons[END_REF]. The powerful algebraic methods used in [START_REF] Godelle | Parabolic subgroups of Garside groups II: ribbons[END_REF] are based in the theory of Garside categories and seem to allow further generalization of Theorem 1 to dual Garside structures [16], although this does not appear in the literature. Here we present simple, geometric proofs, for the particular but important cases of the two well known structures of the braid groups.

Our proof of Theorem 1 can be sketched as follows. We show that if C is round and We assume the usual Artin generator σ i , for i = 1, . . . , n -1, to be the counterclockwise half Dehn twist along the segment [i, i + 1]. We will see that if [C] ∆ p x1 is not round, then a portion of (a suitable representative of) [C] ∆ p x1 crosses the real line in the way shown in Figure 1, for some i < j < k. Moreover, x 1 σ j is simple, meaning that the strands of x 1 ending at j and j + 1 do not cross. We then prove by recursion that, as x xm must cross the real line as in Figure 1, for some i m < j m < k m , where x m σ jm is simple. This implies in particular that [C] This proof can be adapted to the dual setting as follows. First, in order to work with the dual Garside structure of the braid group, it is more convenient to parametrize D n as the unit disk in C with set of punctures { 1 2 e

[C] ∆ p x1 is not round, then [C] ∆ p x1

-2ikπ n

, k = 1, . . . , n}. Throughout the paper the puncture 1 2 e -2ikπ n will be denoted k for brevity, and the disk D n with this parametrization will be denoted D * n . We remark that, if one defines standard curves as isotopy classes of geometric circles, as above, then the analogue of Theorem 1 is not true in the dual setting (see Example 12). Hence we need a different definition for standard curves, adapted to the dual Garside structure: Definition 2 (See also Definition 13). A simple closed curve in D * n is called standard if it is isotopic to a curve which can be expressed, in polar coordinates, as a function ρ = ρ(θ), for θ ∈ [0, 2π[. See Figure 2(a). An isotopy class [C] is said to be standard if some (hence every) representative is standard.

The main result of this paper is the following: The proof of this result parallels the one of Theorem 1 and its simplicity allows to outline it right here. First recall that the usual generators of the dual Garside structure are the braids a i,j = a j,i corresponding to the counterclockwise half Dehn twist along the chord segment joining the punctures i and j, for each pair 1 i, j n, with i ̸ = j.

Theorem 3. Let C be a standard curve in D * n . Let x = δ p x 1 • • • x r be a braid in dual left normal form. If [C] x is standard, then [C] δ p x1•••xm is standard for m = 1, . . . , r.
We will see that if [C] δ p x1 is not standard, then a portion of (a suitable representative of) [C] δ p x1 crosses the circle of radius 1 2 in the way shown in Figure 2(b), for some 1 i, j n. Moreover, x 1 a i,j is simple. We then prove by recursion that, as x 1 • • • x r is in left normal form, the above properties must hold for [C] δ p x1•••xm , for m = 1, . . . , r. That is, some portion of (a suitable representative of) [C] δ p x1•••xm must cross the circle of radius 1 2 as in Figure 2(b), for some 1 i m , j m n, where x m a im,jm is simple. This implies in particular that [C] δ p x1•••xr is not standard, showing Theorem 3.

The following two sections contain the detailed proofs of Theorems 1 and 3, respectively. The last section contains a new algorithm for solving the reducibility problem in braid groups, based on Theorem 3.

Acknowledgements. Thanks to Eddy Godelle for his receptiveness to questions about his papers on parabolic subgroups of Garside groups. The author also wishes to thank Juan González-Meneses for countless helpful comments on earlier versions of the paper.

The classical Artin-Garside case

This section deals with the classical case. Throughout the section, we thus assume D n to be parametrized as the disk with diameter [0, n + 1] in C and with the points 1, 2, . . . , n removed.

In order to make rigorous the idea expressed in Figure 1, we associate to each isotopy class of simple closed curves [C] in D n a unique reduced word W ([C]) which we shall define in Subsection 2.1. The word W ([C]) allows to describe carefully the action of positive braids on [C]; this will be achieved in Subsection 2.2. Subsection 2.3 will be devoted to the proof of Theorem 1.

2.1.

From curves to words. We will always assume that the curves under consideration are nondegenerate, simple and closed, so unless otherwise stated, the word "curve" alone will mean "nondegenerate simple closed curve". Let C be a curve in D n , and suppose that it has a finite number of crossings with the real line, which are all transverse. We shall associate to C a word W (C) that we now define.

Our strategy to define the word W (C) already mostly appears in [START_REF] Fenn | Ordering the braid groups[END_REF], Appendix A. We proceed as follows. Choose a point * of C which lies on the real line as well as an orientation for C. Running along C following the chosen orientation, starting and ending at * , determines a word in the alphabet X = { , , 0, . . . , n} as follows: each arc through the upper half plane contributes a letter to the word, each arc through the lower half plane, a letter , and each intersection with the segment ]i, i + 1[ yields the letter i.

The number corresponding to * can be chosen to be either at the beginning or at the end of the word. The word obtained in this way will be denoted W (C) and we call it the word associated to C. Choosing another point * or putting the letter determined by the intersection point * at the beginning or at the end of W (C) corresponds to a cyclic permutation of the letters in W (C), whereas choosing the reverse orientation of C yields the reverse of W (C). Hence, words associated to curves are to be considered up to cyclic permutation of their letters and up to reverse. Notice that two curves related by an isotopy of D n fixing the real diameter setwise have the same associated word.

We say that the word associated to C is reduced if it does not contain any subword of the form i i or i i. We say that a curve C is reduced if its associated word W (C) is reduced.

Notice that reduced curves are exactly those which do not bound any bigon (see [START_REF] Fenn | Ordering the braid groups[END_REF]) together with the real line. According to [START_REF] Fenn | Ordering the braid groups[END_REF], every curve C is isotopic to a reduced one C red , which is unique up to isotopy of D n fixing the real diameter setwise. We thus may finally define, for each isotopy class of curves [C] in D n , its associated reduced word as

W ([C]) = W (C red ).

The action of positive braids. Let B +

n be the submonoid of B n generated by σ 1 , . . . , σ n-1 , called the monoid of positive braids. Instead of the usual Artin generators, it will be convenient to work with the following bigger generating set of B + n :

{Σ p,k = σ p-1 . . . σ p-k , 1 k < p n}.
According to our previous convention that σ i is a counterclockwise half-Dehn twist along the segment [i, i + 1], the braid Σ p,k corresponds to a move of the puncture numbered p through the upper half plane up to the position p -k while the punctures p -1, p -2, . . . , p -k are shifted one position to the right.

Given the isotopy class of a curve C and a positive braid x we are going to describe which transformations have to be performed on the word W = W ([C]) in order to obtain the word W ([C] x ). To this purpose we first focus on the case x = Σ p,k .

Since the generators we are considering are mainly moves of punctures through the upper half plane, we assume that the action of the braids Σ p,k mainly modifies the upper arcs occuring in C red (from which new lower arcs can arise) while the lower arcs are only modified by translating their endpoints. This will be described by the following formulae, which are also depicted in figure 4.

We define, for i < j:

(i j) Σ p,k =        i j if [p -k, p[ ∩ {i, j} = ∅ (F 1) (i + 1) (j + 1) if [p -k, p[ ∩ {i + 1, j} = {i + 1, j} (F 2) (i + 1) (p -k) (p -k -1) j if [p -k, p[ ∩ {i, j} = {i} (F 3) i (p -k -1) (p -k) (j + 1) if [p -k, p[ ∩ {i + 1, j} = {j} (F 4)
We can define in the same way (j i) Σ p,k . It suffices to take the reverse of the above formulae (the picture is exactly the same than in Figure 4). Notice that W is not necessarily reduced, so that W ([C] Σ p,k ) and W are possibly not the same. The following explains how to turn W into the reduced word W ([C] Σ p,k ):

Lemma 5. Let [C] be an isotopy class of curves and W = W ([C]

). Let 1 k < p n. Let W be as above, and let W Σ p,k be the word obtained from W by removing all instances of subwords of the form p p . Then

W Σ p,k = W ([C] Σ p,k ).
Proof. We observe that the formulae defining (i j) Σ p,k do not contain any subword of the form c c nor c c. Thus the possible instance of such a subword in W necessarily arises from the transformation of a lower arc. Notice that a lower arc c d can only be transformed into an arc k l, with k ∈ {c, c + 1} and l ∈ {d, d + 1}. Hence the latter arc forms a bigon with the horizontal axis if and only if c + 1 = d, k = c + 1 and l = d (up to reverse we may suppose that c < d), in which case W contains the subword d d. By the formulae defining (i j) Σ p,k , this happens if and only if d = p, that is c d = (p -1) p. In particular we have shown that no subword of the form c c can arise in W . We now claim that removing the subwords p p is sufficient in order to turn W into a reduced word; that is, every sequence p p in W is a subsequence of a larger one, of the form a p p b, with a ̸ = b. Let h (p -1) p l be a subword of W . As C is a simple curve, h, l must be in one of the following three cases (see Figure 5):

1) h < p -1 < p < l, 2) l h < p -1 < p, 3) p -1 < p < l h.
We shall show that in the three cases, the subword a p p b of W yielded by the action of Σ p,k satisfies

a ∈ {p -k, . . . , p -1} and b ∈ {0, . . . , p -k -1} ∪ {p + 1, . . . , n},
and thus a ̸ = b, as claimed. In the first two cases the arc (h p) will yield a = h+1 or a = p-k, satisfying a ∈ {p-k, . . . , p-1}. In the third case we will also obtain a = p -k. On the other hand, in the first and third cases we

have b = l (hence b > p); in the second case, either b = l if p -k l + 1 or b = p -k -1 otherwise (and b < p -k).
This achieves the proof of the lemma.

We thus can associate to each isotopy class of curves [C] and each braid Σ p,k the word W ([C]) Σ p,k defined thanks to the above construction. We are now able to define, for each isotopy class of curves [C], the image of its associated reduced word W = W ([C]) under the action of some positive braid x. Indeed, if x is expressed as a product x = ∏ r m=1 Σ pm,km , then by Lemma 5, the inductive formula

W x = (W ∏ r-1 m=1 Σ pm,km ) Σ pr ,kr
defines a word on X which is the reduced word associated to [C] x (hence it does not depend on the chosen decomposition of x in terms of braids Σ p,k ). This can be written

W ([C] x ) = W ([C]) x .
2.3. Proof of Theorem 1. By abuse of notation, instead of speaking about isotopy classes of simple closed curves, we will simply speak about curves, meaning that we are considering the reduced representatives. Consequently, the letter C will denote the reduced representative of the isotopy class of the curve C and its associated reduced word will be denoted W (C).

We introduce a class of curves which is larger than the class of round curves: Definition 6. A curve C will be called almost-round if the word W (C) (up to cyclic permutation of its letters and reversing) can be written as W (C) = w 1 w 2 , where the arcs in w 1 are oriented from left to right whereas those in w 2 are oriented from right to left. (See Figure 6). The reduced words associated to almost-round curves satisfy the following necessary condition:

Lemma 7.
Let C be an almost-round, not round curve. Then W (C) (up to cyclic permutation and reversing) must contain a subword of the form i j l for some 0 i < j < l n.

Proof. Before proving Theorem 1, we introduce some more notation. We will say that some 0 < j < n is a bending point for (or bends) a curve C if the reduced word W (C) admits (up to cyclic permutation of its letters and up to reverse) a subword of the form i j l, for some 0 i < j < l n. Lemma 7 thus asserts that an almost-round, not round curve admits at least one bending point. Given a simple braid s, and a bending point j for some curve C, j will be said to be compatible with s if the strands j and j + 1 of s do not cross in s.

The following is the key for Theorem 1: Lemma 8. Let s 1 , s 2 be two simple braids such that s 1 • s 2 is in left normal form, and let C be a curve. Let j be a bending point for C, compatible with s 1 . Then there exist some bending point j ′ for C s1 , which is compatible with s 2 .

In order to prove this lemma, we recall that simple braids are those positive braids in which any pair of strands crosses at most once (see [START_REF] Elrifai | Algorithms for positive braids[END_REF]). We also recall the following well-known fact (see [START_REF] De La Harpe | An invitation to Coxeter groups[END_REF]): any simple braid s can be decomposed in a unique manner as a product

s = n ∏ p=2 Σ p,kp ,
where 0 k p < p and Σ p,0 = 1. This allows us to see s as a sequence of moves of the strands numbered from 2 to n (in this order), each of them getting k p positions to the left (the number k p depending on s). Notice that the pth strand of s does not cross (in s) any of the strands being to the left of the position p -k p .

Proof of Lemma 8. As s 1 • s 2 is in left normal form, it is sufficient to show that there exist some bending point j ′ for C s1 such that the strands numbered j ′ and j ′ + 1 at the end of s 1 do not cross in s 1 ([7]).

Let

s 1 = ( j-1 ∏ p=2 Σ p,kp ) • Σ j,kj • Σ j+1,kj+1 • ( n ∏ p=j+2 Σ p,kp )
be the decomposition of s 1 in terms of the braids Σ p,k . Acting on C by s 1 following the above factorization, we will be able to find at each step a bending point ι for the resulting curve such that the strands at respective positions ι and ι + 1 at the end of this step have not yet crossed in s 1 .

The first factor ∏ j-1 p=2 Σ p,kp involves moves of punctures which are to the left of j. Hence, due to Formulae (F 1) and (F 3) and Lemma 5, j is still a bending point for the resulting curve. Moreover strands ending at j and j + 1 have not crossed in this first factor.

The factor Σ j,kj moves the jth puncture to the left; once again due to Formulae (F 1) and (F 3) and Lemma 5, the resulting reduced word admits a subword i ′ j (with i ′ < j); j is still a bending point for the corresponding curve. Notice that the former jth puncture lies (now at position j -k j ) under the arc i ′ j and that strands ending at positions j and j + 1 have not crossed.

Then, the factor Σ j+1,kj+1 moves the j + 1st puncture to the left. As j was a bending point for C, compatible with s 1 , the strands which started at positions j and j + 1 do not cross in s 1 , hence this movement of the j + 1st puncture cannot exceed the position j -k j : it ends at the diameter of the upper arc i ′ j and by Formula (F 4) and Lemma 5, the position ι j+1 := j -k j+1 is now a bending point for the resulting curve (see Figure 7). Moreover by construction the strands ending at ι j+1 and ι j+1 + 1 have not crossed in s 1 .

Figure 7. The action of the braid Σ j+1,kj+1 ; as the strands represented as bold crosses cannot cross, the point j + 1 is "blocked" in its move to the left by the position j -k j .

Finally, observe that for j + 2 q n, the movement of the qth puncture yields some bending point ι q for the resulting curve: this is ι q = ι q-1 + 1 if this movement ends to the left of ι q-1 and ι q = ι q-1 otherwise. Thus j ′ := ι n is a bending point of C s1 and strands numbered j ′ and j ′ + 1 at the end of s 1 do not cross in s 1 by construction. This shows that the bending point j ′ of C s1 is compatible with s 2 .

2

We can now achieve the proof of Theorem 1. Recall that we consider a braid x = ∆ p x 1 . . . x r in left normal form which sends a round curve C to another round curve C ′ = C x . Since the braid ∆ corresponds to a rotation of D n , the curve C ∆ t is round for all integers t and we may suppose, up to multiplication by ∆ -p , that the left normal form of x is just x 1 . . . x r . In order to prove Theorem 1, it is sufficient to prove that the curve C x1 is round, since once this fact is proven, we are given a braid x 2 . . . x r in left normal form whose action transforms the round curve C x1 into the round curve C ′ , and the result follows by induction on the number of factors in the left normal form of x. We shall give a proof by contradiction, assuming that the curve C x1 is not round. However it is almost round, as x 1 is a simple braid, see [START_REF] González-Meneses | Reducible braids and Garside theory[END_REF].

According to Lemma 7, there must exist a bending point j 1 for C x1 . Notice that punctures which lie above the curve C x1 come from strands which started to the right of C at the beginning of x 1 as well as punctures which lie below C x1 come from strands which started to the left of C. By Lemma 7, the puncture j 1 lies either below the curve C x1 or is enclosed by it whereas the puncture j 1 + 1 lies either above, or is enclosed by C x1 . Moreover, at most one of them is enclosed by C x1 . This implies that the strands in positions j 1 and j 1 + 1 at the end of x 1 do not cross in x 1 . In other words, j 1 is a bending point for C x1 compatible with x 2 (as

x 1 • x 2 is in left normal form).
Now, by induction on m = 1, . . . , r-1 and by Lemma 8, there exist j 2 , . . . , j r-1 such that j m bends the curve C x1...xm and is compatible with x m+1 , for all m = 1, . . . , r -1. The proof of Lemma 8 also ensures that C x1...xr has a bending point j r .

In particular we have shown that the curve C x1...xr = C x is not round, provided that C x1 is not round. This is a contradiction; therefore C x1 is round and Theorem 1 is shown.

The dual case, proof of Theorem 3

In this section we shall be interested in the dual (or BKL) Garside structure of the braid group [START_REF] Birman | A new approach to the word problem in the braid groups[END_REF]. We thus adopt the parametrization D * n of the n-times punctured disk defined in the Introduction. We shall associate to each isotopy class of simple closed curves [C] in D * n , a unique reduced word W ([C]) to be defined in Subsection 3.1. Reduced words will be a tool for describing the action of dual positive braids on simple closed curves in D * n in Subsection 3.2. Finally, Theorem 3 will be proved in Subsection 3.3.

From curves in D *

n to reduced words. As in the previous section, all the curves we are considering are nondegenerate, simple and closed. We shall speak about "curves" when we really mean "nondegenerate simple closed curves". Notation : the circle of radius 1 2 in D * n centered at the origin will be denoted by Γ. Given i, j ∈ {1, . . . , n}, the move of the puncture i clockwise along Γ up to the position j describes an arc of Γ which we will denote by (i, j). The arc (i, i) is just the puncture i.

Let C be a curve in D * n , and suppose that it has a finite number of crossings with the circle Γ, which are all transverse. We shall associate to C a word W (C) that we now define.

Choose a point * of C which lies on the circle Γ and choose an orientation for C. Running along C following the chosen orientation, starting and ending at * , determines a word in the alphabet Y = { , , , , , 1, . . . , n} as follows. Each arc through the inner component of D * n -Γ contributes a letter to the word. Each intersection of C with the arc (i, i + 1) of Γ contributes a letter i. Finally arcs of C through the outer component of D * n -Γ contribute a letter , ( respectively) if they are oriented clockwise (counterclockwise, respectively); except those as in Figure 8 (a) (thus having their endpoints in the same arc (i, i + 1) of Γ) which contribute a letter or , in a a natural way according to their orientation.

The number corresponding to the intersection point * can be chosen to be either at the beginning or at the end of the word. The word obtained in this way will be denoted by W (C) and we call it the word associated to C. Choosing another point * or putting the letter determined by * at the beginning or at the end of W (C) corresponds to a cyclic permutation of the letters in W (C), whereas choosing the reverse orientation of C yields the reverse of W (C), exchanging with each other the letters and ( and , respectively). Hence, words associated to curves are to be considered up to cyclic permutation of their letters and up to reverse, exchanging the orientation of outer arcs. Notice that two curves related by an isotopy of D * n fixing the circle Γ setwise have the same associated word.

We say that the word associated to C is reduced if it does not contain any subword of the form i i, i i or i i. We say that a curve C is reduced if its associated word W (C) is reduced. Notice that reduced curves are exactly those which do not bound any bigon (see [START_REF] Fenn | Ordering the braid groups[END_REF]) together with the circle Γ. According to [START_REF] Fenn | Ordering the braid groups[END_REF], every curve C is isotopic to a reduced one C red , which is unique up to isotopy of D * n fixing the circle Γ setwise. We finally define, for each isotopy class of curves [C] in D * n , its associated reduced word as

W ([C]) = W (C red ).
3.2. The action of dual positive braids. Let B + * n be the submonoid of B n generated by the braids a i,j , 1 i, j n, i ̸ = j, called the monoid of dual positive braids. We shall use a bigger generating set, namely

P =    a i1,i2 a i2,i3 . . . a ir-2,ir-1 a ir-1,ir ,
2 r n, i 1 , . . . , i r all distinct and placed in this order, following the circle Γ clockwise from i 1 to i r    .

The elements of P are naturally called polygons according to their geometric representation in D * n ; and they correspond to a counterclockwise rotation of a neighborhood of the convex polygon in D * n whose vertices are i 1 , i 2 , . . . , i r , following the cyclic permutation [i r , i r-1 , . . . , i 2 , i 1 ].

The following are well-known (see [START_REF] Birman | A new approach to the word problem in the braid groups[END_REF]):

• Let P 1 , P 2 ∈ P, seen as polygons in B * n . If their respective convex hulls are disjoint, then P 1 P 2 = P 2 P 1 . In this situation, we will say that P 1 and P 2 are disjoint.

• Let i 1 , . . . , i r be r (2 r n) punctures placed in this order following Γ clockwise from i 1 up to i r . Then all the braid words obtained as the concatenation of r -1 consecutive letters taken from the sequence (a i1,i2 , a i2,i3 , . . . , a ir-2,ir-1 , a ir-1,ir , a ir,i1 ) in this order, up to cyclic permutation, are representatives of the same braid P , which is an element of P. Moreover, for each pair 1 d < e r, the letter a i d ,ie is a prefix of P .

Let us now consider the isotopy class of a curve C and its associated reduced word W = W ([C]). Given a dual positive braid x, we are going to describe which transformations have to be performed on the word W in order to obtain the word W ([C] x ). To this purpose we first focus on the case x = P ∈ P.

We assume that the action of P will mainly modify the inner arcs whereas the outer arcs are only modified by shifting their endpoints along the circle Γ.

We first observe that any inner arc i j separates the punctures into two disjoint subsets: one containing the punctures i + 1 and j, and the other containing the punctures j + 1 and i. We say that P is disjoint from the arc i j if all the vertices of P lie in only one of these sets. If it is so, we set (i j) P = i j.

Otherwise we say that P is transverse to the arc i j. In that case, let p i,j be the rightmost vertex of P lying in (j + 1, i); similarly let q i,j be the rightmost vertex of P lying in (i + 1, j). By abuse of notation, we will write i ∈ P to mean that i is a vertex of P . Observe that p i,j = i ⇔ i ∈ P and q i,j = j ⇔ j ∈ P . A priori, looking at the pictures, we would define:

(i j) P = i p i,j (p i,j -1) (q i,j -1) q i,j j (F ′ ) Formula (F ′ ) is depicted in Figure 9.

Figure 9. In this example P (in dotted lines) can be expressed as the product α 1 . . . α 4 ; α 1 α 2 acts trivially on the arc i j, α 3 yields the arc in the middle part, and the action of α 4 on the latter yields the arc in the right-hand side, which is (i j) P , the image under the action of P of the arc i j.

But notice that application of Formula (F ′ ) produces bigons with the circle Γ if either i ∈ P or j ∈ P . That is why we set:

(i j) P =        i p i,j
(p i,j -1) (q i,j -1) q i,j j if i / ∈ P and j / ∈ P (F ′ 0) i p i,j (p i,j -1) (j -1) if i / ∈ P and j ∈ P (F ′ 1) (i -1) (q i,j -1) q i,j j if i ∈ P and j / ∈ P (F ′ 2) (i -1) (j -1) if i ∈ P and j ∈ P (F ′ 3)

Later, we shall need the following:

Remark 10. The image of an inner arc i j under the action of a polygon P lies (up to deformation) in a neighborhood of the union of i j with P .

Let us replace each subword (i j) in W by the corresponding subword (i j) P as defined above. This transforms the outer arcs in W by shifting their endpoints along the circle Γ. Moreover, letters ( , respectively) need to be transformed into ( , respectively) if they correspond in W to an arc (c+1) c, where c+1 is shifted up to the position c (or if they correspond to an arc (c -1) c, where c is shifted up to the position c -1, respectively). See Figure 10 (a). Similarly, letters ( , respectively) need to be transformed into ( , respectively) if the endpoints of the arc do not coincide any more after the suitable translation. Let us denote by W the word on Y obtained in this way. Notice that W is not necessarily reduced, so that W ([C] P ) needs not to be the same as W . The following is the analogue of Lemma 5 in the dual setting: Lemma 11. Let [C] be an isotopy class of curves, and W = W ([C]). Let P ∈ P be a polygon. Let W be as above, and let W P be the word obtained from W by removing all instances of subwords of the form (p -1) (p -1) (or (p -1) (p -1) ) where p is a vertex of P whereas p -1 is not. Then we have W P = W ([C] P ).

Proof. We observe that formulae defining (i j) P do not contain any subword of the form c c nor c c (by we mean either or ). Thus the possible instance of such a subword in W necessarily arises from the transformation of an outer arc of W . According to the formulae above, the translations of punctures involved in such a transformation turn the extremities c and d of an outer arc into k ∈ {c, c -1} and l ∈ {d, d -1} respectively.

First notice that transformations of outer arcs of the form c c or c c cannot yield subwords of the form c c. Now, an arc c d of W (thus with c ̸ = d) will produce a bigon with the circle Γ only if the above k and l are the same and |c -d| = 1. There are two possibilities (up to reverse), shown in Figure 10. Finally, by the formulae defining (i j) P , a necessary condition for a transformation as in Figure 10 (b) to happen is that c + 1 is a vertex of P whereas c is not. We have shown in particular that no subword of the form c c can arise in W , and that the only subwords of the form c c which possibly arise are (p -1) (p -1), where p is a vertex of P whereas p -1 is not.

We now claim that removing all the instances of these subwords is sufficient in order to turn W into a reduced word; that is, every sequence (p -1) (p -1) in W is a subsequence of a larger one of the form a (p -1) (p -1) b, with a ̸ = b. Let r (p -1) p v be a subword of W to be transformed into the above one under the action of P (hence P is transverse to the arc p v). See Figure 11. p v and an edge of P in dashed line.

We have (since p -1 is not a vertex of P ) (r (p -1)

) P = { r (p -1) if P disjoint from r (p -1), . . . (q r,p-1 -1)
q r,p-1 (p -1) otherwise, and

(p v) P = { (p -1) (q p,v -1) if v ∈ P , (p -1) (q p,v -1) q p,v v if v / ∈ P .
Thus it remains to be proved that the values a ∈ {r, q r,p-1 } and b = q p,v -1 are distinct. Notice that by definition, q p,v lies in (p + 1, v) so that b = q p,v -1 lies in (p, v -1). On the other hand, as C red is simple and W is its associated reduced word, r ∈ (v, p -2) (see Figure 11). Finally by definition, q r,p-1 lies in (r + 1, p -1) ⊂ (v + 1, p -1). Hence in any case, a ∈ (v, p -1); this shows that a ̸ = b and achieves the proof of Lemma 11.

We thus can associate to each isotopy class of curves [C] and each braid P ∈ P the word W ([C]) P defined thanks to the above construction. We are now able to define, for each isotopy class of curves [C], the image of its associated reduced word W = W ([C]) under the action of some dual positive braid x. Indeed, if x is expressed as a product x = ∏ r m=1 P m (where each factor lies in P), then by Lemma 11, the inductive formula

W x = (W ∏ r-1 m=1 Pm ) Pr
defines a word on Y which is the reduced word associated to [C] x (hence it does not depend on the chosen decomposition of x in terms of braids in P). This can be written W (

[C] x ) = W ([C]) x .
3.3. Proof of Theorem 3. By abuse of notation, we will speak about curves instead of isotopy classes of curves, meaning that we are always considering the reduced representatives. Consequently the letter C will denote the reduced representative of the isotopy class of the curve C and its associated reduced word will be denoted by W (C).

We shall now prove the analogue of Theorem 1 in the dual setting. As mentioned in the introduction, the statement of Theorem 1, with round curves defined as circles surrounding a set of consecutive punctures is false in this setting, as shows the following example:

Example 12. Let n = 4. Consider the braid x = a 1,2 .a 1,4 which is in left normal form as written.

Figure 12 shows that roundness is not preserved after each factor of the left normal form if we define it as the property of being homotopic to a geometric circle surrounding a set of consecutive punctures.

12. The curve in the middle part fails to be homotopic to a circle surrounding a set of consecutive punctures, although it is the image, under the first factor of the left normal form of x, of a "round" curve which is sent by x to a "round" curve.

We then need to define a suitable class of curves, which will play the role played by round curves in the classical setting. It is the following: Definition 13. A curve C will be called standard if W (C) only admits letters in {1, . . . , n, , } (or in {1, . . . , n, , }, up to reverse).

Notice that this is equivalent to Definition 2, in the Introduction. It turns out that Theorem 1 holds in the dual setting, if we replace round curves by standard curves. This is the statement of Theorem 3, which we will now prove.

First, we recall from [START_REF] Birman | A new approach to the word problem in the braid groups[END_REF] the following facts: Claim 14. Every dual simple braid can be written in a unique manner (up to permutation of the factors) as a product of pairwise disjoint elements of P (hence a commutative product).

Therefore, by Remark 10, studying the action of dual simple braids on the arc i j of the curve C through the decomposition s = P 1 . . . P g can be done quite easily, since all inner arcs obtained by application of Formulae (F ′ ) for some P l are invariant under the action of P k , k ̸ = l. See Figure 13.

Lemma 15. [3]

1) If s is a dual simple braid and s = P 1 . . . P t is its decomposition into pairwise disjoint factors in P, then sa i,j is simple if and only if P u a i,j is simple for every u = 1, . . . , t. 2) If P ∈ P, then P a i,j is simple if and only if no word representative of P can be written with a letter a k,l such that k ∈ (j + 1, i) and l ∈ (i + 1, j).

Then, we introduce, as in the previous section, some further notation. Given a curve C in D * n , and an unordered pair i, j of punctures, we say that i, j is a bending pair for C if the reduced word W (C) contains a sequence of the form i j or i j (up to reverse). Also, given a dual simple element s, a bending pair i, j for a curve C will be said to be compatible with s if a i,j is not a prefix of s.

We are now able to state and prove the key-lemma, aiming to Theorem 3; it is the analogue of Lemma 8 in the dual setting: Lemma 16. Let s 1 , s 2 be two dual simple braids such that s 1 • s 2 is in dual left normal form. Let C be a curve and assume that C admits some bending pair i, j, which is compatible with s 1 . Then there exist a bending pair i ′ , j ′ for C s1 which is compatible with s 2 .

Proof. Consider the decomposition s 1 = P 1 . . . P g of s 1 into pairwise disjoint elements of P. Notice that, as s 1 • s 2 is in left normal form, a i ′ ,j ′ is not a prefix of s 2 whenever s 1 a i ′ ,j ′ is simple, thus it is sufficient to find a bending pair i ′ , j ′ for C s1 such that s 1 a i ′ ,j ′ is simple.

By hypothesis, a i,j is not a prefix of s. Thus, none of the polygons P 1 , . . . , P g has both i and j as vertices. First, the action of any polygon among P 1 , . . . , P g which is disjoint from the arc i j in W (C) results in a curve for which i, j is still a bending pair (see Lemma 11). Moreover, by Lemma 15, if P is such a polygon, then P a i,j is simple.

Figure 13. The braid s 1 , as a product of pairwise disjoint polygons, is depicted in dashed lines. We can see, from left to right, the action of this braid on the arc i j.

Then, the action of the polygons among P 1 , . . . , P g which are transverse to the arc i j can be studied as follows (see Figure 13). The involved polygons can be ordered by running along the arc i j starting at (i, i + 1): Q 1 , . . . , Q h . For t = 1, . . . , h, if p t is the rightmost vertex of Q t in (j + 1, i) and q t is the rightmost vertex of Q t in (i + 1, j), then by Formulae (F ′ ) and Lemma 11, under consideration are not periodic and the curves, simple, closed and nondegenerate, will be considered up to isotopy so that the term curve will be applied to the isotopy class of a curve.

From Theorem 3, it follows that if a braid preserves a family of standard curves, then its cyclic sliding [START_REF] Gebhardt | The cyclic sliding operation in Garside groups[END_REF] also preserves a family of standard curves (see Proposition 4.2. and Corollary 4.3. in [START_REF] González-Meneses | On reduction curves and Garside properties of braids[END_REF]). Therefore: Proposition 17. Let x be a reducible braid. Then there exist some y in SC BKL (x) (the set of sliding circuits of x [START_REF] Gebhardt | The cyclic sliding operation in Garside groups[END_REF], with respect to the dual structure) which preserves a family of (pairwise disjoint) standard curves.

The main result of this section asserts that this last condition, i.e. preserving a (non-empty) family of pairwise disjoint standard curves, is checkable in polynomial time. In fact, we shall prove: Theorem 18. There is an algorithm which decides whether a given n-strand braid x in dual left normal form x = δ p x 1 . . . x ℓ admits a standard invariant curve. Moreover this algorithm takes time O(ℓ • n 4 ).

Proof. Given a subset I 0 of {1, . . . , n} (of cardinality 2 #(I 0 ) n -1) and a braid x in dual left normal form x = δ p x 1 . . . x ℓ , the main task of the algorithm is to construct a bigger set S(I 0 , x) of punctures, which must be enclosed by the image under x of any standard curve C surrounding punctures in I 0 , provided C x is standard. This is achieved by the following, which is the key-result: Lemma 19. Suppose we are given a dual simple braid s, decomposed into pairwise disjoint polygons s = P 1 . . . P g , together with a proper subset I of {1, . . . , n} of cardinality at least 2, whose elements are enumerated p 1 , . . . , p k in this order (up to cyclic permutation) running along the circle Γ clockwise (we will put p k+1 = p 1 ). For each i = 1, . . . , k, consider all the polygons among P 1 , . . . , P g having at least one vertex, but not all, in (p i + 1, p i+1 -1) and which does not have p i as a vertex, and take for each of them its leftmost vertex in the arc (p i + 1, p i+1 -1). Let I ′ be the union of I with all punctures collected in the above way. If C is a standard curve surrounding the punctures in I (and possibly other punctures) such that C s is standard, then C surrounds the punctures in I ′ . Moreover, if I ′ = I, the standard curve whose set of inner punctures is exactly I is sent to a standard curve by s. Proof. First observe that the image of a standard curve under s is standard if and only if the image of this curve under each of the P i 's is standard (Remark 10). Now, for a polygon Q and a standard curve C oriented counterclockwise, according to Formulae (F ′ ), the following are equivalent:

• C Q is standard. • for each inner arc a b of C which is transverse to Q, the puncture b is a vertex of Q.
Let C be a standard curve (oriented counterclockwise) surrounding the punctures in I, such that C s is standard. The punctures in I ′ -I (that is the punctures added to I by the process of Lemma 19) cannot lie in the outer component of D * n -C. Indeed, the belonging of such a puncture to the outer component of D * n -C would yield some inner arc a b of C, transverse to a polygon in s not having b as a vertex; this would be in contradiction with the above remarks. See Figure 14. Now let C I be the standard curve surrounding exactly the punctures in I (oriented counterclockwise) and suppose that the process of Lemma 19 yields I ′ = I. Then, whenever a polygon in s and an inner arc a b of C I are transverse, the puncture b is a vertex of the involved polygon (otherwise some puncture in (b + 1, a) would be added to I by the process of Lemma 19). By the remark above, it follows that C s I is standard, as claimed.

The set I ′ of the above lemma depends only on I and s, and we will denote by S(I, s) the set ρ(s)(I ′ ) (where ρ is the natural morphism B n -→ S n ).

Lemma 19 says that if C is a standard curve surrounding the punctures in I and C s is standard, then C s must surround the punctures in S(I, s). Using Theorem 3 and an induction on the number of non-δ factors in the left normal form of x allows to construct the set S(I 0 , x).

First, we set S(I 0 , δ p x 1 ) = S(ρ(δ p )(I 0 ), x 1 ). Then we define, for i = 1, . . . , ℓ -1, S(I 0 , δ p x 1 . . . x i+1 ) = S(S(I 0 , δ p x 1 . . . x i ), x i+1 ).

Notice that the set S(I 0 , x) can be computed in time O(ℓ • n). Notice also that, in virtue of Lemma 19, the equality S(I, x) = I implies that the curve whose set of inner punctures is exactly I is x-invariant.

The last step in the proof of Theorem 18 is the following: Proof. The algorithm does the following:

• set I 0 = {a, b},

• for m = 1, . . . , n -2, compute the set I m = S(I m-1 , x) ∪ I m-1 .

Remark that I i-1 ⊂ I i for all i. If I n-2 = {1, . . . , n}, then the algorithm answers negatively; otherwise, the standard curve which surrounds exactly the punctures in I n-2 is x-invariant. Indeed, in the latter case, there must exist some k, 1 k < n -2, such that #(I k ) = #(I k+1 ), and therefore I k = I k+1 = I n-2 . This means that S(I k , x) = I k and therefore the standard curve whose set of inner punctures is exactly I n-2 (and thus contains a and b) is x-invariant. The complexity of the algorithm is O(ℓ • n 2 ), according to the above estimation about the computation of S(I 0 , x).

Iterating the above algorithm for each pair of points in {1, . . . , n} yields the algorithm in the statement of Theorem 18, since the number of pairs of points in {1, . . . , n} is n•(n-1)

2

, so that the complexity of the whole algorithm is O(l • n 4 ) as claimed.

We observe that this algorithm is the analogue of Theorem 2.9. in [START_REF] González-Meneses | Reducible braids and Garside theory[END_REF]. The result in [START_REF] González-Meneses | Reducible braids and Garside theory[END_REF] deals with the classical structure, but needs an additional hypothesis about the triviality of the "inner" braid; we believe that obtaining the same conclusion without this hypothesis could be an indication that the dual structure is more adapted for the kind of problems we are dealing with.

The algorithm deciding the Nielsen-Thurston type of a given braid x is as follows:

1. Test whether x is periodic [START_REF] Birman | Conjugacy in Garside groups III: Periodic braids[END_REF], and if it is so, return "periodic", and stop. 2. Compute the set of sliding circuits (for the dual structure) of x [START_REF] Gebhardt | Solving the conjugacy problem in Garside groups by cyclic sliding[END_REF]. 3. For each element y of SC BKL (x), for each k = 1, . . . , n 2 , apply the algorithm of Theorem 18 to the braid y k (a curve belonging to a family fixed by y must indeed be fixed by some power y k , with k = 1, . . . , n 2 ). 4. Stop whenever a positive answer is found, and return "reducible"; otherwise return "pseudo-Anosov".

The complexity of this algorithm is not bounded above by a polynomial in n and ℓ since the size of the set of sliding circuits is known to be exponential in general [START_REF] Gebhardt | The cyclic sliding operation in Garside groups[END_REF], [START_REF] Prasolov | Small braids having a big Ultra Summit Set[END_REF].

Figure 1 .

 1 Figure 1. Strands which arrive at positions j and j + 1 (depicted as crosses) have not crossed in x 1 .

Figure 2 .

 2 Figure 2. (a) shows a standard curve as in Definition 2. (b) A part of a nonstandard curve.

Example 4 .

 4 Let C be the curve depicted in Figure 3. If we fix * in the interval ]1, 2[ and the

Figure 3 .

 3 Figure 3. The curve C of Example 4.

Figure 4 .

 4 Figure 4. How the action of the braid Σ p,k does transform upper arcs? In dashed lines is represented the trace of the move of the puncture initially numbered p (up to the position p -k). In continuous lines the arc i j on the left hand side and its image (i j) Σ p,k on the right hand side.

Figure 5 .

 5 Figure 5. The possible cases of subwords h (p -1) p l appearing in W , with the distinct possible moves of the point p.

Figure 6 .

 6 Figure 6. The curve in Part (a) is almost-round since its associated reduced word can be written as W = w 1 w 2 = (1 4 6 9 10 11)( 8 6 4 3 ). The curve in Part (b) is not almost-round.

Figure 8 . 9 Example 9 .

 899 Figure 8. (a) An arc i i. (b) The curve of Example 9

Figure 10 .

 10 Figure 10. Transformation of an outer arc c d of W , with |c -d| = 1, into an outer arc of W having the same extremities (in dashed line is depicted an edge of P ). (a) The transformation c (c + 1) c c was already mentioned: no bigon is formed. (b) The only way (up to reverse) to get a bigon: c (c + 1) c c.

Figure 11 .

 11 Figure 11. The arc r (p -1) p v and an edge of P in dashed line.

Figure 14 .

 14 Figure 14. An example that illustrates Lemma 19: here the simple braid s is decomposed as s = P 1 . . . P 4 , and I is made of 4 points p 1 , p 2 , p 3 , p 4 . Then running along each arc (p i , p i+1 ) allows the construction of I ′ , which consists of adding punctures depicted as crosses to I.

Proposition 20 .

 20 Let a, b be any pair of punctures in {1, . . . , n}. There is an algorithm which decides whether a given n-braid x of length ℓ admits a standard invariant reduction curve surrounding the punctures a and b. Moreover this algorithm runs in time O(ℓ • n 2 ).

if h = 1 and j is not a vertex of Q 1 is a bending pair for the curve C s1 . Moreover, by Lemma 15, in any case the braid s 1 a i ′ ,j ′ is simple. It follows that i ′ , j ′ is a bending pair for the curve C s1 compatible with s 2 .

We now achieve the proof of Theorem 3. Recall that we consider a braid x = δ p x 1 . . . x r in dual left normal form which sends some standard curve C to another standard curve C ′ = C x . Since the braid δ corresponds to a rotation of D * n , and thus sends standard curves to standard curves, up to multiplication by a power of δ we may assume that x is a dual positive braid, whose left normal form is x 1 . . . x r . By a direct induction on the number of factors in the left normal form of x, it is sufficient to show that C x1 is standard. We shall give a proof by contradiction, assuming that C x1 is nonstandard. We will see (by induction on m = 1, . . . , r) that none of the curves C x1...xm for 1 m r can be standard, contradicting the fact that C x is standard.

Let x 1 = P 1 . . . , P g be the decomposition of x 1 into pairwise disjoints elements of P. On the other hand suppose that W (C) is written only with letters in { , , 1, . . . , n} (that is choose the counterclockwise orientation for C).

As C x1 is nonstandard, so must be C Pt for at least one of the P t 's (according to Remark 10). We may assume that C P1 is nonstandard. Therefore, there exist an arc i j in W (C) such that P 1 is transverse to i j and j is not a vertex of P 1 (hence W (C P1 ) has at least one letter , see Formulae (F ′ 0) and (F ′ 2)). Consider all the polygons among P 1 , . . . , P g which are transverse to the arc i j. By Remark 10, only these polygons witness the action of x 1 on i j. Running along the arc i j starting at (i, i + 1) allows us to order them in a natural way (see the proof of Lemma 16): Q 1 , . . . , Q h . Let q 1 be the rightmost vertex of Q 1 in (i + 1, j) (so that q 1 ̸ = j) and if h > 1 let p 2 be the rightmost vertex of Q 2 in (j + 1, i).

Then we set i 1 = q 1 and j 1 = p 2 if h > 1, j 1 = j otherwise. Formulae (F ′ ) and Lemma 11 now imply that W (C x1 ) contains the subword i 1 j 1 (or i 1 j 1 ). By construction, the braid x 1 a i1,j1 is simple according to Lemma 15.

In other words we saw that i 1 , j 1 is a bending pair for C x1 , and since x 1 • x 2 is in left normal form, this bending pair is compatible with x 2 . It follows by induction on m and Lemma 16 that one can find, for each m = 1, . . . , r -1, a bending pair i m , j m for the curve C x1...xm which is compatible with x m+1 . The existence of a bending pair i r , j r for the curve C x1...xr also follows from the proof of Lemma 16.

We proved in particular that the curve C x1...xr = C x is not standard. This is a contradiction which achieves the proof of Theorem 3.

Deciding the dynamical type of braids

The above results give rise to an algorithm for deciding the Nielsen-Thurston type of a given braid, in the spirit of [START_REF] Lee | A Garside theoretic approach to the reducibility problem in braid groups[END_REF], [START_REF] González-Meneses | Reducible braids and Garside theory[END_REF], using the dual structure. Thurston's classification Theorem [START_REF] Thurston | On the geometry and dynamics of diffeomorphisms of surfaces[END_REF] asserts that the elements of Mapping Class Groups of surfaces (and therefore, in particular, braids) split into three mutually exclusive types, according to their dynamical properties: periodic, reducible non-periodic and pseudo-Anosov. The reader is referred to [START_REF] Thurston | On the geometry and dynamics of diffeomorphisms of surfaces[END_REF], [START_REF] Fathi | Travaux de Thurston sur les surfaces[END_REF] for details and precise definitions. We restrict ourselves to recalling that reducible braids are those preserving a family of pairwise disjoint isotopy classes of nondegenerate simple closed curves in the n-times punctured disk. Periodic braids being easy to detect [START_REF] Birman | Conjugacy in Garside groups III: Periodic braids[END_REF], the main problem to be solved is to decide whether a non-periodic braid is pseudo-Anosov or reducible. In what follows, we will assume that the braids


O Gout

MD C Lebrun Frenay

MD P Labauge

PhD ; G E Lepage

MD P Clavelou

PhD ; S Allouche

MD Pedias * Group

Prior suggestive symptoms in one-third of patients consulting for a

"The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non exclusive for government employees) on a worldwide basis to the BMJ Publishing Group Ltd and its Licensees to permit this article (if accepted) to be published in the Journal of Neurology, Neurosurgery & Psychiatry editions and any other BMJPGL products to exploit all subsidiary rights, as set out in our licence (http://jnnp.bmjjournals.com/ifora/licence.pdf)." ABSTRACT Objective: To evaluate the prevalence of prior inflammatory events in patients consulting for a first inflammatory neurological event, and improve early diagnosis of multiple sclerosis.

Methods: During the initial visit, the neurologist gave patients a self-administered questionnaire containing 72 questions regarding previous symptoms lasting more than 24 hours. During the follow-up visit the neurologist validated the symptoms and collected information about the current attack.

Results:

The cohort included 178 patients (74% women, mean age (±SD) 33.7 (±10.1) years).

The main reason for the initial visit was visual disturbance and sensory troubles in limbs.

Mean (±SD) global EDSS score was 1.4 (±1.1), 46% of brains MRIs were positive according to Barkhof-Tintoré criteria, 41% had abnormal WBC count in CSF and 71% had IgG oligoclonal bands. Prior symptoms suggestive of demyelination were reported by 79 patients (44%), validated by the neurologist for 70% (55 patients) and identified only by the neurologist in four patients. Sequelae were observed in 14 patients with validated prior symptoms (26%). The self-administered questionnaire showed an overall sensitivity of 93% and specificity of 80% for identifying patients with prior symptoms suggestive of demyelination.

Conclusion:

A patient-administered questionnaire subsequently validated by the neurologist demonstrated that 33% of patients consulting for a first demyelinating event had prior symptoms suggestive of CNS demyelination that had gone unnoticed, and almost 70% had either sequelae of prior demyelination or McDonald criteria for dissemination in space. Such a questionnaire could be a useful tool for earlier diagnosis of multiple sclerosis.

INTRODUCTION

In multiple sclerosis (MS) the notion of first attack is still vague. Before the clinical event that leads to consultation in a hospital's Neurology department, there are often preexisting symptoms suggestive of MS such as fatigue, limb weakness, blurred vision, paresthesias, vertigo, and urinary disturbances [START_REF] Compston | Multiple sclerosis[END_REF]. These initial symptoms correspond to the Clinically Isolated Syndrome (CIS). In the study Optic Neuritis Treatment Trial (ONTT) [START_REF] Neuritis | The 5-year risk of MS after optic neuritis. Experience of the optic neuritis treatment trial[END_REF][START_REF] Neuritis | High-and Low-risk profiles for the development of multiple sclerosis within 10 years after optic neuritis: experience of the optic neuritis treatment trial[END_REF] antecedents such as those described above (mostly paresthesia) were identified in 45 of 149 (30%) patients who had abnormal cerebral Magnetic Resonance Imaging (MRI) and 12 of 202 (5.9%) patients with normal MRI. For all patients, whatever the MRI result, presence of such an antecedent was a risk factor for developing clinically definite MS within 5 years. Furthermore, according to McDonald's new criteria for MS diagnosis, [START_REF] Mcdonald | Recommended diagnostic criteria for multiple sclerosis: guidelines from the international panel on the diagnosis of multiple sclerosis[END_REF][START_REF] Polman | Diagnostic criteria for multiple sclerosis: 2005 revisions to the "McDonald Criteria[END_REF] patients who presented at the first attack with sequelae of a preexisting symptom suggestive of a demyelinating event (i.e., lasting at least for 24 hours), can be considered as having MS, as can patients who have had 2 attacks in addition to clinical evidence of one lesion and criteria for dissemination in space, which is demonstrated by MRI or at least 2T2 lesions and a positive cerebrospinal fluid (CSF). Therefore, the primary objective of the present study was to evaluate the prevalence of patients presenting with antecedents suggestive of demyelinating event(s) that went unnoticed among patients consulting or hospitalised in a Neurology department for a first inflammatory demyelinating event and to initiate the development of a specific screening questionnaire to accurately identify these suggestive antecedents.

METHODS

Patients

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Patients included in the study were male and female adults who had been hospitalised or seen in the Neurology outpatient clinics for a first inflammatory neurological event.

Study design

This was a prospective, multicenter, observational pilot study, conducted in 14 French reference hospital neurology departments specialising in MS. The objective was to collect data on neurological antecedents in patients hospitalised for a first demyelinating event (i.e., CIS) as well as data on environmental factors and family history that could affect the management and evolution of the disease. Patients were included in the study after being informed about the study by the neurologist. At the initial visit, the neurologist gave a selfadministered questionnaire to patients about antecedents suggestive of a first demyelinating event. Patients were invited to fill out the questionnaire at home and to return it at the next planned visit. Questions were about demographic data (age, sex, marital status), sociogeographic characteristics (occupation, educational level, residence area), and neurologically relevant medical and family history. In particular, the patient had to describe all previous symptoms lasting more than 24 hours that could be related to a CNS demyelination: fatigue, visual disturbance (diplopia, blurred vision, and ocular pain), sensations (tingling, burning, numbness, electrical shock in one or several limbs), clumsiness (hand clumsiness, impaired walking, and limb muscle weakness), bladder problems (urinary urgency or retention), loss of balance, vertigo, hearing disturbance, and facial paralysis. During the normal follow-up visit (usually one to two months after the initial visit), the patient returned the completed questionnaire to the neurologist, who then reviewed it with the patient. The neurologist then completed part of the questionnaire, indicating: onset of the current attack, supposed localisation of the neurological lesion, Expanded Disability Status Scale (EDSS) score, [START_REF] Kurtzke | Rating neurologic impairment in multiple sclerosis: an expanded disability status scale (EDSS)[END_REF] and signs and symptoms attesting to the antecedents reported by the patient. The neurologist also reported all available results of complementary tests concerning this current attack, including brain MRI, Barkhof-Tintoré criteria [START_REF] Barkhof | Comparison of MRI criteria at first presentation to predict conversion to clinically definite multiple sclerosis[END_REF][START_REF] Tintoré | Isolated demyelinating syndromes: comparison of different MR imaging criteria to predict conversion to clinically definite multiple sclerosis[END_REF] evaluation, and lumbar puncture. All brain MRI was performed on 1.5-Tesla scanner and IgG oligoclonal bands were detected by isoelectric focusing.

Statistical methods

Descriptive statistics were performed for all patients or events, and for selected sub-groups.

For quantitative variables, the total number, mean, standard deviation (SD), median, quartiles, and range were calculated. For qualitative data, the total number, percentages, and 95% Confidence Interval were calculated. Comparisons between groups (Chi 2 , Fisher and Wilcoxon tests) and correlations were performed considering α risk at level 5%. An analysis of sensitivity and specificity was performed on the questionnaire.

RESULTS

Population description

A total of 178 patients were included: 131 women (74%) and 47 men (26%). Mean (±SD) age at baseline was 33.7 (±10.1) years, with a range of 15-60 years. A family history of MS was reported by 11% of the patients. Most patients (72%) were referred by a specialist, usually an ophthalmologist, neurologist, or emergency department physician. The median time between the first episode and the patient's initial visit was 29 days (range, 0-686 days). According to the neurologist, the affected area of the CNS was the optical nerve (in 33% of cases), spinal cord (37.5%), brainstem (12.5%), or brain (11%). Six percent of patients presented with two different affected sites. Mean EDSS score was 1.4 (±1.1).

Brain MRI results

MRI was performed for 99% of the patients, and with gadolinium infusion for 94% of patients. The mean delay from onset was 27 (± 24) days (range, 0-87 days). MRI was abnormal in 87% of cases and 46% were positive according to Barkhof-Tintoré criteria (presenting with at least 3 of the 4 criteria).

Lumbar puncture -Cerebrospinal fluid (CSF) test

Lumbar puncture was performed for 87.5% of the patients. The mean white blood cell (WBC) count was 7 cells/mm 3 (range 0-100). Immunoglobin G (IgG) oligoclonal bands were present in 71% of patients.

Prior symptoms suggestive of demyelination

A total of 139 prior symptoms suggestive of demyelination were reported by 79 patients (44%), and at least one of these suggestive symptoms was validated by the neurologist for 55 patients (70%) (fig 1). In addition, neurologists identified 12 prior symptoms in 10 patients that had not been reported by the patients. Therefore, neurologists identified 59 patients (33%) with at least one prior symptom suggestive of demyelination: 36 patients presented with a single event and 23 patients presented with several events, usually two or three. The mean time period between the first suggestive symptom and the first demyelinating event leading to a Neurology department consult was 46 months (range, 1 to 202 months). The most frequent events were tingling sensations (9.6% of all patients), visual disturbances (6.2%) and muscular weakness in one or several limbs (6.2%). Partial numbness, impaired walking, and loss of balance were each suggestive of prior CNS demyelination for 9 patients (5.1%). Six to eight patients presented with electrical shock sensation (Lhermitte's sign), diplopia, fatigue, or urinary urgency. Most demyelinating events led to consultation with a general practitioner (GP) or a specialist (or both, for 26 patients) but no further investigation had been performed to diagnose CNS demyelination. In summary, in terms of identifying patients with at least one prior symptom suggestive of demyelination, the self-administered questionnaire had an overall sensitivity of 93% and specificity of 80% (table 1). Sensitivity was then analyzed for each symptom reported by the patient or identified by the neurologist. Sensitivity was excellent but the rate of false positives was high for vertigo (77%), hearing disturbance (75%), bladder problems (63%), and fatigue (58%). Conversely, positive predictive value was good (over 70%) for facial paralysis, sensation, and visual disturbance. Comparison of MRI findings with the presence or absence of prior symptoms suggestive of demyelination showed that abnormal MRI was more frequent in patients with prior symptoms (95% compared to 84%, p=0.039).

Sequelae attesting to prior demyelinating events were reported for 14 patients (8% of all patients and 26% of patients reporting validated prior symptoms). Finally, according to McDonald criteria, 70% of patients with prior symptoms had MS (fig 1).

DISCUSSION

New MS diagnosis criteria based on MRI findings [START_REF] Tintoré | New diagnostic criteria for multiple sclerosis: application in first demyelinating episode[END_REF] have good sensitivity and specificity. [START_REF] Dalton | Application of the new McDonald criteria to patients with clinically isolated syndromes suggestive of multiple sclerosis[END_REF][START_REF] Miller | Clinically isolated syndromes suggestive of multiple sclerosis, part I: natural history, pathogenesis, diagnosis, and prognosis[END_REF][START_REF] Korteweg | MRI criteria for dissemination in space in patients with clinically isolated syndromes: a multicentre follow-up study[END_REF][START_REF] Tintore | Baseline MRI predicts future attacks and disability in clinically isolated syndromes[END_REF] In this pilot study, paraclinical results were taken into account to standardise diagnosis of a first event of inflammatory demyelination. In addition, a selfadministered questionnaire was tested to identify previous neurological inflammatory events suggestive of MS and possible sequelae present at examination in order to fulfil criteria for MS. Characteristics of the cohort of patients included in this study were consistent with those previously described for patients with inflammatory demyelinating disease.

Neurologists could identify previous relevant events suggestive of CNS demyelination in 33% of patients coming to the neurology department for a perceived initial inflammatory demyelinating event. Half of these patients had even reported several suggestive prior symptoms. There was an average interval of 46 months between these antecedents and the current episode leading to neurology department consult. Interesting findings were that most of these previous events led patients to visit a GP, a specialist, or both, but did not lead to further investigation and special follow-up. The self-administered questionnaire was revealed to be a useful tool with which to identify patients with prior symptoms suggestive of demyelination. Reported prior symptoms were validated by a neurologist for 70% of patients.

Neurologists identified only four more patients with suggestive prior symptoms that were not reported in the self-administered questionnaire. Finally, symptoms or signs at examination attesting to these prior demyelinating events were reported for 24% of patients and criteria for dissemination in space were fulfilled in another 46%, leading to a diagnosis of MS in 70% of these patients.

ACKNOWLEDGMENTS

  This study was sponsored by Foundation Rothschild and received financial support from Teva Pharmaceuticals and Sanofi Aventis. Thanks to the members of the PEDIAS Group: Heinzlef O., CHI Poissy/St Germain -Poissy; Roullet E., Hôpital Tenon -Paris; Hautecoeur P., CH St Philibert -Lomme; De Seze J., CHU de Strasbourg, Hôpital civil -Strasbourg; Camu W., Hôpital G. de Chauliac -Montpellier; Defer G., CHU Côte de Nacre -Caen; Pelletier J., CHU La Timone -Marseille; Debouverie M., Hôpital Central -Nancy; Barroso B.,, Hôpital F. Mitterrand -Pau; Coustans M., Centre hospitalier de Cornouaille -Quimper; Laplaud D., Hôpital Laënnec -Nantes; Rumbach L., Hopital J. Minjoz -Besançon; Lyon-caen O., Tourbah A., Hôpital de la Pitié-Salpétrière -Paris; Lebiez P.E., CH Louis Pasteur -Cherbourg; Moreau T., CHU Dijon -Dijon; Magy L., CHU Dupuytren -Limoges.

Figure

  Figure 1 Flow chart

Table 1

 1 Sensitivity and specificity of the self-administered questionnaire (Analysis in terms of patients)

	1 Flow chart
	178 patients with a «first demyelinating event» completed the self-administered
	questionnaire
	139 prior events reported by 79 Patients	12 prior events, not reported by
				patients, were identified by
				neurologists in 10 patients
	58 prior	81 neurologist-validated	6 prior events in 4 new patients
	events not	prior events in 55
	validated	patients
			59 patients had prior events validated
			or identified by a neurologist
	119 patients had CIS	45 patients	14 patients had
	(68 had either MRI	(27 had either MRI	sequelae of prior
	dissemination in space	dissemination in space	events (MS patients)
	or 2T2 lesion and	or 2T2 lesion and
	IgGOB)		IgGOB)
	68 had possible MS	27 had MS and 28 had
	and 51 had CIS		possible MS

Author receiving correspondence: Olivier Gout, MD, Service de Neurologie, Fondation Ophtalmologique Adolphe de Rothschild, Paris Disclosure of any personal or financial support:

The authors report no conflicts of interest, except for S. Allouche.

Key words (Search Terms)

Multiple sclerosis [41], clinical trial observational study [23], all demyelinating disease (CNS) [40],