Closures of positive braids and the Morton-Franks-Williams inequality

arXiv:1308.1250v1 [math.GT] 6 Aug 2013 Closures of positive braids and the Morton-Franks-Williams inequality J. Gonz´alez-Meneses∗and P. M. G. Manch´on 6 August 2013 Abstract We study the Morton-Franks-Williams inequality for closures of simple braids (also known as positive permutation braids). This allows to prove, in a simple way, that the set of simple braids is a orthonormal basis for the inner product of the Hecke algebra of the braid group defined by K´alm´an, who first obtained this result by using an interesting connection with Contact Topology. We also introduce a new technique to study the Homflypt polynomial for closures of positive braids, namely resolution trees whose leaves are simple braids. In terms of these simple resolution trees, we characterize closed positive braids for which the Morton-Franks-Williams inequality is strict. In particular, we determine explicitly the positive braid words on three strands whose closures have braid index three. 1 Introduction Let PL(v, z)∈Z[v±1, z±1] be the two-variable Homflypt polynomial, isotopy invariant of oriented links with normalization P(v, z) = 1 and determined by the Homflypt skein relation v−1P(v, z)−vP (v, z) = zP (v, z). In the following we will use the notation P(L) instead of PL(v, z). Note that, for braids, the Homflypt skein relation can be written as v−1σi−vσ−1 i=z, or equivalently, as the quadratic skein relation σ2 i=vzσi+v2. ∗Both authors partially supported by MTM2010-19355 and FEDER. First author partially supported by FQM-P09-5112 and the Australian Research Councils Discovery Projects funding scheme (project number DP1094072). 1 We first recall and fix terminology about the Morton-Franks-Williams (MFW) bounds and inequalities. If L=b bis the closure of a braid b∈Bnwith nstrands and writhe w= wr(b), then w−n+ 1 ≤∂− v(P(L)) and ∂+ v(P(L)) ≤w+n−1 are the known MFW lower and upper inequalities [9, 5], where ∂− v(P(L)) (resp. ∂+ v(P(L))) is the lowest (resp. highest) v-degree of P(L). We refer to w−n+ 1 (resp. w+n−1) as the MFW lower (resp. upper) bound of b. It follows that, if we define MF W (L) = spanv(P(L)) 2+ 1 = ∂+ v(P(L)) −∂− v(P(L)) 2+ 1, we have the celebrated MFW inequality MF W (L)≤n. In particular MF W (L)≤ s(L) where s(L) is the braid index (or Seifert circle index) of L. Let a, b ∈Bnbe two braids with nstrands. Then ha, biRis, by definition, the coefficient of vw+n−1in the two-variable polynomial (−z)n−1P(c ab∗), where b∗ is the reverse braid of band w= wr(ab∗). This product can be extended to the whole of the Hecke algebra Hn(z), obtaining a symmetric bilinear form. This is the inner product introduced by K´alm´an in [6]. Recall that Hn(z) can be seen as the linear combinations of braids in Bnwith coefficients in Z[z±1], quotiented by the Homflypt skein relation with v= 1. Given a permutation α∈Snon {1,2,...,n}, there is exactly one positive braid Tαwhich determines the permutation αon its endpoints, and such that every two strands of it cross at most once. The braid Tαis said to be the simple braid associated to α(originally called positive permutation braid in [4]). Note that wr(Tα) = l(α), the length of the permutation α. We will write T∗ αfor (Tα)∗. It is a well-known result that the set of simple braids on nstrands is a basis of Hn(z). Moreover, the main theorem in [6] states that it is an orthonormal basis for the above inner product. The original proof is based on Contact Topology: it constructs a Legendrian representative of the link \ TαT∗ β, and uses a result by Rutherford [12] that relates the ruling polynomial of a front projection of a Legendrian link with its Homflypt polynomial. In this paper we relate all the above notions, namely we study how simple braids behave with respect to the MFW inequalities, and we apply the obtained results to K´alm´an’s inner product, and to closures of positive braids on three strands. More precisely, in Section 2 we prove that, among all the closures of simple braids, the MFW upper bound is reached only for the closure of the identity braid. This is used in Section 3 to give a simple proof of K´alm´an’s result: the set of simple braids is an orthonormal basis for K´alm´an’s inner product. In particular, our proof contains implicitly an algorithm for calculating this inner product. Further, in Section 4 we introduce the notion of simple resolution trees, as positive resolution trees whose leaves are simple braids. By using them, we will obtain a characterization of the closed positive braids for which the MFW 2 inequality is sharp (Theorem 3 and Corollary 5). Note that, when working with closures of positive braids, the MFW lower bound is always reached (a fact that we easily reproved by using again simple resolution trees), hence the MFW inequality is sharp if and only if the MFW upper bound is reached. In particular, this technique allows us to determine explicitly in Section 5 which positive braid words on three strands have closures of braid index three. Acknowledgements: Part of this work was done during a stay of the first author at the Centre de Recerca Matem`atica (CRM) in Bellaterra (Barcelona, Spain) and at the Department of Applied Mathematics-EUITI, Universidad Polit´ecnica de Madrid (Spain), and also during a stay of the second author at the Department of Algebra, Universidad de Sevilla (Spain). We thank these institutions for their hospitality. 2 MFW inequality for simple braids In this section we show the key result in this paper: the MFW upper bound is reached, among closures of simple braids, only for the identity braid. Proposition 1. Let α∈Snbe a permutation with length w=l(α). Then ∂+ v(P(c Tα)) = w+n−1if and only if α= id, and the coefficient of vw+n−1in P(c Tid)is (−z)1−n. Proof. If α= id ∈Snthen w= 0, Tα= 1n=|| n ...|and c Tαis a collection of n unlinked trivial knots n ... with Homflypt polynomial P(c Tα) = δn−1=v−1−v zn−1 =z1−nv1−n+···+ (−z)1−nvn−1. So the result holds for α= id, even in the extreme case when n= 1. We will prove the result by induction on n, the number of strands of the braid Tα. As the trivial braid is the only braid on 1 strand, we have already shown the case n= 1. Assume now the statement for α∈Skwith k= 1,...,n−1 and suppose α∈Sn. Consider the inclusion i:Sn−1֒→Snω7→ ω⊗1. We distinguish two cases: •If α∈Sn\i(Sn−1), there is a unique permutation α′∈Sn−1and a unique natural number k < n such that α=α′sn−1sn−2...sk. It turns out that Tα=Tα′σn−1σn−2. . . σk, as shown in Figure 1 (see for example [8], page 167). Let b=Tα′σn−2. . . σk∈Bn−1. Clearly c Tα=b b, hence by the MFW upper inequality applied to the Homflypt polynomial of b b, one has: ∂+ v(P(c Tα)) = ∂+ v(P(b b)) ≤(w−1) + (n−1) −1 = w+n−3< w +n−1. 3 Tα′ Figure 1: Tα=Tα′σn−1σn−2. . . σk, with n= 5 and k= 2 •If α=α′⊗1 with α′∈Sn−1, then Tα=Tα′⊗1, c Tα=c Tα′⊔and P(c Tα) = δP (c Tα′). As we have already shown the result when αis trivial, we can assume that α6= id and then α′6= id. Since δ=v−1−v z, applying the induction hypothesis to Tα′which has n−1 strands and wcrossings, it follows that ∂+ v(P(c Tα)) = ∂+ v(P(c Tα′)) + 1 <(w+ (n−1) −1) + 1 = w+n−1. At this point, one could ask for an analogous result for the MFW lower inequality for closures of simple braids. However, it is known that, for closed positive braids (and simple braids are positive braids) the MFW lower bound is always reached (see for example [7], comment after Example 1·8). In spite of this, we will give in Section 4 a direct proof of this fact, working with simple resolution trees. 3 Inner products and the Homflypt skein relation Recall, from the introduction, the inner product h· ,·iRdefined by K´alm´an on the Hecke algebra Hn(z). The following result was first obtained by K´alm´an [6], who proved it by using an interesting connection with Contact Topology (more details in the introduction). Here we give a simple proof of it, based on the Homflypt skein relation and on properties of the simple braids. Theorem 2. (K´alm´an) The set of simple braids {Tα}α∈Snis an orthonormal basis for h· ,·iR. Proof. We want to prove that, for any permutations α, β ∈Sn, hTα, TβiR=1 if β=α, 0 otherwise. This is equivalent to show that for all α, β ∈Sn, the coefficient of vw+n−1 in P(\ TαT∗ β) is (−z)1−nif β=α, and 0 otherwise, where w= wr(TαT∗ β) = 4 wr(Tα) + wr(Tβ) = l(α) + l(β). Note that, by the MFW inequality, to say that the coefficient of vw+n−1in P(\ TαT∗ β) is zero is equivalent to say that ∂+ v(P(\ TαT∗ β)) < w +n−1. The proof is by induction on the length l(β) of the permutation β. If l(β) = 0, then β= id, Tβ= 1n∈Bn,\ TαT∗ β=c Tαand w=l(α). Then the result follows from Proposition 1. Assume now that l(β)≥1. Let β=κsiwith l(β) = l(κ)+1, hence Tβ=Tκσiand T∗ β=σiT∗ κ. 1. If Tασiis a simple braid (equivalently Tασi=Tαsi), then we will see that α6=βand hTα, TβiR= 0. Indeed, α=βwould imply Tκσiσi= Tβσi=Tασito be a simple braid, a contradiction since in Tκσiσithe strands ending in positions iand i+ 1 cross at least twice. In particular, α6=κsihence αsi6=κ. Then (see Figure 2) TαT∗ β=TασiT∗ κ=TαsiT∗ κ hence hTα, TβiR=hTαsi, TκiR= 0 where we have applied induction in the last equality since l(κ)< l(β). TαTαTαsi =σi= T∗ βT∗ κT∗ κ Figure 2: Case Tασisimple: TαT∗ β=TασiT∗ κ=TαsiT∗ κ 2. If Tασiis a non-simple braid, then l(αsi) = l(α)−1 and there exists a reduced expression α=α1siof αending with siand Tα=Tα1σi. Tα T∗ β Tα1Tα1 = = σi σiσ2 i T∗ κT∗ κ Figure 3: Case Tασinon-simple: TαT∗ β=Tα1σ2 iT∗ κ Then TαT∗ β=Tα1σ2 iT∗ κ(see Figure 3) and, by the quadratic relation σ2 i=vzσi+v2, P(\ TαT∗ β) = P(\ Tα1σ2 iT∗ κ) = vzP (\ TαT∗ κ) + v2P(\ Tα1T∗ κ). Multiplying the above equality by (−z)n−1and considering the coefficients of vw−n+1, it follows that: hTα, TβiR=zhTα, TκiR+hTα1, TκiR. 5 Note that α1=κ⇔α1si=κsi⇔α=β. We finally distinguish two cases; induction will be applicable since l(κ)< l(β): •Assume α=β. Then α6=κand α1=κ, hence hTα, TβiR=zhTα, TκiR+hTα1, TκiR=z·0 + 1 = 1. •Assume α6=β. In particular α16=κ. Moreover, α6=κsince, otherwise, Tασi=Tκσi=Tβwould be a simple braid. Hence hTα, TβiR=zhTα, TκiR+hTα1, TκiR=z·0 + 0 = 0. 4 Positive braids and the Morton-Frank-Williams inequality Suppose that bis a positive braid whose closure is the oriented link L. Based on the quadratic relation σ2 i=vzσi+v2, in order to calculate the Homflypt polynomial of Lwe can construct a resolution tree of b. That is, a binary tree with root band where each ramification has the following form, with Pand Q positive braids: P σ2 iQ v2vz P Q P σiQ ❅❅ ❅    Figure 4: Parent P σ2 iQ, left child P Q and right child P σiQ It is known (see the proof of Theorem 3) that simple braids are precisely those positive braids which cannot be written as P σ2 iQ, with Pand Qpositive braids. This means that if a simple braid appears in a resolution tree, it must necessarily be a leaf. A resolution tree is called simple if all the leaves are simple braids. As far as we know, simple resolution trees have not been considered yet; positive resolution trees have been used for example in [11]. As an example, we show in Figure 5 a simple resolution tree for the braid b= 32322323 (meaning σ3σ2σ3σ2σ2σ3σ2σ3), with writhe w= 8 and n= 4 strands. This tree shows in an explicit way that the Homflypt polynomial of b b is a combination of Homflypt polynomials of closures of simple braids, with coefficients in N[z, v] given by the product of the edge labels in the path going from each leaf to the root b. Collecting the leaves which correspond to the same 6 simple braid, in this example we obtain P(b b) = (1 + z2)·v8·P(b I4) + (z+z3)·v7·P(cσ2) + (2z+z3)·v7·P(cσ3) + (2z2+z4)·v6·P([σ2σ3) + (2z2+z4)·v6·P([σ3σ2) + (z+ 3z3+z5)·v5·P(\σ3σ2σ3). vz 4 I4 323 3 2 23 32 323 3 23 32 323 3 2 23 32 323 v2 323 v2 v2 v2 v2 v2 v2 v2 v2 v2 v2 v2 v2 v2 v2 v2 v2 32322323 323323 3232323=2322323 3223 32323=23223 33 233 2323=3233 223 23323 2323=3233 232323=323323 3223 33 32323=23223 233 2323=3233 vz vz vz vz vzvzvz vzvzvzvzvzvz vz vz vz I Figure 5: A simple resolution tree for the braid b= 32322323 We now prove that simple resolution trees always exist and that we can directly see, from a simple resolution tree, whether a closed positive braid reaches the MFW upper bound. Theorem 3. Let L=b bbe a link obtained as the closure of a positive braid bof nstrands and writhe w. Then badmits a simple resolution tree and, moreover, the MFW upper bound is sharp for L, that is, ∂+ v(P(L)) = w+n−1, if and only if at least one leaf in this simple resolution tree is the identity braid. Proof. That any positive braid has a simple resolution tree follows from the following well known fact: a positive braid βis not simple if and only if we can decompose it as β=P σ2 iQ, with Pand Qpositive braids (see, for example, Lemma 2.5 and following remark in [4]). As relations in the braid group are homogeneous, the lengths of the braids P Q and P σiQare strictly smaller than the length of β. Therefore, starting with the root b, we can iteratively decompose every node which is not simple into two smaller nodes. Clearly this process terminates, yielding a simple resolution tree for b. Let Tα1,...,Tαkbe the (not necessarily distinct) simple braids corresponding to the kleaves of a simple resolution tree of b. For i= 1,...,k, let zaivw−l(αi) be the monomial obtained by multiplying the edge labels of the path that goes 7 from the leaf Tαito the root b. Note that aiis the number of right children in this path. Then P(L) = k X i=1 zai·vw−l(αi)·P(d Tαi).(1) Since wr(Tαi) = l(αi), by Proposition 1 the highest v-degree of each summand is (w−l(αi)) + (wr(Tαi) + n−1) = w+n−1 if and only if αi= id. Then, if no leaf is the identity braid, ∂+ v(P(L)) < w +n−1. Reciprocally, if at least one leaf of the simple resolution tree is trivial, then the coefficient of vw−n+1 in P(L) is, again by Proposition 1, the sum of the monomials zaj(−z)1−nthat correspond to αj= id, a sum which is obviously nonzero. As stated in the introduction, it is well known that the MFW lower inequality is actually an equality for the closure of any positive braid (see for example [7], comment after Example 1·8). Here we reprove this result by making use of Equation (1) derived from a simple resolution tree, and following the steps in the proof of Proposition 1. Rather than give complete details, we prefer to explain some historical remarks about the positiveness of the Homflypt polynomial. Recall that a nonzero (Laurent) polynomial in zis said to be positive if all its coefficients are nonnegative. Answering positively a question by V. F. R. Jones, Cromwell and Morton [3] proved that, for positive links, the evaluation of the Homflypt polynomial P(L)(v, z) in any v∈(0,1) provides a positive Laurent polynomial in z. If v= 1 we obtain the Conway polynomial, also positive except that it can be zero if the original link is split. Proposition 4. Let L=b bbe a link obtained as the closure of a positive braid b of nstrands and writhe w. Then the MFW lower inequality is sharp for L, that is, ∂− v(P(L)) = w−n+ 1. Moreover, the coefficient of vw−n+1 in P(L)is a positive Laurent polynomial in z. Proof. Following a double induction, first on the number of strands and then on the writhe, we will see that the coefficient qL(z) of vw−n+1 in P(L)(v, z) is a positive Laurent polynomial in the variable z. If n= 1, then w= 0 and the closure of the braid is the trivial knot with polynomial 1, so the result holds. For n > 1 we follow the steps in the proof of Proposition 1 to see first that the result is true for any simple braid Tαwith nstrands. If α∈Sn\i(Sn−1) then c Tα=b dfor a positive braid dwith n−1 strands and writhe w−1, as given in the proof of Proposition 1. Thus P(c Tα) = P(b d) and (w−1)−(n−1)+1 = w−n+1. Since dis positive (although non-simple) and has less than nstrands, induction can be applied. If α∈i(Sn−1) then α=α′⊗1 and P(c Tα) = δP (c Tα′) where Tα′is simple, with the same writhe as Tαand one less strand (again, see the proof of Proposition 1). Clearly, qL(z) = 1 zqd Tα′(z), so the result holds for every simple braid with nstrands. Finally, once we have proved the result for the closure of simple braids with n 8 strands, the result for a positive braid with nstrands follows from considering Equation (1), derived from a simple resolution tree. According to Proposition 4, for closures of positive braids the MFW inequality is sharp if and only if the MFW upper bound is reached. Then the following result is a nice consequence of Theorem 3: Corollary 5. The MFW inequality is sharp for a closed positive braid if and only if one (hence all) of its braid word representatives can be obtained from the empty word by a finite sequence of transformations of the following types: 1. Inserting σ2 ifor some i= 1,...,n−1, 2. doubling a letter σifor some i= 1,...,n−1, and 3. applying positive braid relations. Proof. Starting with the empty word, a sequence of the above transformations builds a branch of a simple resolution tree for the corresponding positive braid. Since the leaf of this branch is the identity, the MFW upper bound is sharp according to Theorem 3. Reciprocally, suppose that L=b breaches the MFW upper bound, and construct a simple resolution tree for b. By Theorem 3 at least one of its leaves is the identity. Ascending in the tree from such a leaf provides the sequence of transformations of the above types which define a braid word for b. We now enumerate some examples which can be deduced from Corollary 5: Corollary 6. Let wbe a positive word representing a braid b. Then the MFW inequality is sharp for the oriented link L=b bif the word wis in the following list: 1. Words which are product of positive powers of the generators, where all the exponents are greater than or equal to two, that is, w=Qkσek ikwith ek≥2for all k. For example, σ2 3σ5 2σ2 1σ3 2σ3 3. 2. Even positive palindromic braid words, that is, positive words with an even number of letters that reads the same backwards as forwards. For example, σ3σ2σ2 1σ2σ3. 3. Any word of the form uw0vwhere u, v are positive words and w0is any positive word representing the square of the half twist ∆∈Bn. Proof. Words in the first item can be obtained by a finite number of transformations of type 1 and 2 in Corollary 5. Words in the second item can be obtained by a finite number of transformations of type 1 in Corollary 5. 9