A sharp signature bound for positive four-braids

A sharp signature bound for positive four-braids Abstract We provide the optimal linear bound for the signature of positive four-braids in terms of the three-genus of their closures. As a consequence, we improve previously known linear bounds for the signature in terms of the first Betti number for all positive braid links. We obtain our results by combining bounds for positive three-braids with Gordon and Litherland’s approach to signature via unoriented surfaces and their Goeritz forms. Examples of families of positive four-braids for which the bounds are sharp are provided. 1. Introduction This paper is concerned with positive braid knots and links—the knots and links obtained as the closure β^ of positive braids β—and the following two link invariants: the first Betti number b1(L) of a link L—the minimal first Betti number of oriented surfaces in R3 with oriented boundary L—and the signature σ(L) of a link L as introduced by Trotter [17] and (for links) by Murasugi [13]. Let a link L be the closure of a non-trivial positive braid—a positive braid such that its closure is not an unlink. We suspect (as conjectured in [7]) that   b1(L)≥−σ(L)>b1(L)2, (1.1)where the first inequality is immediate from the definition of the signature. This article establishes (1.1) for closures of positive 4-braids. Theorem 1.1 Let β be a positive 4-braid such that its closure β^is not an unlink, then  −σ(β^)>b1(β^)2. By an observation [7, Reduction Lemma], which we recall in the appendix for the reader’s convenience, Theorem 1.1 implies the following bound for all positive braids: Theorem 1.2 Let β be a positive braid such that its closure β^is not an unlink, then  −σ(β^)>b1(β^)8. In other words, up to a factor of 4, (1.1) holds. The main technical ingredient in the proof of Theorem 1.1 is Gordon and Litherland’s approach of using (non-oriented) checkerboard surfaces and the associated Goeritz form to calculate the signature [10]. Let us shortly put Theorems 1.1 and 1.2 in context. Links that arise as closures of positive braids are a well-studied class of links containing important families such as (positive) torus links, algebraic links and Lorenz links, while themselves being a subclass of positive links. Rudolph established that closures of positive braids have strictly negative signature [14]. For positive 4-braids, previous results by Stoimenow [16, Theorem 4.2] and the author [7, Main Proposition] provided linear bounds of the signature in terms of the first Betti number with factor 211 and 512, respectively. For the more general class of positive links, Baader, Dehornoy and Liechti provide a linear bound for the signature in terms of the first Betti number with factor 148 [3, Theorem 2]. The novelty of Theorem 1.1 is that the factor 12 of the linear bound is optimal; compare the discussion in Section 5. The linear bound with factor 18 for all positive braids as provided in Theorem 1.2 is a clear improvement over the best-known previous bound; compare [7] and [3, Theorem 2]. A geometric consequence of the signature bounds are lower bounds for the topological slice genus g4top(K) of a knot K—the minimal genus among all locally flat oriented surfaces in the unit 4-ball B4 with boundary K⊂S3=∂B4. Indeed, combining Theorems 1.1 and 1.2 with Kauffman and Taylor’s results that ∣σ(K)∣≤2g4top(K) for all knots K [11] yields the following. For a knot K, denote the three-genus—half the first Betti number of K—by g(K). Corollary 1.3. For a knot K that is not the unknot and that is the closure of a positive braid, one has  g(K)≥g4top(K)>g(K)8.Furthermore, if K is the closure of a positive 4-braid, then  g(K)≥g4top(K)>g(K)2. Remark 1.4 There are knots K that are closures of positive braids for which g(K) is strictly larger than g4top(K), which is surprising since in the smooth setting, the smooth slice genus is equal to the three-genus by Kronheimer and Mrowka’s resolution of the Thom conjecture [12, Corollary 1.3]. Indeed, Rudolph observed that the topological slice genus of the torus knot T5,6 is strictly less than 10=g(T5,6). In fact, there are infinite families of positive braid knots for which the topological slice genus can be linearly bounded away from the three-genus by a significant amount [4, 15, Theorem 2]. This justifies interest in a linear lower bound as provided in Corollary 1.3. We conclude the introduction by outlining the strategy of the proof of Theorem 1.1, as provided in Section 3. We will use Gordon and Litherland’s approach to the signature via Goeritz forms to show the following. For every link L that is the closure of a non-trivial positive 4-braid, there exists a link L′ that is the closure of a positive 3-braid with   b1(L′)=b1(L)+1and∣σ(L)−σ(L′)∣≤1.This will allow us to reduce Theorem 1.1 to the following proposition. Proposition 1.5 Let β be a positive 3-braid with b1(β^)≥2, then  −σ(β^)≥b1(β^)2+1. Proposition 1.5 improves Stoimenow’s result that −σ(β^)>b1(β^)2 for positive non-trivial 3-braids [16, Theorem 4.1]. We provide a proof for Proposition 1.5 which is independent of Stoimenow’s techniques; see Section 4. Optimality of Theorem 1.1 and Proposition 1.5 is discussed in Section 5. 2. Setup: signatures of links via Goeritz forms and positive braids We set up notions and recall facts about braids and the signature of links. 2.1. Signature of links and Goeritz forms For a link L—an oriented smooth embedding of a non-empty finite union of circles in S3—the signature, denoted by σ(L), is defined to be the signature of the symmetrized Seifert form on H1(F), where F is any compact and oriented surface in S3 with oriented boundary L; compare Trotter and Murasugi [13, 17]. In particular, one has that −b1≤σ≤b1 holds for all links. Unifying Trotter’s approach to the signature and work of Goeritz [9], Gordon and Litherland [10] introduced the following procedure to calculate the signature. For any link diagram DL of L—the image of a generic project of the link L to a standard 2–sphere R2∪{∞} in S3 together with crossing information—one has   σ(L)=σ(SL)−μ(SL),where SL is a non-oriented surface with boundary L given as one of the two checkerboard surfaces ( SL is contained in DL⊂S3 away from neighborhoods of crossings. In a neighborhood of a crossing, SL is given by a small ‘half-twisted’ band. We refer to Fig. 4 for an illustrative example and to Gordon and Litherland’s original work [10] for more details) of DL and σ(SL) and μ(SL) are defined as follows. To every crossing c of DL, one associates a type (I or II) and a sign η(c) (1 or −1) by the rule specified in Fig. 1. Then one defines   μ(SL)=∑c∈crossingsofLoftypeIIη(c). (2.1)To define σ(SL), pick a basis [δ1],…,[δk] of H1(SL) represented by simple closed curves δi⊂SL and let the matrix GL={gij} be given by gij=lk(δi,δj±). Here δj± denotes the link in S3⧹SL obtained from δj⊂SL by a small push-off in both normal directions of SL and lk denotes the linking number in S3. Then one sets σ(SL)=σ(GL), where σ(GL) denotes the signature of GL—the number of positive eigenvalues minus the number of negative eigenvalues counted with multiplicities. The bilinear form defined by GL is called the Goeritz form. This fits into the setting of the more general Gordon–Litherland pairing, where one uses any (in general non-orientable) surface SL with (unoriented) boundary L rather than a checkerboard surface and −μ(SL) is replaced by half the Euler number of SL; see [10, Corollary 5]. A warning concerning sign conventions is in order: the above definition of σ(L) has opposite sign of that given in [3, 14, 16]. The present convention agrees with the convention in [10, 13, 17] and appears to be the standard one. For example,   σ(T2,n+1)=−n(ratherthann)forallpositiveintegersn. (2.2) Figure 1. View largeDownload slide Left: sign associated to a crossing. Right: type associated to a crossing. The type only depends on whether or not the surface SL (gray) can locally be given an orientation that induces the orientation of the link. In particular, the type is independent of the crossing information. Figure 1. View largeDownload slide Left: sign associated to a crossing. Right: type associated to a crossing. The type only depends on whether or not the surface SL (gray) can locally be given an orientation that induces the orientation of the link. In particular, the type is independent of the crossing information. Figure 4. View largeDownload slide The checkerboard surfaces Sβ and Sα (gray) associated with the braid words β=a1a3a2a1a3a2a2a1a3a2 (left) and α=a1a1a2a1a1a2a2a1a1a2 (right). The curves γi and δi constitute bases of H1(Sβ) and H1(Sα), respectively. Figure 4. View largeDownload slide The checkerboard surfaces Sβ and Sα (gray) associated with the braid words β=a1a3a2a1a3a2a2a1a3a2 (left) and α=a1a1a2a1a1a2a2a1a1a2 (right). The curves γi and δi constitute bases of H1(Sβ) and H1(Sα), respectively. The following properties of the signature of a link follow rather directly from both the original definition and Gordon and Litherland’s approach. If a link L′ can be obtained from a link L by one saddle move, then   ∣σ(L)−σ(L′)∣≤1[13,Lemma7.1]. (2.3) Here a saddle move is defined as changing the link in a 3-ball as described on the left-hand side in Fig. 2. If L′ can be obtained from L by smoothing of a crossing, defined on the right-hand side in Fig. 2, then (2.3) also holds since a smoothing of a crossing corresponds to a saddle move on the link. The signature is additive under split union and connected sum: for all links L and L′  σ(L⊔L′)=σ(L♯L′)=σ(L)+σ(L′)[13,Lemmas7.2and7.3]. (2.4) Note that the connected sum L♯L′ is only well defined if one specifies which components of the two links L and L′ are connected via the connect sum operation; however, (2.4) holds for all such specifications. Figure 2. View largeDownload slide Left: a saddle move. Right: a smoothing of a crossing, which is obtained by applying a saddle move to the sphere indicated by the dotted ellipse. Figure 2. View largeDownload slide Left: a saddle move. Right: a smoothing of a crossing, which is obtained by applying a saddle move to the sphere indicated by the dotted ellipse. 2.2. Positive braids Let Bb be Artin’s braid group on b strands [1]:   Bb=⟨a1,…,ab−1∣aiaj=ajaifor∣i−j∣≥2,aiajai=ajaiajfor∣i−j∣=1⟩.A braid β in Bb corresponds to a (geometric) braid, represented via braid diagrams, and yields a link β^ via the closure operation. Generators ai (respectively ai−1) correspond to braids given by the braid diagram where the ith and (i+1)st strands cross once positively (respectively, negatively). This yields that a braid word—a word in the generators of Bb—defines a diagram for the braid it encodes. We illustrate this in Fig. 3 for one example to set conventions and refer to [6] for a detailed account. Figure 3. View largeDownload slide A braid diagram (left) coming from the braid word a1−1a2a2a1a3a2 and a link diagram (right) for the corresponding braid closure. Figure 3. View largeDownload slide A braid diagram (left) coming from the braid word a1−1a2a2a1a3a2 and a link diagram (right) for the corresponding braid closure. A positive braid on b strands is an element β in Bb that can be written as positive braid word as1as2…asl(β) with si∈{1,…,b−1}, where l(β) is called the length or writhe of β. Note that l(β) is an invariant of positive braids β since it is independent of the choice of a positive braid word for β. The first Betti number of the closure of positive braids is understood as a consequence of Bennequin’s inequality [5, Theorem 3], which implies the following formula:   b1(β^)=l(β)−b+cforeverypositivebraidβ, (2.5)where b is the number of strands of β and c equals 1 plus the number of generators ai that are not used in a positive braid word for β. For example, for all positive integers n and m, the torus link Tn,m is a positive braid link with first Betti number (n−1)(m−1) since the closure of the braid (a1⋯an−1)m∈Bn is Tn,m. 3. Proof of Theorem 1.1 In this section, we prove that if a link L with b1(L)≥1 is the closure of a positive 4-braid, then   −σ(L)≥b1(L)2+12>b1(L)2.Besides the notions introduced in Section 2, the proof uses Proposition 1.5, which is proved in Section 4. Proof of Theorem 1.1 For the entire proof, let L be the closure of a non-trivial positive 4-braid given by a positive 4-braid word β. Without loss of generality all generators a1, a2 and a3 appear in β at least twice; in particular, b1(L)≥3 by (2.5). Indeed, otherwise L is a connected sum or split union of links that are closures of positive braids on 3 or less strands, for which the statement follows from Proposition 1.5, (2.2) and the fact that signature is additive on connect sums and split unions. We define a 4-braid word α by replacing all a3 in β by a1. For example, if β=a1a3a2a1a3a2a2a1a3a2, then α=a1a1a2a1a1a2a2a1a1a2. From the given braid word β, we get a braid diagram and a corresponding link diagram Dβ (compare Section 2.2) representing L=β^. Similarly, the braid word α yields a link diagram Dα for α^. Next we checkerboard color the two diagrams Dβ and Dα (we choose the convention that the unbounded region of a diagram is white) and we denote the black checkerboard surfaces by Sβ and Sα, respectively. See Fig. 4, where this is illustrated for an example. We note that the diagrams Dβ and Dα and the checkerboard surfaces Sβ and Sα depend on the braid words β and α, respectively (rather than just depending on the braids that β and α represent in B4). Let k denote the number of a2 in the braid word β. The number of a2 in the braid word α is also k. The checkerboard surfaces Sβ and Sα have first Betti number k+1. This is best seen by observing that the Betti number of Sβ and Sα are equal to the number of bounded white regions in Dβ and Dα, respectively, of which there are k+1. We describe this in a bit more detail and set up a one-to-one correspondence between the bounded white regions of Dβ and Dα, which will be useful below. Every a2 in β corresponds to a crossing in Dβ, which is touched by two white regions (one from above and one from below). We label the generators a2 in β by 1, 2,…, k in the order of appearance from the left and assign the same label to the unique white region touching the corresponding crossings from above. Only one bounded white region remains. We label this bounded white region by k+1. The same labeling procedure applied to α yields a labeling for the white bounded regions of Dα and this sets up a one-to-one correspondence between the bounded white regions of Dβ and Dα. See Fig. 4, where this labeling is illustrated. Following Goeritz and Gordon–Litherland [9, 10], we choose bases ([γ1],…,[γk]) and ([δ1],…,[δk]) for H1(Sβ) and H1(Sα), respectively, as follows. The boundary of a white region in Dβ (respectively, Dα) labeled i defines a curve γi (respectively, δi) in Sβ (respectively, Sα). We orient the γi and δi counterclockwise and let [γi] and [δi] denote the corresponding homology classes in H1(Sβ) and H1(Sα), respectively. Let Gβ and Gα denote the Goeritz matrices of Sβ and Sα with respect to the bases ([γ1],…,[γk]) and ([δ1],…,[δk]), respectively. The above one-to-one correspondence between the white bounded regions of Dβ and Dα is set up such that for all i,j≤k,   lk(γi,γj±)=lk(δi,δj±). In other words, all entries of Gα not in the last row or column coincide with the corresponding entries of Gβ. Since the signatures of two real-valued symmetric (k+1)×(k+1) matrices that are identical except in the last column and row differ by at most one, we have   −σ(Gβ)≥−σ(Gα)−1. (3.1) Furthermore, μ(Sβ)=μ(Sα), since μ(Sβ) (respectively, μ(Sα)) is equal to the number of a1 and a3 in β (respectively, α) by (2.1). We note that α^ is the split union of the unknot and the link L′ obtained as the closure of the 3-braid given by interpreting α as a 3-braid word. Therefore, σ(L′)=σ(α^) by (2.4). Note also that b1(L′)=b1(L)+1 by (2.5). With all of the above and Gordon and Litherland’s   σ(L)=σ(Gβ)−μ(Sβ)andσ(α^)=σ(Gα)−μ(Sα),we calculate   −σ(L)=−σ(Gβ)+μ(Sβ)≥−σ(Gα)−1+μ(Sβ)=−σ(Gα)−1+μ(Sα)=−σ(α^)−1=−σ(L′)−1≥b1(L′)2+1−1=b1(L)+12,where we used (3.1), μ(Sβ)=μ(Sα), Proposition 1.5 and (2.5) in the second, third, sixth and last line, respectively. This concludes the proof of   −σ(L)≥b1(L)+12>b1(L)2,for all closures L of non-trivial positive 4-braids.□ 4. Proof of Proposition 1.5 In this section, we prove that for all positive 3-braids β with b1(β^)≥2, we have   −σ(β^)≥b1(β^)2+1. (4.1) Proof of Proposition 1.5 Let β be a positive 3-braid with b1(β^)≥2. Denote by Δ the positive half-twist on 3-strands a1a2a1=a2a1a2. Let nβ be the maximum of all non-negative integer n′ such that β=Δn′α for a positive 3-braid α. Note that nβ exists since the n′ are less or equal than a third of the length l(β). Since we are interested in the signature and the first Betti number of the closure of β, there is no loss of generality by assuming that nβ is maximal among all such n′ for β′=Δn′α′, where β′ is any positive braid conjugate to β. This maximality of nβ is assumed in the entire proof. For example, the braid β=a1a2a2a2a2, which has nβ=0, is not considered since it is conjugate to a2a1a2a2a2=Δa2a2. Furthermore, we assume in the entire proof that β^ is not the closure of a 2-braid since for such closures one has   −σ(β^)=(2.2)l(β)−1=(2.5)b1(β^)≥b1(β^)2+1.For example, for all positive integers l, the braid β=Δa1l is not considered since its closure is the braid index two torus link T2,l+2. If −σ(β^)≥b1(β^)2 holds for a positive 3-braid β, then one also has   −σ((Δ4)k^β)≥b1((Δ4)k^β)2+1,for all positive integers k. This can be seen as follows. Gambaudo and Ghys [8, Lemma 4.1] established that   −σ((Δ4)k^β)=−σ(β^)+8k[8,Lemma4.1], (4.2)for all 3-braids β and all integers k. Thus, −σ(β^)≥b1(β^)2 yields   −σ((Δ4)k^β)=(4.2)−σ(β^)+8k≥b1(β^)2+8k≥(2.5)b1((Δ4)k^β)2+2k≥b1((Δ4)k^β)2+1.Therefore, it suffices to establish (4.1) for positive 3-braids β with b1(β^)≥2 and nβ=0,1,2, or 3. Let us first consider the case nβ=0. After possibly a conjugation (given by cyclic permutation of positive braid words), we have that β is given by a positive braid word starting with a1 and ending with a2, i.e.   β=a1l0a2l1a1l2…a1lc−1a2lcfor some odd integer c≥1 and positive integers li. Furthermore, we have that for all i∈{0,…,c} the li are integers larger than or equal to 2 since otherwise (up to cyclic permutation) β will contain a1a2a1 or a2a1a2 as a subword which is impossible by the definition of nβ. Note that b1(β^)≥2c by (2.5). By applying c−12 saddle moves on β^, we get a link L that is a connected sum of c+32 torus links of braid index two; in fact, a connect sum of the torus link T2,leven, where leven=∑i=0c−12l2i, and the torus links T2,lj for odd j. This is possible by making saddle moves that ‘separate’ all but one of the a2li-blocks as illustrated in Fig. 5. Thus, we have   −σ(L)=(2.4),(2.2)b1(L)=b1(T2,leven)+∑i=0c−12b1(T2,l2i+1)=b1(β^)−c−12.And, therefore, we get   −σ(β^)≥(2.3)−σ(L)−c−12=b1(β^)−c+1≥b1(β^)−b1(β^)2+1. If nβ=1, we can assume (after possibly a conjugation) that β=Δα with   α=a1l0a2l1a1l2…a2lc−1a1lcfor some even integer c≥2, where for all i∈{0,…,c}, the li are integers larger than or equal to 2, by a similar argument as in the case of nβ=0. Indeed, for the first part of the statement, if a positive braid word for α starts with a power of a1 ( a2) and ends with a power of a2 ( a1), then a cyclic permutation and the braid relation a2kΔ=Δa1k (the braid relation Δa2k=a1kΔ and a cyclic permutation) allow to find a conjugate of β for which α starts and ends with powers of a1. And c≥2 can be assumed since the case c=0, that is β=Δa1l (which has closure T2,2+l), was dealt with already. For the second part of the statement, we observe that li=1 for at least one i allows to find a positive braid word for a conjugate of β that starts with Δ2, which is impossible by the definition of nβ. Note that b1(β^)=b1(α^)+3≥2c+3. By a similar argument as in the case of nβ=0, we get a link L that is a connected sum of c2+1 torus links of braid index two by c2 saddle moves on β^ that separate all of the a2li blocks in α and we have   −σ(L)=b1(L)=b1(β^)−c2. And, therefore, we get   −σ(β^)≥(2.3)−σ(L)−c2=b1(β^)−c≥b1(β^)−b1(β^)−32≥b1(β^)2+32. Similar arguments work for nβ=2 or 3. A small difference occurs: the link L will be a connected sum of braid index two torus knots and Δ2a1l^ for some positive integer l (instead of just braid index two torus knots); however, −σ(Δ2a1l^)=b1(Δ2a1l^) (see, for example [2]) and so the argument remains the same.□ Figure 5. View largeDownload slide Top: the link α^ with c−12 spheres (dotted circles) that indicate the saddle moves which yield L. Bottom: the link L with c+12 spheres (dotted circles) that indicate how L is separated into c+32 summands. Figure 5. View largeDownload slide Top: the link α^ with c−12 spheres (dotted circles) that indicate the saddle moves which yield L. Bottom: the link L with c+12 spheres (dotted circles) that indicate how L is separated into c+32 summands. 5. Sharpness of the signature bounds: examples of positive 3-braids and 4-braids of small signature In this section, we provide examples that show that the linear bounds provided in Theorem 1.1 and Proposition 1.5 are essentially optimal. For every positive integer n, we study the following families of braids. The positive 3-braids   αn=(a12a22)2n+1andαn′=a2(a12a22)2n+1and the positive 4-braids   βn=(a1a3a22)2n+1andβn′=a2(a1a3a22)2n+1.We remark that the closure of βn′ is a knot for all n. Proposition 5.1 For all positive integers n, the signature of the closures of αn, αn′, β and βn′are equal to the bounds provided in Theorem1.1and Proposition1.5, respectively:   −σ(αn^)=4n+2=8n+22+1=b1(αn^)2+1,−σ(αn′^)=4n+3=⌈8n+32+1⌉=⌈b1(αn′^)2+1⌉,−σ(βn^)=4n+1=8n+12+12=b1(βn′^)2+12,and−σ(βn′^)=4n+2=⌈8n+22+12⌉=⌈b1(βn′^)2+12⌉. Proposition 5.1 provides infinitely many examples of 3-braids and 4-braids of even and odd Betti numbers for which the bounds of Theorem 1.1 and Proposition 1.5, respectively, are realized. This proves that the bounds are optimal among all linear expressions in the Betti number with coefficients in the half integers. However, there is still room for improving the bounds from Theorem 1.1 and Proposition 1.5. For example, for closures of positive 3-braids and 4-braids (in fact, for all positive braid links) with Betti number 5 or less, the Betti number equals ∣σ∣ (see [2]); but this is not reflected in the bounds provided. More interestingly, what about other Betti numbers that do not occur as Betti numbers of the closures of the above families? Proof of Proposition 5.1 First, we calculate σ(βn^). The link βn^ is the (unoriented) boundary of the embedded annulus A⊂R3 obtained from the blackboard-framed standard link diagram for the T2,2n+1 torus knot. In other words, A is the framed knot of knot type T2,2n+1 and framing −4n−2 (where the zero-framing is identified with the homological framing). For the next bit, we use the terminology of [10]: the Euler number e¯(A) equals twice the framing of A and the signature σ(A)=σ(GA) equals 1. Therefore, we have   σ(βn^)=σ(GA)+e¯(A)2=1−4n−2=−4n−1. To calculate the signature of βn′^, we observe that βn′ is obtained from βn by adding one generator a2. In other words, we can smooth one crossing in βn′^ to obtain βn^. Therefore, we have   −σ(βn′^)≤(2.3)−σ(βn^)+1=4n+2,which yields −σ(βn′^)=4n+2 since −σ(βn′^)≥4n+2 by Theorem 1.1. To calculate σ(αn^) and σ(αn′^), we note that αn and αn′ are obtained from βn and βn′, respectively, by replacing all generators a3 with a1. Following the argument in the proof of Theorem 1.1, this yields   4n+1=−σ(βn^)≥−σ(αn^)−1and4n+2=−σ(βn′^)≥−σ(αn′^)−1. (5.1)This finishes the proof since the inequalities in (5.1) are equalities by Proposition 1.5.□ Funding The author gratefully acknowledges support by the Swiss National Science Foundation Grant 155477. Acknowledgements I thank Josh Greene for sharing his work on checkerboard surfaces, which led me to consider to use them to prove Theorem 1.1. Thanks also to Sebastian Baader for helpful remarks. I thank both of them for the fun hours we spent calculating signatures. References 1 E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg  4 ( 1925), 47– 72. Google Scholar CrossRef Search ADS   2 S. Baader, Positive braids of maximal signature, Enseign. Math.  59 ( 2013), 351– 358. ArXiv:1211.4824[math.GT]. Google Scholar CrossRef Search ADS   3 S. Baader, P. Dehornoy and L. Liechti, Signature and concordance of positive knots, ArXiv e-prints, ArXiv:1503.01946 [math.GT]. 4 S. Baader, P. Feller, L. Lewark and L. Liechti, On the topological 4-genus of torus knots, Trans. Amer. Math. 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Math.  76 ( 1962), 464– 498. Google Scholar CrossRef Search ADS   Appendix A. Reduction of Theorem 1.2 to Theorem 1.1 By [7, Reduction Lemma], Theorem 1.1 implies Theorem 1.2. For the reader’s convenience and the sake of completeness, we recall the argument. The idea of the proof is first to smooth crossings in a given positive braid link such that a connected sum of closures of positive braids on 4 or fewer strands remains, and then to apply Theorem 1.1 to these summands. Proof of Theorem 1.2 We fix a positive integer n and let β be a non-trivial positive braid in Bn. Without loss of generality, β is not a non-trivial split union of links; in other words, every generator ai with 1≤i≤n−1 is contained in β at least once. For i in {1,2,3,4}, we denote by β(i) the braid obtained from β by smoothing the crossings corresponding to all but one (say the leftmost) ak for all k in {i,i+4,i+8,i+12,…} as illustrated in Fig. 6. The closure of such a β(i) is a connected sum of closures of positive braids on 4 or fewer strands. Since we have b1(β^)=(2.5)∑k=1n−1(♯{akinβ}−1), there exists an i such that   b1(β(i)^)≥34b1(β^). (1.2)We fix such an i. Let L1,…,Ll be closures of positive braids on at most 4 strands such that the closure of β(i) is the connected sum of the Lj. Therefore, we have   −σ(β(i)^)=(2.4)−∑j=1lσ(Lj)>Theorem1.1∑j=1lb1(Lj)2=b1(β(i)^)2≥(1.2)3b1(β^)8.The braid β(i) is obtained from β by smoothing b1(β^)−b1(β(i)^)≤14b1(β^) crossings. By (2.3), smoothing a crossing changes the signature by at most ±1; thus, we get   −σ(β^)≥−14b1(β^)−σ(β(i)^)>−14b1(β^)+3b1(β^)8=b1(β^)8.□ Figure 6. View largeDownload slide A diagram of a positive 12-braid β (left) with indications (dotted circles) which crossings to smooth to obtain β(4) (right). The closure of β(4) is a connected sum of the closures of three 4-braids. Figure 6. View largeDownload slide A diagram of a positive 12-braid β (left) with indications (dotted circles) which crossings to smooth to obtain β(4) (right). The closure of β(4) is a connected sum of the closures of three 4-braids. © 2017. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Quarterly Journal of Mathematics Oxford University Press

A sharp signature bound for positive four-braids

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Abstract

Abstract We provide the optimal linear bound for the signature of positive four-braids in terms of the three-genus of their closures. As a consequence, we improve previously known linear bounds for the signature in terms of the first Betti number for all positive braid links. We obtain our results by combining bounds for positive three-braids with Gordon and Litherland’s approach to signature via unoriented surfaces and their Goeritz forms. Examples of families of positive four-braids for which the bounds are sharp are provided. 1. Introduction This paper is concerned with positive braid knots and links—the knots and links obtained as the closure β^ of positive braids β—and the following two link invariants: the first Betti number b1(L) of a link L—the minimal first Betti number of oriented surfaces in R3 with oriented boundary L—and the signature σ(L) of a link L as introduced by Trotter [17] and (for links) by Murasugi [13]. Let a link L be the closure of a non-trivial positive braid—a positive braid such that its closure is not an unlink. We suspect (as conjectured in [7]) that   b1(L)≥−σ(L)>b1(L)2, (1.1)where the first inequality is immediate from the definition of the signature. This article establishes (1.1) for closures of positive 4-braids. Theorem 1.1 Let β be a positive 4-braid such that its closure β^is not an unlink, then  −σ(β^)>b1(β^)2. By an observation [7, Reduction Lemma], which we recall in the appendix for the reader’s convenience, Theorem 1.1 implies the following bound for all positive braids: Theorem 1.2 Let β be a positive braid such that its closure β^is not an unlink, then  −σ(β^)>b1(β^)8. In other words, up to a factor of 4, (1.1) holds. The main technical ingredient in the proof of Theorem 1.1 is Gordon and Litherland’s approach of using (non-oriented) checkerboard surfaces and the associated Goeritz form to calculate the signature [10]. Let us shortly put Theorems 1.1 and 1.2 in context. Links that arise as closures of positive braids are a well-studied class of links containing important families such as (positive) torus links, algebraic links and Lorenz links, while themselves being a subclass of positive links. Rudolph established that closures of positive braids have strictly negative signature [14]. For positive 4-braids, previous results by Stoimenow [16, Theorem 4.2] and the author [7, Main Proposition] provided linear bounds of the signature in terms of the first Betti number with factor 211 and 512, respectively. For the more general class of positive links, Baader, Dehornoy and Liechti provide a linear bound for the signature in terms of the first Betti number with factor 148 [3, Theorem 2]. The novelty of Theorem 1.1 is that the factor 12 of the linear bound is optimal; compare the discussion in Section 5. The linear bound with factor 18 for all positive braids as provided in Theorem 1.2 is a clear improvement over the best-known previous bound; compare [7] and [3, Theorem 2]. A geometric consequence of the signature bounds are lower bounds for the topological slice genus g4top(K) of a knot K—the minimal genus among all locally flat oriented surfaces in the unit 4-ball B4 with boundary K⊂S3=∂B4. Indeed, combining Theorems 1.1 and 1.2 with Kauffman and Taylor’s results that ∣σ(K)∣≤2g4top(K) for all knots K [11] yields the following. For a knot K, denote the three-genus—half the first Betti number of K—by g(K). Corollary 1.3. For a knot K that is not the unknot and that is the closure of a positive braid, one has  g(K)≥g4top(K)>g(K)8.Furthermore, if K is the closure of a positive 4-braid, then  g(K)≥g4top(K)>g(K)2. Remark 1.4 There are knots K that are closures of positive braids for which g(K) is strictly larger than g4top(K), which is surprising since in the smooth setting, the smooth slice genus is equal to the three-genus by Kronheimer and Mrowka’s resolution of the Thom conjecture [12, Corollary 1.3]. Indeed, Rudolph observed that the topological slice genus of the torus knot T5,6 is strictly less than 10=g(T5,6). In fact, there are infinite families of positive braid knots for which the topological slice genus can be linearly bounded away from the three-genus by a significant amount [4, 15, Theorem 2]. This justifies interest in a linear lower bound as provided in Corollary 1.3. We conclude the introduction by outlining the strategy of the proof of Theorem 1.1, as provided in Section 3. We will use Gordon and Litherland’s approach to the signature via Goeritz forms to show the following. For every link L that is the closure of a non-trivial positive 4-braid, there exists a link L′ that is the closure of a positive 3-braid with   b1(L′)=b1(L)+1and∣σ(L)−σ(L′)∣≤1.This will allow us to reduce Theorem 1.1 to the following proposition. Proposition 1.5 Let β be a positive 3-braid with b1(β^)≥2, then  −σ(β^)≥b1(β^)2+1. Proposition 1.5 improves Stoimenow’s result that −σ(β^)>b1(β^)2 for positive non-trivial 3-braids [16, Theorem 4.1]. We provide a proof for Proposition 1.5 which is independent of Stoimenow’s techniques; see Section 4. Optimality of Theorem 1.1 and Proposition 1.5 is discussed in Section 5. 2. Setup: signatures of links via Goeritz forms and positive braids We set up notions and recall facts about braids and the signature of links. 2.1. Signature of links and Goeritz forms For a link L—an oriented smooth embedding of a non-empty finite union of circles in S3—the signature, denoted by σ(L), is defined to be the signature of the symmetrized Seifert form on H1(F), where F is any compact and oriented surface in S3 with oriented boundary L; compare Trotter and Murasugi [13, 17]. In particular, one has that −b1≤σ≤b1 holds for all links. Unifying Trotter’s approach to the signature and work of Goeritz [9], Gordon and Litherland [10] introduced the following procedure to calculate the signature. For any link diagram DL of L—the image of a generic project of the link L to a standard 2–sphere R2∪{∞} in S3 together with crossing information—one has   σ(L)=σ(SL)−μ(SL),where SL is a non-oriented surface with boundary L given as one of the two checkerboard surfaces ( SL is contained in DL⊂S3 away from neighborhoods of crossings. In a neighborhood of a crossing, SL is given by a small ‘half-twisted’ band. We refer to Fig. 4 for an illustrative example and to Gordon and Litherland’s original work [10] for more details) of DL and σ(SL) and μ(SL) are defined as follows. To every crossing c of DL, one associates a type (I or II) and a sign η(c) (1 or −1) by the rule specified in Fig. 1. Then one defines   μ(SL)=∑c∈crossingsofLoftypeIIη(c). (2.1)To define σ(SL), pick a basis [δ1],…,[δk] of H1(SL) represented by simple closed curves δi⊂SL and let the matrix GL={gij} be given by gij=lk(δi,δj±). Here δj± denotes the link in S3⧹SL obtained from δj⊂SL by a small push-off in both normal directions of SL and lk denotes the linking number in S3. Then one sets σ(SL)=σ(GL), where σ(GL) denotes the signature of GL—the number of positive eigenvalues minus the number of negative eigenvalues counted with multiplicities. The bilinear form defined by GL is called the Goeritz form. This fits into the setting of the more general Gordon–Litherland pairing, where one uses any (in general non-orientable) surface SL with (unoriented) boundary L rather than a checkerboard surface and −μ(SL) is replaced by half the Euler number of SL; see [10, Corollary 5]. A warning concerning sign conventions is in order: the above definition of σ(L) has opposite sign of that given in [3, 14, 16]. The present convention agrees with the convention in [10, 13, 17] and appears to be the standard one. For example,   σ(T2,n+1)=−n(ratherthann)forallpositiveintegersn. (2.2) Figure 1. View largeDownload slide Left: sign associated to a crossing. Right: type associated to a crossing. The type only depends on whether or not the surface SL (gray) can locally be given an orientation that induces the orientation of the link. In particular, the type is independent of the crossing information. Figure 1. View largeDownload slide Left: sign associated to a crossing. Right: type associated to a crossing. The type only depends on whether or not the surface SL (gray) can locally be given an orientation that induces the orientation of the link. In particular, the type is independent of the crossing information. Figure 4. View largeDownload slide The checkerboard surfaces Sβ and Sα (gray) associated with the braid words β=a1a3a2a1a3a2a2a1a3a2 (left) and α=a1a1a2a1a1a2a2a1a1a2 (right). The curves γi and δi constitute bases of H1(Sβ) and H1(Sα), respectively. Figure 4. View largeDownload slide The checkerboard surfaces Sβ and Sα (gray) associated with the braid words β=a1a3a2a1a3a2a2a1a3a2 (left) and α=a1a1a2a1a1a2a2a1a1a2 (right). The curves γi and δi constitute bases of H1(Sβ) and H1(Sα), respectively. The following properties of the signature of a link follow rather directly from both the original definition and Gordon and Litherland’s approach. If a link L′ can be obtained from a link L by one saddle move, then   ∣σ(L)−σ(L′)∣≤1[13,Lemma7.1]. (2.3) Here a saddle move is defined as changing the link in a 3-ball as described on the left-hand side in Fig. 2. If L′ can be obtained from L by smoothing of a crossing, defined on the right-hand side in Fig. 2, then (2.3) also holds since a smoothing of a crossing corresponds to a saddle move on the link. The signature is additive under split union and connected sum: for all links L and L′  σ(L⊔L′)=σ(L♯L′)=σ(L)+σ(L′)[13,Lemmas7.2and7.3]. (2.4) Note that the connected sum L♯L′ is only well defined if one specifies which components of the two links L and L′ are connected via the connect sum operation; however, (2.4) holds for all such specifications. Figure 2. View largeDownload slide Left: a saddle move. Right: a smoothing of a crossing, which is obtained by applying a saddle move to the sphere indicated by the dotted ellipse. Figure 2. View largeDownload slide Left: a saddle move. Right: a smoothing of a crossing, which is obtained by applying a saddle move to the sphere indicated by the dotted ellipse. 2.2. Positive braids Let Bb be Artin’s braid group on b strands [1]:   Bb=⟨a1,…,ab−1∣aiaj=ajaifor∣i−j∣≥2,aiajai=ajaiajfor∣i−j∣=1⟩.A braid β in Bb corresponds to a (geometric) braid, represented via braid diagrams, and yields a link β^ via the closure operation. Generators ai (respectively ai−1) correspond to braids given by the braid diagram where the ith and (i+1)st strands cross once positively (respectively, negatively). This yields that a braid word—a word in the generators of Bb—defines a diagram for the braid it encodes. We illustrate this in Fig. 3 for one example to set conventions and refer to [6] for a detailed account. Figure 3. View largeDownload slide A braid diagram (left) coming from the braid word a1−1a2a2a1a3a2 and a link diagram (right) for the corresponding braid closure. Figure 3. View largeDownload slide A braid diagram (left) coming from the braid word a1−1a2a2a1a3a2 and a link diagram (right) for the corresponding braid closure. A positive braid on b strands is an element β in Bb that can be written as positive braid word as1as2…asl(β) with si∈{1,…,b−1}, where l(β) is called the length or writhe of β. Note that l(β) is an invariant of positive braids β since it is independent of the choice of a positive braid word for β. The first Betti number of the closure of positive braids is understood as a consequence of Bennequin’s inequality [5, Theorem 3], which implies the following formula:   b1(β^)=l(β)−b+cforeverypositivebraidβ, (2.5)where b is the number of strands of β and c equals 1 plus the number of generators ai that are not used in a positive braid word for β. For example, for all positive integers n and m, the torus link Tn,m is a positive braid link with first Betti number (n−1)(m−1) since the closure of the braid (a1⋯an−1)m∈Bn is Tn,m. 3. Proof of Theorem 1.1 In this section, we prove that if a link L with b1(L)≥1 is the closure of a positive 4-braid, then   −σ(L)≥b1(L)2+12>b1(L)2.Besides the notions introduced in Section 2, the proof uses Proposition 1.5, which is proved in Section 4. Proof of Theorem 1.1 For the entire proof, let L be the closure of a non-trivial positive 4-braid given by a positive 4-braid word β. Without loss of generality all generators a1, a2 and a3 appear in β at least twice; in particular, b1(L)≥3 by (2.5). Indeed, otherwise L is a connected sum or split union of links that are closures of positive braids on 3 or less strands, for which the statement follows from Proposition 1.5, (2.2) and the fact that signature is additive on connect sums and split unions. We define a 4-braid word α by replacing all a3 in β by a1. For example, if β=a1a3a2a1a3a2a2a1a3a2, then α=a1a1a2a1a1a2a2a1a1a2. From the given braid word β, we get a braid diagram and a corresponding link diagram Dβ (compare Section 2.2) representing L=β^. Similarly, the braid word α yields a link diagram Dα for α^. Next we checkerboard color the two diagrams Dβ and Dα (we choose the convention that the unbounded region of a diagram is white) and we denote the black checkerboard surfaces by Sβ and Sα, respectively. See Fig. 4, where this is illustrated for an example. We note that the diagrams Dβ and Dα and the checkerboard surfaces Sβ and Sα depend on the braid words β and α, respectively (rather than just depending on the braids that β and α represent in B4). Let k denote the number of a2 in the braid word β. The number of a2 in the braid word α is also k. The checkerboard surfaces Sβ and Sα have first Betti number k+1. This is best seen by observing that the Betti number of Sβ and Sα are equal to the number of bounded white regions in Dβ and Dα, respectively, of which there are k+1. We describe this in a bit more detail and set up a one-to-one correspondence between the bounded white regions of Dβ and Dα, which will be useful below. Every a2 in β corresponds to a crossing in Dβ, which is touched by two white regions (one from above and one from below). We label the generators a2 in β by 1, 2,…, k in the order of appearance from the left and assign the same label to the unique white region touching the corresponding crossings from above. Only one bounded white region remains. We label this bounded white region by k+1. The same labeling procedure applied to α yields a labeling for the white bounded regions of Dα and this sets up a one-to-one correspondence between the bounded white regions of Dβ and Dα. See Fig. 4, where this labeling is illustrated. Following Goeritz and Gordon–Litherland [9, 10], we choose bases ([γ1],…,[γk]) and ([δ1],…,[δk]) for H1(Sβ) and H1(Sα), respectively, as follows. The boundary of a white region in Dβ (respectively, Dα) labeled i defines a curve γi (respectively, δi) in Sβ (respectively, Sα). We orient the γi and δi counterclockwise and let [γi] and [δi] denote the corresponding homology classes in H1(Sβ) and H1(Sα), respectively. Let Gβ and Gα denote the Goeritz matrices of Sβ and Sα with respect to the bases ([γ1],…,[γk]) and ([δ1],…,[δk]), respectively. The above one-to-one correspondence between the white bounded regions of Dβ and Dα is set up such that for all i,j≤k,   lk(γi,γj±)=lk(δi,δj±). In other words, all entries of Gα not in the last row or column coincide with the corresponding entries of Gβ. Since the signatures of two real-valued symmetric (k+1)×(k+1) matrices that are identical except in the last column and row differ by at most one, we have   −σ(Gβ)≥−σ(Gα)−1. (3.1) Furthermore, μ(Sβ)=μ(Sα), since μ(Sβ) (respectively, μ(Sα)) is equal to the number of a1 and a3 in β (respectively, α) by (2.1). We note that α^ is the split union of the unknot and the link L′ obtained as the closure of the 3-braid given by interpreting α as a 3-braid word. Therefore, σ(L′)=σ(α^) by (2.4). Note also that b1(L′)=b1(L)+1 by (2.5). With all of the above and Gordon and Litherland’s   σ(L)=σ(Gβ)−μ(Sβ)andσ(α^)=σ(Gα)−μ(Sα),we calculate   −σ(L)=−σ(Gβ)+μ(Sβ)≥−σ(Gα)−1+μ(Sβ)=−σ(Gα)−1+μ(Sα)=−σ(α^)−1=−σ(L′)−1≥b1(L′)2+1−1=b1(L)+12,where we used (3.1), μ(Sβ)=μ(Sα), Proposition 1.5 and (2.5) in the second, third, sixth and last line, respectively. This concludes the proof of   −σ(L)≥b1(L)+12>b1(L)2,for all closures L of non-trivial positive 4-braids.□ 4. Proof of Proposition 1.5 In this section, we prove that for all positive 3-braids β with b1(β^)≥2, we have   −σ(β^)≥b1(β^)2+1. (4.1) Proof of Proposition 1.5 Let β be a positive 3-braid with b1(β^)≥2. Denote by Δ the positive half-twist on 3-strands a1a2a1=a2a1a2. Let nβ be the maximum of all non-negative integer n′ such that β=Δn′α for a positive 3-braid α. Note that nβ exists since the n′ are less or equal than a third of the length l(β). Since we are interested in the signature and the first Betti number of the closure of β, there is no loss of generality by assuming that nβ is maximal among all such n′ for β′=Δn′α′, where β′ is any positive braid conjugate to β. This maximality of nβ is assumed in the entire proof. For example, the braid β=a1a2a2a2a2, which has nβ=0, is not considered since it is conjugate to a2a1a2a2a2=Δa2a2. Furthermore, we assume in the entire proof that β^ is not the closure of a 2-braid since for such closures one has   −σ(β^)=(2.2)l(β)−1=(2.5)b1(β^)≥b1(β^)2+1.For example, for all positive integers l, the braid β=Δa1l is not considered since its closure is the braid index two torus link T2,l+2. If −σ(β^)≥b1(β^)2 holds for a positive 3-braid β, then one also has   −σ((Δ4)k^β)≥b1((Δ4)k^β)2+1,for all positive integers k. This can be seen as follows. Gambaudo and Ghys [8, Lemma 4.1] established that   −σ((Δ4)k^β)=−σ(β^)+8k[8,Lemma4.1], (4.2)for all 3-braids β and all integers k. Thus, −σ(β^)≥b1(β^)2 yields   −σ((Δ4)k^β)=(4.2)−σ(β^)+8k≥b1(β^)2+8k≥(2.5)b1((Δ4)k^β)2+2k≥b1((Δ4)k^β)2+1.Therefore, it suffices to establish (4.1) for positive 3-braids β with b1(β^)≥2 and nβ=0,1,2, or 3. Let us first consider the case nβ=0. After possibly a conjugation (given by cyclic permutation of positive braid words), we have that β is given by a positive braid word starting with a1 and ending with a2, i.e.   β=a1l0a2l1a1l2…a1lc−1a2lcfor some odd integer c≥1 and positive integers li. Furthermore, we have that for all i∈{0,…,c} the li are integers larger than or equal to 2 since otherwise (up to cyclic permutation) β will contain a1a2a1 or a2a1a2 as a subword which is impossible by the definition of nβ. Note that b1(β^)≥2c by (2.5). By applying c−12 saddle moves on β^, we get a link L that is a connected sum of c+32 torus links of braid index two; in fact, a connect sum of the torus link T2,leven, where leven=∑i=0c−12l2i, and the torus links T2,lj for odd j. This is possible by making saddle moves that ‘separate’ all but one of the a2li-blocks as illustrated in Fig. 5. Thus, we have   −σ(L)=(2.4),(2.2)b1(L)=b1(T2,leven)+∑i=0c−12b1(T2,l2i+1)=b1(β^)−c−12.And, therefore, we get   −σ(β^)≥(2.3)−σ(L)−c−12=b1(β^)−c+1≥b1(β^)−b1(β^)2+1. If nβ=1, we can assume (after possibly a conjugation) that β=Δα with   α=a1l0a2l1a1l2…a2lc−1a1lcfor some even integer c≥2, where for all i∈{0,…,c}, the li are integers larger than or equal to 2, by a similar argument as in the case of nβ=0. Indeed, for the first part of the statement, if a positive braid word for α starts with a power of a1 ( a2) and ends with a power of a2 ( a1), then a cyclic permutation and the braid relation a2kΔ=Δa1k (the braid relation Δa2k=a1kΔ and a cyclic permutation) allow to find a conjugate of β for which α starts and ends with powers of a1. And c≥2 can be assumed since the case c=0, that is β=Δa1l (which has closure T2,2+l), was dealt with already. For the second part of the statement, we observe that li=1 for at least one i allows to find a positive braid word for a conjugate of β that starts with Δ2, which is impossible by the definition of nβ. Note that b1(β^)=b1(α^)+3≥2c+3. By a similar argument as in the case of nβ=0, we get a link L that is a connected sum of c2+1 torus links of braid index two by c2 saddle moves on β^ that separate all of the a2li blocks in α and we have   −σ(L)=b1(L)=b1(β^)−c2. And, therefore, we get   −σ(β^)≥(2.3)−σ(L)−c2=b1(β^)−c≥b1(β^)−b1(β^)−32≥b1(β^)2+32. Similar arguments work for nβ=2 or 3. A small difference occurs: the link L will be a connected sum of braid index two torus knots and Δ2a1l^ for some positive integer l (instead of just braid index two torus knots); however, −σ(Δ2a1l^)=b1(Δ2a1l^) (see, for example [2]) and so the argument remains the same.□ Figure 5. View largeDownload slide Top: the link α^ with c−12 spheres (dotted circles) that indicate the saddle moves which yield L. Bottom: the link L with c+12 spheres (dotted circles) that indicate how L is separated into c+32 summands. Figure 5. View largeDownload slide Top: the link α^ with c−12 spheres (dotted circles) that indicate the saddle moves which yield L. Bottom: the link L with c+12 spheres (dotted circles) that indicate how L is separated into c+32 summands. 5. Sharpness of the signature bounds: examples of positive 3-braids and 4-braids of small signature In this section, we provide examples that show that the linear bounds provided in Theorem 1.1 and Proposition 1.5 are essentially optimal. For every positive integer n, we study the following families of braids. The positive 3-braids   αn=(a12a22)2n+1andαn′=a2(a12a22)2n+1and the positive 4-braids   βn=(a1a3a22)2n+1andβn′=a2(a1a3a22)2n+1.We remark that the closure of βn′ is a knot for all n. Proposition 5.1 For all positive integers n, the signature of the closures of αn, αn′, β and βn′are equal to the bounds provided in Theorem1.1and Proposition1.5, respectively:   −σ(αn^)=4n+2=8n+22+1=b1(αn^)2+1,−σ(αn′^)=4n+3=⌈8n+32+1⌉=⌈b1(αn′^)2+1⌉,−σ(βn^)=4n+1=8n+12+12=b1(βn′^)2+12,and−σ(βn′^)=4n+2=⌈8n+22+12⌉=⌈b1(βn′^)2+12⌉. Proposition 5.1 provides infinitely many examples of 3-braids and 4-braids of even and odd Betti numbers for which the bounds of Theorem 1.1 and Proposition 1.5, respectively, are realized. This proves that the bounds are optimal among all linear expressions in the Betti number with coefficients in the half integers. However, there is still room for improving the bounds from Theorem 1.1 and Proposition 1.5. For example, for closures of positive 3-braids and 4-braids (in fact, for all positive braid links) with Betti number 5 or less, the Betti number equals ∣σ∣ (see [2]); but this is not reflected in the bounds provided. More interestingly, what about other Betti numbers that do not occur as Betti numbers of the closures of the above families? Proof of Proposition 5.1 First, we calculate σ(βn^). The link βn^ is the (unoriented) boundary of the embedded annulus A⊂R3 obtained from the blackboard-framed standard link diagram for the T2,2n+1 torus knot. In other words, A is the framed knot of knot type T2,2n+1 and framing −4n−2 (where the zero-framing is identified with the homological framing). For the next bit, we use the terminology of [10]: the Euler number e¯(A) equals twice the framing of A and the signature σ(A)=σ(GA) equals 1. Therefore, we have   σ(βn^)=σ(GA)+e¯(A)2=1−4n−2=−4n−1. To calculate the signature of βn′^, we observe that βn′ is obtained from βn by adding one generator a2. In other words, we can smooth one crossing in βn′^ to obtain βn^. Therefore, we have   −σ(βn′^)≤(2.3)−σ(βn^)+1=4n+2,which yields −σ(βn′^)=4n+2 since −σ(βn′^)≥4n+2 by Theorem 1.1. To calculate σ(αn^) and σ(αn′^), we note that αn and αn′ are obtained from βn and βn′, respectively, by replacing all generators a3 with a1. Following the argument in the proof of Theorem 1.1, this yields   4n+1=−σ(βn^)≥−σ(αn^)−1and4n+2=−σ(βn′^)≥−σ(αn′^)−1. (5.1)This finishes the proof since the inequalities in (5.1) are equalities by Proposition 1.5.□ Funding The author gratefully acknowledges support by the Swiss National Science Foundation Grant 155477. Acknowledgements I thank Josh Greene for sharing his work on checkerboard surfaces, which led me to consider to use them to prove Theorem 1.1. Thanks also to Sebastian Baader for helpful remarks. I thank both of them for the fun hours we spent calculating signatures. References 1 E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg  4 ( 1925), 47– 72. Google Scholar CrossRef Search ADS   2 S. Baader, Positive braids of maximal signature, Enseign. Math.  59 ( 2013), 351– 358. ArXiv:1211.4824[math.GT]. Google Scholar CrossRef Search ADS   3 S. Baader, P. Dehornoy and L. Liechti, Signature and concordance of positive knots, ArXiv e-prints, ArXiv:1503.01946 [math.GT]. 4 S. Baader, P. Feller, L. Lewark and L. Liechti, On the topological 4-genus of torus knots, Trans. Amer. Math. Soc., Accepted for publication. ArXiv:1509.07634 [math.GT]. 5 D. Bennequin, Entrelacements et équations de Pfaff, Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), Vol. 107 of Astérisque, Soc. Math. France, Paris, 1983, pp. 87–161. 6 J. S. Birman, Braids, Links, and Mapping Class Groups , Princeton University Press, Princeton, NJ, 1974, Annals of Mathematics Studies, No. 82. 7 P. Feller, The signature of positive braids is linearly bounded by their first Betti number, Int. J. Math.  26 ( 2015), PP. 1550081, 14. ArXiv:1311.1242 [math.GT]. Google Scholar CrossRef Search ADS   8 J.-M. Gambaudo and É. Ghys, Braids and signatures, Bull. Soc. Math. France  133 ( 2005), 541– 579. Google Scholar CrossRef Search ADS   9 L. Goeritz, Knoten und quadratische Formen, Math. Z.  36 ( 1933), 647– 654. Google Scholar CrossRef Search ADS   10 C. M. Gordon and R. A. Litherland, On the signature of a link, Invent. Math.  47 ( 1978), 53– 69. Google Scholar CrossRef Search ADS   11 L. H. Kauffman and L. R. Taylor, Signature of links, Trans. Amer. Math. Soc.  216 ( 1976), 351– 365. Google Scholar CrossRef Search ADS   12 P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces, I, Topology  32 ( 1993), 773– 826. Google Scholar CrossRef Search ADS   13 K. Murasugi, On a certain numerical invariant of link types, Trans. Amer. Math. Soc.  117 ( 1965), 387– 422. Google Scholar CrossRef Search ADS   14 L. Rudolph, Nontrivial positive braids have positive signature, Topology  21 ( 1982), 325– 327. Google Scholar CrossRef Search ADS   15 L. Rudolph, Some topologically locally-flat surfaces in the complex projective plane, Comment. Math. Helv.  59 ( 1984), 592– 599. Google Scholar CrossRef Search ADS   16 A. Stoimenow, Bennequin’s inequality and the positivity of the signature, Trans. Amer. Math. Soc.  360 ( 2008), 5173– 5199. Google Scholar CrossRef Search ADS   17 H. F. Trotter, Homology of group systems with applications to knot theory, Ann. Math.  76 ( 1962), 464– 498. Google Scholar CrossRef Search ADS   Appendix A. Reduction of Theorem 1.2 to Theorem 1.1 By [7, Reduction Lemma], Theorem 1.1 implies Theorem 1.2. For the reader’s convenience and the sake of completeness, we recall the argument. The idea of the proof is first to smooth crossings in a given positive braid link such that a connected sum of closures of positive braids on 4 or fewer strands remains, and then to apply Theorem 1.1 to these summands. Proof of Theorem 1.2 We fix a positive integer n and let β be a non-trivial positive braid in Bn. Without loss of generality, β is not a non-trivial split union of links; in other words, every generator ai with 1≤i≤n−1 is contained in β at least once. For i in {1,2,3,4}, we denote by β(i) the braid obtained from β by smoothing the crossings corresponding to all but one (say the leftmost) ak for all k in {i,i+4,i+8,i+12,…} as illustrated in Fig. 6. The closure of such a β(i) is a connected sum of closures of positive braids on 4 or fewer strands. Since we have b1(β^)=(2.5)∑k=1n−1(♯{akinβ}−1), there exists an i such that   b1(β(i)^)≥34b1(β^). (1.2)We fix such an i. Let L1,…,Ll be closures of positive braids on at most 4 strands such that the closure of β(i) is the connected sum of the Lj. Therefore, we have   −σ(β(i)^)=(2.4)−∑j=1lσ(Lj)>Theorem1.1∑j=1lb1(Lj)2=b1(β(i)^)2≥(1.2)3b1(β^)8.The braid β(i) is obtained from β by smoothing b1(β^)−b1(β(i)^)≤14b1(β^) crossings. By (2.3), smoothing a crossing changes the signature by at most ±1; thus, we get   −σ(β^)≥−14b1(β^)−σ(β(i)^)>−14b1(β^)+3b1(β^)8=b1(β^)8.□ Figure 6. View largeDownload slide A diagram of a positive 12-braid β (left) with indications (dotted circles) which crossings to smooth to obtain β(4) (right). The closure of β(4) is a connected sum of the closures of three 4-braids. Figure 6. View largeDownload slide A diagram of a positive 12-braid β (left) with indications (dotted circles) which crossings to smooth to obtain β(4) (right). The closure of β(4) is a connected sum of the closures of three 4-braids. © 2017. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oup.com

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The Quarterly Journal of MathematicsOxford University Press

Published: Mar 1, 2018

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