On reduction curves and Garside properties of braids

arXiv:1006.2258v1 [math.GT] 11 Jun 2010 On reduction curves and Garside properties of braids Juan Gonz´alez-Meneses April 21, 2010 Abstract In this paper we study the reduction curves of a braid, and how they can be used to decompose the braid into simpler ones in a precise way, which does not correspond exactly to the decomposition given by Thurston theory. Then we study how a cyclic sliding (which is a particular kind of conjugation) affects the normal form of a braid with respect to the normal forms of its components. Finally, using the above methods, we provide the example of a family of braids whose sets of sliding circuits (hence ultra summit sets) have exponential size with respect to the number of strands and also with respect to the canonical length. 1 Introduction Braids can be seen as isotopy classes of orientation-preserving automorphisms of the n-times punctured disc Dn, that is, the braid group on nstrands Bn is isomorphic to the mapping class group M(Dn). Attending to the NielsenThurston classification of mapping classes, braids can be periodic, reducible or pseudo-Anosov. In this paper we shall study reducible braids, which are those braids preserving a family of disjoint, non-degenerate, simple closed curves in Dn. From the algebraic point of view, the braid group Bnhas a well known lattice structure. The submonoid B+ n⊂Bnconsists of those elements of Bnwhich can be written as positive powers of the standard generators σ1,...,σn−1. This monoid defines a partial order of Bngiven by a4b⇔a−1b∈B+ n. This is a lattice order, which is invariant under left multiplication. The triple (Bn, B+ n,∆), where ∆ = σ1(σ2σ1)···(σn−1σn−2···σ1) is the braid known as half twist or Garside element, determines a Garside structure of the braid group Bn[8]. This structure, first discovered by Garside [13], has been very useful for showing many properties of Bn, as well as for providing a substantial number of solutions to 1 the word problem and the conjugacy problem in this group. These algorithms can also be used in other groups sharing the same algebraic properties, which are known under the common name of Garside groups [8]. One of the latest solutions to the conjugacy problem in Garside groups, thus in braid groups, is given in [15] (see also [16]). In that paper, the cyclic sliding is defined as a special conjugation that can be applied to any given braid. Iterated application of cyclic sliding conjugates any braid to another one which has minimal length in its conjugacy class, and has some other good algebraic properties [15]. But in the case of braid groups, reducible braids can behave not so nicely with respect to cyclic sliding, as we shall see. Hence, if one is interested in conjugacy properties of braids, a deeper study of the relation between the geometric and algebraic properties of braids is needed. It is well known that the essential reduction curves of a given braid decompose it into ‘pieces’, or components, in the spirit of Thurston’s decomposition of a mapping class. If the decomposition is done in the appropriate way, each component is again a braid, with fewer number of strands. These components are, in general, not carefully defined in the papers dealing with reducible braids: A reference to Thurston’s decomposition of a mapping class is given instead. But the decomposition in the case of braids is not exactly the same, at least if one needs each of the resulting components to be a braid. In Section 2 we will explain the notion of reducible braid, and the decomposition of such a braid into braid components. In Section 3 we briefly recall the notions we need from Garside theory. In particular we will define cyclic sliding and the set of sliding circuits of a braid. These sets are the ones computed in [15] to solve the conjugacy problem in Garside groups. The normal form of a braid obtained from the Garside structure of Bnis related to the normal form of each of its components, and this relation is more clear in the case in which the reduction curves are isotopic to geometric circles. We will study this in Section 4, and we will see how the application of a cyclic sliding may transform the normal form of a reducible braid, and of each of its components. Finally, using the results from previous sections we will provide, in Section 5, an example of a family of braids whose sets of sliding circuits have exponential size, with respect to the number of strands and also with respect to the length of the braids. We believe this is the first example of this kind. 2 2 Reducible braids 2.1 Reduction curves Let Dnbe the n-times punctured closed disc D2\{P1,...,Pn}. For simplicity, we will assume that D2is embedded in the complex plane C, that its boundary is a geometric circle, and the npunctures are the first nnatural numbers. A simple closed curve Cin Dnis said to be non-degenerate if it is not isotopic to a puncture or to the boundary of Dn, that is, if it encloses more than one and less than npunctures. We will consider such curves up to isotopy, so we will denote by [C] the isotopy class of a curve C. The term curve in this paper will be applied to either a particular non-degenerate, simple closed curve, or its isotopy class. From now on, a family of curves Fwill mean a family of disjoint, non-degenerate, simple closed curves in Dn. Its isotopy class will be denoted [F]. We will say that a curve Cis round if it is isotopic in Dnto a geometric circle. A family of curves Fis round if each of its curves is round, or equivalently, if Fis isotopic to a family of geometric circles. A braid, being an isotopy class of automorphisms of Dn, acts on the set of (isotopy classes of) simple closed curves in Dn. Given a curve Cand a braid β∈Bn, we will denote by [C]βthe curve obtained from [C] after the action induced by β. Similarly, we use the notation [F]β, where [F] is a family of curves. A braid β∈Bnis said to be reducible if there exists a family of curves [F] such that [F]β= [F]. Equivalently, βis reducible if there exist a curve Cand a positive integer msuch that [C]βm= [C] and the curves {C,Cβ,...,Cβm−1}are pairwise disjoint. Such curves are called reduction curves of β. For instance, the braid β= (σ1σ2σ3σ4σ5)2∈B6is a reducible braid, as it preserves the family of round curves F={C1,2,C3,4,C5,6}, where Ci,j is the round curve determined by a geometric circle enclosing punctures ito j(see Figure 1). In this example β3preserves Fcurve-wise. Reduction curves are useful for decomposing braids into simpler ones. But a given braid may admit many (even infinite) distinct families of reduction curves, so the mentioned decompositions are a priori not unique. Nevertheless, there is a special, uniquely defined family of reduction curves of a braid, formed by the so called essential reduction curves [7]. In order to define them, we recall that the geometric intersection number of two curves Cand Din a manifold, denoted i(C,D), is the smallest cardinal of a set in {C′∩D′;C′∈[C],D′∈[D]}. Given β∈Bn, a curve [C] is said to be an essential reduction curve of βif: 1. [C] is a reduction curve of β. 3 Figure 1: The braid (σ1σ2σ3σ4σ5)2preserves two families of reduction curves. 2. If [D] is such that i(C,D)>0, then [D]βm6= [D] for every integer m > 0. The set of essential reduction curves of βis called the canonical reduction system of β, and it is denoted CRS(β). Notice that CRS(β) is a family of disjoint reduction curves of β. Some good properties of CRS(β) are that CRS(β) = CRS(βm) for every m6= 0, and that CRS(α−1βα) = CRS(β)αfor every α, β ∈ Bn. Also, CRS(β) = ∅if and only if βis either periodic or pseudo-Anosov. For instance, in the example of Figure 1, CRS(β) = ∅: The curves of F= {C1,2,C3,4,C5,6}are not essential, since they have nonzero geometric intersection with the round reduction curves C2,3and C4,5. Actually, in that example βis periodic, as β3= ∆2(recall that a braid is periodic if it has a nontrivial power belonging to the center of Bn, that is, to h∆2i). 2.2 Decomposition of a reducible braid Let βbe a non-periodic, reducible braid, so that CRS(β)6=∅. This canonical reduction system can be used to decompose βinto smaller braids, as we shall now see. The basic idea is to consider the action induced by βon the connected components of Dn\{CRS(β)}. Roughly speaking, each of these components is a punctured disc, so these restrictions are homeomorphisms of punctured discs, up to isotopy, and they can be considered as braids, which are usually called the components of β. But there are several ambiguities in this definition, which the reader probably noticed. We will discuss about them in this section, so that we will be able to give a precise definition for the components of a braid β∈Bn. Using the theory of mapping classes, Thurston’s decomposition theorem (together with the definition of canonical reduction systems in [7]) states that CRS(β) decomposes Dninto two (not necessarily connected) invariant subsurfaces, such that βrestricted to one of them is pseudo-Anosov, and restricted to the second one has finite order (up to isotopy in a collar neighborhood of CRS(β)). We recall that braid groups have no torsion, but this does not cause 4 any conflict with the above statement: Thurston’s decomposition theorem deals with mapping classes in which the admissible isotopies fix the boundary setwise (and the theorem allows isotopy in a collar neighborhood of CRS(β)), while the admissible isotopies for braids fix the boundary of Dnpointwise. For instance ∆2is trivial when considered as a mapping class in Thurston’s sense, as it corresponds to a Dehn twist along a curve parallel to the boundary ∂(Dn), while it is certainly not a trivial braid. Hence ∆ has finite order as a mapping class, but not as a braid. In general, we do not want to decompose βinto a couple of mapping classes, as above, since in that case the resulting subsurfaces are not necessarily punctured discs, and the resulting mapping classes do not correspond to braids. We prefer to treat each connected component of Dn\CRS(β) independently. Notice first that there is only one component of Dn\CRS(β) which is isomorphic to a punctured closed disc, namely the one containing the boundary of Dn. All other components are isomorphic to punctured open discs, that is, to punctured spheres. In order to avoid this situation, one can paste each curve of CRS(β) to a connected component, in the following way. Let F=CRS(β)∪ {∂(Dn)}. As Dnembeds in the complex plane, and we are dealing with simple closed curves, we can rigourously talk about parts of Dnenclosed by these curves. Then, for each curve C ∈ F, there is exactly one component XCof Dn\F which is enclosed by C, and such that C ⊂ XC. We can then define DC=XC∪ C, which is homeomorphic to a punctured closed disc with at least two punctures, and we can decompose Dnas: Dn=G C∈F DC. Now notice that βpreserves CRS(β) set-wise, but not necessarily curve-wise. For each C ∈ CRS(β), if βsends Cto C′∈CRS(β), then the punctured closed disc DCis sent to DC′. Hence, the restriction of our braid βto this component is a homeomorphism βC:DC→DC′. Both DCand DC′are homeomorphic to closed discs with the same number of punctures, so βCcan be considered to be a braid, but not in a canonical way. Distinct homeomorphisms taking DCand DC′to the same punctured disc, yield distinct braids representing βC. Hence βCis not well defined in this way, unless we are able to find a canonical way to send each DCto a standard punctured disc. Even if we find such a canonical way, there is another problem: Braids are defined up to isotopy fixing the boundary and the punctures, but one is allowed to move a curve in the interior of Dn. For instance, suppose that [C]6= [C′]. Now consider two curves C1and C2in a collar neighborhood of C, both parallel to and disjoint from C, one enclosing Cand the other one enclosed by C. Let τ1and τ2be Dehn twists along C1and C2, respectively. Then the map γ=τ−1 2◦τ1◦β is isotopic to β, so βand γrepresent the same braid. But their restrictions to 5 DCdo not coincide: They differ by a Dehn twist along the boundary, that is, one equals the other multiplied by the half twist ∆. In the above example, if [C] = [C′], that is, if βsends [C] to itself, the restrictions of βand γto DCare conjugate by ∆. Actually, if CRS(β) is preserved by β curve-wise, the restrictions of βto each DCare well defined up to conjugacy. But even in this case we are not happy enough. An alternative approach could be to consider βas a collection of strands: The three dimensional representation of a motion of npunctures in the disc D. This motion can be extended to an isotopy of Dnwhich sends the set of npunctures to itself. Hence the curves in CRS(β) can also be thought as moving curves, which trace a kind of tube enclosing some punctures. We could try to define the restriction of βto DCby considering only some strands starting inside the tube corresponding to C. More precisely, consider DC\DC. This is a family of points and curves. The curves, C1,...,Cr, are the outermost curves in CRS(β) enclosed by C. The points correspond to the punctures of Dnenclosed by Cbut not by C1,...,Cr. We could then try to define the braid βCas the braid formed by the strands corresponding to those punctures, together with the fat strands corresponding to the tubes formed by C1,...,Cr. The problem here is that the curve Ciis not necessarily round, so its corresponding tube can be deformed in such a way that it is not clear to see how one could consider it as a strand. We can try to solve this problem in the following way. Given a braid β, we can consider a subbraid by erasing some of its strands, that is, by filling some of the punctures of Dn. This is a priori not well defined, as the set of punctures that remain, say I, is not necessarily preserved by the braid. Actually, we would obtain a partial braid, an element of the fundamental groupoid of the configuration space of mpoints in D([2], see also [10]). There is a canonical element αIin this groupoid, sending the points {1,...,m}to I, in which the punctures lie all the time along their motion in the diameter of Dcorresponding to the real line (or in which the image of the diameter is the diameter). Then, if we denote by e βIthe element of the groupoid obtained from βby keeping only the strands starting at I, we can define βI=αIe βIα−1 β(I), in which both the starting and ending points are {1,...,m}, so βIis a well defined braid on mstrands, that we will call the subbraid of βcorresponding to I. Notice that if one draws βas a flat diagram, then βI∈Bmis the braid whose strands cross exactly in the same way as the strands starting at Icross in β. See Figure 2. We now go back to our original braid βsuch that CRS(β)6=∅. Recall that we are trying to define a component βCassociated to C ∈ CRS(β)∪ {∂(Dn)}. Recall also the curves C1,...,Crdefined above. We can now define a subset I⊂ {1,...,n}containing all punctures enclosed by Cand not enclosed by any Ci, together with one puncture enclosed by Ci, for i= 1,...,r. We could then define βCas the subbraid βI, as this fits with our intuitive idea. Unfortunately, 6 Figure 2: The subbraid βI, where β= (σ1σ2σ3σ4σ5)2and I={2,3,5}. this is not well defined, as the following example shows. Consider, the braid β=σ2σ2σ1σ3∈B4, whose canonical reduction system CRS(β) = {C1,C2}is drawn in Figure 3. It is clear that βC1and βC2should be trivial braids on two strands. But if we try to take a subbraid of βto define β∂(D4), this would depend on the choice of punctures: β{1,2}=σ1∈B2, while β{1,4}= 1 ∈B2. This problem comes from the fact that C1and C2are not round curves, otherwise the choice of the strands inside these curves would be irrelevant. Figure 3: The braid β=σ2σ2σ1σ3and its conjugate by σ2,b β=σ2σ1σ3σ2. The solution to the above problem is given by Sang Jin Lee and Eon-Kyung Lee [18]. In this remarkable paper they show that, given an isotopy class [F] of a family of curves in Dn, there is a unique positive braid α∈Bnsuch that [F]αis round, and αhas minimal length among all positive braids satisfying this property. Actually, αis a prefix of any other positive braid transforming F into a family of round curves. Let us call αthe minimal standardizer of F. Denote b β=α−1βα, and notice that αsends the curves C,C1,...,Crto round 7 curves that we denote, respectively, b C,b C1,..., b Cr. We will finally define βCas the subbraid of b βobtained by taking one strand inside each b Ci, plus the strands corresponding to the punctures of Db C. More precisely: Definition 2.1. Let β∈Bnand C ∈ CRS(β)∪{∂(Dn)}. Let I⊂ {1,...,n}be obtained from Db C\Db Cby replacing each curve with a puncture enclosed by that curve. Then we define βC, the component of βassociated to C, as the subbraid (b β)I. In the example given in Figure 3, in order to obtain β∂(Dn)one first needs to conjugate β=σ2σ2σ1σ3by its minimal standardizer α=σ2, to obtain b β=σ2σ1σ3σ2, where we can clearly see that β∂(Dn)=b β{1,3}=b β{1,4}=b β{2,3}= b β{2,4}=σ1∈B2. We remark that this notion of component of a braid, although not stated exactly in this way, is the same as the one given in [18]. Another important remark, to avoid confusion, is that if one decomposes βalong CRS(β) into smaller braids, these braids are not necessarily pseudo-Anosov or periodic, as it happened in Thurston’s decomposition theorem. In order to apply Thurston’s theorem to the case of braids, one needs the following: Definition 2.2. Let β∈Bnand let F=CRS(β)∪ {∂(Dn)}. Given C ∈ F, let Ci∈[C]βifor i≥0, and let mbe the smallest positive integer such that [Cm] = [C0] = [C]. Then we define the interior braid of βassociated to Cto be β◦ C:= (βm)C=βC0βC1···βCm−1. Thanks to this notion of interior braid we can use Thurston’s theorem in the case of braids. More precisely, βCis not necessarily periodic or pseudo-Anosov as Cis not necessarily preserved by β, but if we take a suitable power of β which fixes C, its corresponding component (β◦ C) is indeed either periodic or pseudo-Anosov. We end this section by pointing out that in the definition of βCwe did not make use of the fact that Cis an essential curve, but only that it belongs to a family of curves which is invariant under β. Actually, we can decompose a braid βalong any invariant family of curves F(containing ∂(Dn)), in the same way as above. The notation should be modified in this case, and we will denote β[C∈F]to be the component associated to Cin this decomposition. In particular βC=β[C∈CRS(β)]. In some cases we do not even need the family Fto be preserved by β. If Fis a family of round curves such that [F]βis also a family of round curves, we can just define β[C∈F]as in Definition 2.1 by replacing CRS(β) with F,Db Cwith DCand b βwith β. So in this case β[C∈F]is the subbraid βI, where Iis obtained from DC\DCby replacing each curve with a puncture enclosed by that curve. This subbraid is well defined thanks to the roundness of [F] and [F]β. 8 3 Garside structure: Cyclic sliding and sliding circuits We now briefly recall the notions introduced in [15] to solve the conjugacy problem in Garside groups (in particular in braid groups), and which replace the previous notions of cyclings, decyclings [11] and ultra summit sets [14]. Recall that the braid group (and every Garside group) has a lattice structure, so every two elements α, β ∈Bnadmit an element α∧β, called greatest common prefix, which is their meet with respect to the partial order 4. Using this, one can define a normal form of the elements in Bn, called the left normal form[9, 1, 11, 12], which is a decomposition of any element x∈Bnas ∆px1···xr, where pis the maximal integer such that ∆−pxis positive, each xiis nontrivial, and xi= (xi···xr)∧∆, for i= 1,...,r. The infimum,supremum and canonical length of xare defined, respectively, inf(x) = p, sup(x) = p+rand ℓ(x) = r. The factors x1, . . . , xrin the above decomposition are simple elements, that is, positive prefixes of ∆. If sis simple, denote ∂(s) = s−1∆, the complement of s, which is also simple. It is well known that if aand bare two simple elements, the left normal form of the product ab is equal to (as)t(the first factor is as and the second one is t), where s=∂(a)∧b. Notice that in this case as could be equal to ∆ and tcould be trivial. Let τbe the inner automorphism of Bnassociated to ∆, that is, τ(α) = ∆−1α∆ for any α∈Bn. From the left normal form x= ∆px1···xrwe can define the initial factor of x∈Bnas ι(x) = (x∆−p)∧∆. That is, ι(x) = (τ−p(x1···xr))∧ ∆, so if r= 0 one has ι(x) = 1, while if r > 0 one obtains ι(x) = τ−p(x1). We define the final factor of xas ϕ(x) = (∆p+r−1∧x)−1x, which means ϕ(x) = xr if r > 0 and ϕ(x) = ∆ if r= 0. It is well known that ι(x−1) = ∂(ϕ(x)). The preferred prefix of xis defined by p(x) = ι(x)∧ι(x−1), and the cyclic sliding of xis the conjugate of xby its preferred prefix, that is, s(x) = p(x)−1xp(x). To see why this is a natural definition, see [15]. For the moment, just notice that if xhas nonzero canonical length, p(x) = ι(x−1)∧ι(x) = ∂(xr)∧τ−p(x1), hence the first factor in the left normal form of xrτ−p(x1) is precisely xrp(x). We say that an element y∈Bnis in a sliding circuit if sm(y) = yfor some m > 0. The set of sliding circuits of a braid x, denoted SC(x), is the set of conjugates of xbelonging to a sliding circuit. There is a simple algorithm to solve the conjugacy problem in Bn(and in any Garside group), by using cyclic slidings and by computing sets of sliding circuits [16, Algorithm 0]. In [15] it is shown that application of cyclic sliding will never increase the canonical length of an element. Hence, as the set of conjugates of a given element with bounded canonical length is finite, it follows that any braid xcan be conjugated 9 Lemma 5.1. Given δd1,...,dm, denote u0=d0= 1. One has: 1. S(δd1,...,dm) = {σui;ui+ 1 6=ui+1}. 2. F(δd1,...,dm) = {σdi;di+ 1 6=di+1}. In particular, S(δd1,...,dm)∩F(δd1,...,dm) = ∅. Proof. From [6], the permutation associated to δd1,...,dmis given by the single cycle π= (1 u1u2···ukn dmdm−1···d1). As δd1,...,dmis a simple braid, we recall that its starting set is given by the generators σisuch that π(i)> π(i+1). Suppose that i=djfor some j. Then π(i)< i. In this case, if i+ 1 = dj+1 or i+ 1 = none has π(i+ 1) = dj=i > π(i), and if i+ 1 = ukfor some k, one has π(i+ 1) > i + 1 > π(i). In either case σi6=S(δd1,...,dm). Suppose now that i=ukfor some k. Then π(i)> i. In this case, if i+ 1 = uk+1 one has π(i) = i+ 1 < π(i+ 1), so σi6=S(δd1,...,dm). But if i+ 1 = djfor some jor i+ 1 = n, one has π(i)> i ≥π(i+ 1), so σi∈S(δd1,...,dm). The first claim is then shown. The second claim follows from the first one, once we notice that for every positive braid xone has F(x) = S(←− x), where ←− xis the braid obtained from any positive word representing x, read backwards (or the image of xunder the antiisomorphism of Bnwhich sends each σito itself). Since ←−−−−− δd1,...,dm=δu1,...,uk, the second claim follows. Finally, the only possible element in S(δd1,...,dm)∩F(δd1,...,dm) is σ1, but σ1 belongs either to S(δd1,...,dm) or to F(δd1,...,dm) depending whether 2 = d1or 2 = u1. Since both properties are mutually exclusive, the intersection is empty. We still need to define some other special elements for our example: Lemma 5.2. For every i, j ∈ {1,...,n−1}there is a simple braid αi,j such that S(α) = {σi}and F(α) = {σj}. Proof. It suffices to take, if i≤j,α=σiσi+1 ···σj, and if i≥j,α= σiσi−1···σj. Lemma 5.3. Let i1, i2,...,ik+1 ∈ {1,...,n−1}, and let ηbe a simple conjugate of δsuch that σi1∈F(η). Then the element xη,i1,...,ik+1 = (αi1,i2αi2,i3···αik,ik+1 )−1η(αi1,i2αi2,i3···αik,ik+1 ) is a conjugate of δwhose left normal form is: ∆−k∂−2k+1(αik,ik+1 )···∂−3(αi2,i3)∂−1(αi1,i2)η αi1,i2αi2,i3···αik,ik+1 . 16 Proof. The first statement is evident as xη,i1,...,ik+1 is a conjugate of η, which is a conjugate of δ. Now, it is well known that a product of two simple factors ab is left-weighted if and only if S(b)⊂F(a). Since F(αij−1,ij) = {σij}= S(αij,ij+1 ), the factorization αij−1,ijαij,ij+1 is left-weighted for j= 2,...,k. Also S(αi1,i2) = {σi1} ⊂ F(η), hence the factorization η αi1,i2is also leftweighted. If we know the left normal form of a braid, the left normal form of its inverse is also known [11]. In this case, since αi1,i2αi2,i3···αik,ik+1 is in left normal form as written, it follows that the left normal form of (αi1,i2αi2,i3···αik,ik+1 )−1is equal to ∆−k∂−2k+1(αik,ik+1 )···∂−3(αi2,i3)∂−1(αi1,i2). It only remains to show that ∂−1(αi1,i2)ηis left-weighted. But if aand bare two simple elements such that ab = ∆, one has F(a)∪S(b) = {1,...,n−1}, that is F(∂−1(b)) = {1,...,n−1}\S(a). In this case F(∂−1(αi1,i2)) = {1,...,n− 1}\S(αi1,i2) = {1,...,n−1}\{σi1}. On the other hand ηis a simple conjugate of δ, so Lemma 5.1 tells us that S(η)∩F(η) = ∅. Since σi1∈F(η), it follows that σi1/∈S(η), that is, S(η)⊂ {1,...,n−1}\{σi1}=F(∂−1(αi1,i2)), hence ∂−1(αi1,i2)ηis left-weighted, as we wanted to show. Corollary 5.4. For every i1,...,i2k∈ {1,...,n−1}and every simple conjugate ηof δsuch that σi1∈F(η), the braid ∆2kxη,i1,...,i2kis a conjugate of δnk+1 whose infimum is 1 and whose canonical length is 4k−1. Moreover xη,i1,...,i2k= xγ,j1,...,j2kif and only if η=γand (i1,...,i2k) = (j1,...,j2k). Proof. We saw in the previous result that xη,i1,...,i2khas infimum −2k+ 1 and canonical length 4k−1. If we multiply this braid by ∆2k, we will obtain a braid whose infimum is 1 and whose canonical length is still 4k−1 as stated. Since xη,i1,...,i2kis a conjugate of δand ∆2=δnis a central element of Bn, it follows that ∆2kxη,i1,...,i2kis a conjugate of ∆2kδ=δnk+1. Multiplying an element from the left by a power of ∆ does not modify the non- ∆ factors of its left normal form. This implies that the final 2kfactors of the left normal form of ∆2kxη,i1,...,i2kare precisely ηαi1,i2αi2,i3···αi2k−1,i2k. As the element αi,j is uniquely determined by the indices iand j, it follows that the braid ∆2kxη,i1,...,i2kis uniquely determined by ηand by the indices (i1,...,i2k), as we wanted to show. We already have all the ingredients to provide a family of elements whose set of sliding circuits is exponential both on the number of strands and on the canonical length. Proposition 5.5. For n≥3and k≥1, consider the braid β∈Bn+2 given by: β= (σ1···σn−1)nk+1 (σn+1)4k+1. Then ℓ(β) = 4k+ 1 and #(SC(β)) ≥2n−2(n−1)2k−1. 17 Proof. We remark that SC(β) is much bigger than 2n−2(n−1)2k−1, but we chose this bound for the simplicity of the proof, as it is already exponential on the braid index (n+ 2) and on the canonical length (4k+ 1), as we shall see. Denote δ=σ1···σn−1, so β=δnk+1 (σn+1)4k+1. We recall that as δn= ∆2 none has δnk+1 = ∆2k nδ, so the left normal form of βis (∆nσn+1)2k(δσn+1)(σn+1)2k, which is the left-weighted product of 4k+ 1 simple elements (the parenthesized terms correspond to simple factors). Hence inf(β) = 0 and ℓ(β) = 4k+ 1. Notice also that this braid is non-periodic and reducible: it is non-periodic as it is written as a word in which σndoes not appear, so the same must happen to every power of β, hence no power of βcan be equal to a power of ∆n+2. It is reducible since it preserves curve-wise the family of circles F={C1,n,Cn+1,n+2}. Actually CRS(β) = Fsince β[∂(Dn)∈F]is trivial, β[C1,n∈F]=δnk+1 is periodic and nontrivial, and β[Cn+1,n+2∈F]=σ4k+1 1is also periodic and nontrivial, but we will not make use of this fact. Now, for every simple conjugate ηof δin Bn, we can choose an index iη such that σiη∈F(η). Then for every i2,...,i2k∈ {1,...,n −1}, denote x= (∆n)2kxη,iη,i2,...,i2kand y=x(σn+1)4k+1. We saw in Corollary 5.4 that xis a conjugate of δnk+1 in Bn, so there exists γ∈Bnsuch that γ−1xγ =δ2k+1. We can consider γas a braid in Bn+2, by the standard embedding Bn→Bn+2 that sends σi∈Bnto σi∈Bn+2 (this corresponds to adding two vertical strands to the right of each braid). Then in Bn+2 one has γ−1yγ =γ−1(x(σn+1)4k+1)γ= δnk+1(σn+1)4k+1 =β. Hence yis conjugate to our original braid β. We will now show that ybelongs to a sliding circuit. By Corollary 5.4, x∈ Bnhas infimum 1 and canonical length 4k−1, so its left normal form is ∆ns1···s4k−1for some simple elements s1,...,s4k−1(which are specified in Corollary 5.4). The left normal form of y=x(σn+1)4k+1 ∈Bn+2 is then (∆nσn+1)(s1σn+1)···(s4k−1σn+1)(σn+1). None of these simple factors is equal to ∆n+1, hence inf(y) = 0 and sup(y) = 4k+ 1. The preferred prefix of this element is then p(x(σn+1)4k+1) = ∂(σn+1)∧(∆nσn+1) = ∆n. Notice that this corresponds to the fourth condition in Proposition 4.10, applied to yand C1,n. Since p(y) = ∆n, applying a cyclic sliding to ymerely conjugates its component xby ∆n, that is, s(y) = ∆−1 ny∆n=x′(σn+1)4k+1, where x′= ∆−1 nx∆n. If we define i′ r=n−irfor r= 2,...,2k, also i′ η=n−iηand η′= ∆−1 nη∆n, we see that η′is a simple conjugate of δsuch that σi′ η∈F(η′), hence x′= (∆n)2kxη′,i′ η,i′ 2...,i′ 2k. We can then apply the above argument to s(y) and x′, to conclude that p(s(y)) = ∆n, hence s2(y) = ∆−2 ny∆2 n=y. Therefore ybelongs to a sliding circuit, so y∈SC(β). Finally notice that ywas defined by choosing ηand i2,...,i2k(the index iη is determined by η). Distinct choices of these data yield different values of x=y[C1,n∈F], hence different values of y. Therefore there are at least as many elements in SC(β) as possibilities we have for choosing these values. There are 18 2n−2choices for ηand n−1 choices for each ir, for r= 2,...,2k, hence there are at least 2n−2(n−1)2k−1elements in SC(β). We conclude by noticing that the above example, as well as most examples of these kind, can be treated in a more intelligent way than just computing its set of sliding circuits, provided one needs to know whether a given element is conjugate to it. If one knows its essential reduction curves (as it is the case), one can apply cyclic sliding to each component, defining a smaller subset of SC(β) containing the conjugates of βin which every component belongs to a sliding circuit. In order to do something like that, one needs an efficient algorithm to detect the canonical reduction system of a braid. There are some algorithms to do this: one is given by Bestvina and Handel [5] using the theory of train tracks, and there is another one which will be soon available [17], which is an improvement of the one given in [3] and [4], using Garside theory. 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