THE HOMOLOGY POLYNOMIAL AND THE BURAU REPRESENTATION FOR PSEUDO-ANOSOV BRAIDS By Warren Michael Shultz A DISSERTATION Submitted to Michigan State University in partial fulllment of the requirements for the degree of Mathematics  Doctor of Philosophy 2021 ABSTRACT THE HOMOLOGY POLYNOMIAL AND THE BURAU REPRESENTATION FOR PSEUDO-ANOSOV BRAIDS By Warren Michael Shultz The homology polynomial is an invariant for pseudo-Anosov mapping classes [3]. We study the homology polynomial as an invariant for pseudo-Anosov braids and its connection to the Burau representation. Given a pseudo-Anosov braid β ∈ Bn , we determine necessary and sucient conditions under which the homology polynomial of β is equal to the the characteristic polynomial of the image of β under the Burau representation. In particular, we build upon [1] and show that the orientation cover associated to a pseudo-Anosov braid is equivalent to a quotient to the Burau cover when the measured foliations associated to β have odd-ordered singularities at each puncture and any singularity that occurs in the interior of Dn is even-ordered. We next construct an algorithm which allows us to determine the homology polynomial from the Burau representation for an arbitrary pseudo-Anosov braid. As an application, we show how to easily determine the homology polynomial for large family of pseudo-Anosov braids. ACKNOWLEDGEMENTS First, I want to thank my advisor, Ee Kalfagianni, for her guidance, insight, and extremely generous support. I also want to thank Robert Bell for the time and expertise he shared with me. I also am grateful to Ben Scmidt, Matthew Hedden, and Kristina Hendricks for their involvement in my development as a mathematician. I also want to thank Echo Fields and Wolfgang Rünzi for their guidance. Finally, I want to thank Mollee Shultz for her support, encouragement, and valuable feedback. iii TABLE OF CONTENTS LIST OF FIGURES ...................................... v CHAPTER 1 INTRODUCTION .............................. 1 1.1 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 CHAPTER 2 PRELIMINARIES .............................. 4 2.1 Pseudo-Anosov mapping classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 The Nielsen-Thurston Classication . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Singular measured foliations and pseudo-Anosov mapping classes . . . 5 2.2 Train Tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Measured train tracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The Bestvina-Handel Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 From ecient graph maps to train tracks . . . . . . . . . . . . . . . . . . 14 2.4 The homology polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.1 The orientation cover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4.2 W (G, g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.3 The decomposition h(x) = p(x)s(x) . . . . . . . . . . . . . . . . . . . . . 19 2.5 The Burau representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 CHAPTER 3 THE HOMOLOGY POLYNOMIAL AND ITS CONNECTION TO THE BURAU REPRESENTATION .................... 22 3.1 The Burau estimate quotients of the Burau cover . . . . . . . . . . . . . . . . . 22 3.2 The homology polynomial from the burau representation . . . . . . . . . . . . . 25 3.3 Proof of Theorem 1.0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.2 Odd-ordered singularities in the interior . . . . . . . . . . . . . . . . . . . 27 3.3.3 Even-ordered singularities occurring at punctures . . . . . . . . . . . . . 30 3.3.4 Conclusion and proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 An algorithm for constructing β ′ from β . . . . . . . . . . . . . . . . . . . . . . . 34 CHAPTER 4 EXAMPLES AND APPLICATIONS ................... 35 4.1 Application of Theorem 1.0.2 to a large family of braids . . . . . . . . . . . . . 36 4.2 An example comparing the computation of h(x) from denition and com- puting h(x) using Theorem 1.0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.2.1 An odd ordered singularity in the interior of the disk . . . . . . . . . . . 42 BIBLIOGRAPHY ....................................... 48 iv LIST OF FIGURES Figure 2.1: Saddles for n = 3, 4, 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Figure 2.2: A train track in S0,4 (left) and a 1-complex that is not a train track in S0,3 (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Figure 2.3: τ ⊂ D3 (left) and β(τ ) (right) where β = σ1 σ2−1 . . . . . . . . . . . . . . . . . 9 Figure 2.4: k -junctions for k = 1, 2, 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Figure 2.5: A graph map induced by σ1 σ2−1 . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Figure 2.6: The branches a1 and a2 form a corner. The branch b does not form a corner with either ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Figure 3.1: β (left), β′ (right), τ (above) . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Figure 3.2: Left: ηi , Right: η̂i (edges without labels have weight 0) . . . . . . . . . . . 30 Figure 3.3: τ (left) and τ′ (right) after lling in an order-2 singularity . . . . . . . . . . 33 Figure 4.1: β⋆α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Figure 4.2: β(m1 ,p1 ),...,(mk ,pk ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Figure 4.3: Comparison of βm,p and γm,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Figure 4.4: Trees of star type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Figure 4.5: The tree map gS ∶ TS → TS for S = {(3, 2), (4, 1))}. . . . . . . . . . . . . . . . 41 Figure 4.6: Constructing τ from TS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Figure 4.7: From top to bottom: G, σ1 (G), σ1 σ2 (G), and σ1 σ2 σ3 −1 (G) . . . . . . . . . 44 Figure 4.8: The train track induced by the graph map shown in Figure 4.7. . . . . . . 44 Figure 4.9: The basis element η1 . All other edges are assigned a weight of 0. . . . . . . 45 Figure 4.10: The braid σ1 σ2 σ3 −1 . The dashed line represents the additional strand after declaring the singularity a new puncture resulting in the 5-braid σ1 σ2 σ1 σ3 σ4 −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 v CHAPTER 1 INTRODUCTION The homology polynomial for a pseudo-Anosov mapping class [f ] on a surface S is an integer polynomial invariant h(x) introduced in [3]. If the stable and unstable foliations of [f ] are orientable, then h(x) is associated to the induced action of [f ] on H1 (S, R). It is the product of two additional polynomial invariants h(x) = p(x)s(x) each with topological meaning. By identifying Bn with the mapping class group on the n-punctured disk, the homology polynomial becomes an invariant for pseudo-Anosov braids. However determining the homology polynomial for an arbitrary mapping class can be dicult or impossible in practice. Computing h(x) involves an application of the Bestvina- Handel algorithm ([2]). Software limitations make this impractical for many mapping classes. We will show that for many pseudo-Anosov braids it is trivially easy to determine the homology polynomial using the (reduced) Burau representation. The (reduced) Burau rep- resentation is map Ψ ∶ Bn → GLn−1 (Z[t, 1/t]) The image of β ∈ Bn is ψβ (t) and is called the Burau matrix of β. It is an (n − 1) × (n − 1) matrix with entries from the ring of Laurent polynomials Z[t, 1/t]. Our rst result is a connection between the characteristic polynomial of ψβ (t) and h(x). In Section 3.2 we prove the following: Theorem 1.0.1. Suppose β ∈ Bn is pseudo-Anosov with stretch factor λ and homology polynomial h(β, x). Let (F u , µu ) and (F s , µs ) be the singular measured foliations for β . Finally, let ψβ (t) be the Burau matrix for β and let χ(ψβ (t)) = ∣xI − ψβ (t)∣ Then the following are equivalent 1 (1) χ(ψβ (η)) = h(β, x) for some root of unity η . (2) χ(ψβ (−1)) = h(β, x) and −1 is the only root of unity at which this equality occurs. (3) sr(ψβ (−1)) = λ where sr(ψβ (−1)) is the spectral radius of ψβ (−1). (4) The singularites of F u and F s are odd-ordered if they occur at a puncture and even- ordered if they occur in the interior of Dn . (5) Dn is the orientation double-cover of τ (after attaching a punctured disk to the bound- (2) ary of Dn ). With the above in mind, if Fu and Fs above have an even-ordered singularity at a punc- ture or odd-ordered singularity in the interior we will say that β produces a bad singularity. Suppose that β ∈ Bn is pseudo-Anosov and produces at least one bad singularity. In this case we cannot recover h(x) from ψβ (t) and the direct computation may be dicult. However we can still use the Burau representation to compute h(x) which is our next result. Theorem 1.0.2. Let β0 ∈ Bn be a pseudo-Anosov braid identied with its pseudo-Anosov representative β0 ∶ Dn → Dn . Suppose the measured foliations for β0 have p odd-ordered singularities occurring at interior points x1 , . . . , xp and q even-ordered singularities occurring at punctures p1 , . . . , pq . Let β = β0k where k ≥ 1 is chosen so that β xes each pi and xi pointwise. Identify Dn+q−r with (Dn ∪ {p1 , . . . , pr }) − {x1 , . . . , xq }. Since β xes each xi and pi point- wise it induces a map β ′ ∶ Dn+q−r → Dn+q−r . The braid β ′ ∈ Bn+q−r is pseudo-Anosov with h(β, x) = (1 + x) h(β ′ , x) = ∣xI − ψβ ′ (−1)∣ ε where ε ≥ 0 is the number of order-2 singularities occurring at a puncture. In other words, if β is pseudo-Anosov we can either recover h(x) from ψβ (−1) or we can construct a new braid β′ and recover h(x) from ψβ ′ (−1). 2 Of course, determining the types of singularities produced by a pseudo-Ansov braid is not necessarily an easier task than computing its homology polynomial directly. The usefulness of Theorem 1.0.2 is demonstrated in Chapter 4. In particular, in Section 4.1 we present a large family of pseudo-Anosov braids and use Theorem 1.0.2 to trivialize the computation of h(x) (regardless of the singularity types they produce). 1.1 Organization of Dissertation In Chapter 2 we prove an brief overview of the homology polynomial and the reduced Burau representations along with any necessary preliminaries. In Chapter 3 we prove both Theo- rem 1.0.1 and Theorem 1.0.2. Finally, in Chapter 4 we present examples and applications of Theorem 1.0.1 and Theorem 1.0.2. 3 CHAPTER 2 PRELIMINARIES 2.1 Pseudo-Anosov mapping classes We assume a basic familiarity with braid groups [6] and mapping class groups [4]. Unless stated otherwise, we assume all surfaces are closed, connected, and orientable with a disjoint collection of nitely many points and open disks removed. Denition 2.1.1. Let S be a surface. Let Hom+ (S, ∂S) be the collection of orientatation preserving homeomorphisms on S. The mapping class group on S is Mod(S) = π0 (Hom+ (S, ∂S)) = Hom+ (S, ∂S)/ ∼ where f ∼ g if there is an isotopy from f to g which xes all punctures and boundary components pointwise. If [f ], [g] ∈ ModS then [f ][g] is dened as composition, which is to say [f ][g] = [f ○ g]. Let Dn denote the disk with n≥3 points removed. It is well known that the braid group on n strands Bn is represented as a mapping class group on Dn , that is Bn ≃ Mod(Dn ). For convenience, we identify β∈B with its representative isotopy class in Mod(Dn ). See [6] Section 1.6 for more details. Denition 2.1.2. Let γ ⊂S be a simple closed curve. We say γ is essential if it is not homotopic to a point, a puncture, or a boundary component of S. We say an isotopy class of curves is essential if it has an essential representative γ. For example, if Sg,n is the genus g surface with n points removed, then S0,n has no essential curves for n = 0, 1, 2, 3. On a torus, the meridinal and longitudinal curves are both essential. 4 2.1.1 The Nielsen-Thurston Classication The Nielsen-Thurston classication says that all mapping classes can be classied as one of three types. For convenience we provide the statement below. For more information see [4], Section 13.3. Theorem 2.1.3 (The Nielsen-Thurston Classication). Let S be a compact, orientable sur- face with possibly nitely many punctures and let [f ] ∈ Mod(S). Then there is a representa- tive homeomorphism f ∶ S → S that is periodic , reducible , or pseudo-Anosov . Furthermore if f is pseudo-Anosov then it is neither periodic nor reducible. We say a mapping class [f ] is periodic if there is some positive integer k such that [f k ] has a representative isotopic to the identity. We say [f ] is reducible if there a non-trivial collection of isotopy classes of essential simple closed curves {c1 , . . . , ck } so that {f (c1 ), . . . , f (ck )} = {c1 , . . . , ck } up to isotopy. Our main focus will be on pseudo-Anosov mapping classes. A basic overview of what it means to be pseudo-Anosov is given below. 2.1.2 Singular measured foliations and pseudo-Anosov mapping classes Denition 2.1.4. A singular foliation F on a surface S is a decomposition of S into a disjoint union of subsets of S called leaves along with a nite collection of singular points {x1 , . . . , xm } ⊂ S so that 1. For every nonsingular point p∈S there is a smooth chart from a neighborhood of p to R2 which sends each leaf to a horizontal line segment. 2. For each singluar point xi ∈ S there is a smooth chart from a neighborhood of p to R2 which sends leaves to level sets of a k -pronged saddle with k ≥ 3. A smooth arc α∈S is transverse to F if it is transverse to each leaf of F and is disjoint from the singular points of F. A singular measured foliation is a pair (F, µ) where µ is a 5 Figure 2.1: Saddles for n = 3, 4, 5 measure that assigns a positive value to each smooth arc α transverse to F with µ invariant under any leaf-preserving isotopy. In Chapter 3 we will wish to keep track of the types of singularities that occur. Denition 2.1.5. Let F be a singular foliation on a surface S with a singular point x∈S with a chart sending the leaves in a neighborhood of x to the the level sets of a k -pronged saddle. Then we say x is an order k singularity. See Figure 2.1. Denition 2.1.6. A homeomorphism f ∶ S → S is pseudo-Anosov if there is a pair of transverse measured foliations (F u , µu ) (F s , µs ) on S and a real number λ>1 so that f ⋅ (F u , µu ) = (F u , λµu ) and f ⋅ (F s , µs ) = (F s , λ−1 µs ) A mapping class is pseudo-Anosov if it has a pseudo-Anosov representative. The measured foliations (F u , µu ) and (F s , µs ) are called the unstable and stable foliations respectively. The number λ is called the dilitation or stretch factor of [f ]. 2.2 Train Tracks See [8] Chapter 1 for more information on the denitions given in this section. Denition 2.2.1. Let S be a closed orientable surface of genus g with nitely many punc- tures and let τ ∈S be an embedded smooth, closed, 1-complex. We will refer to the vertices 6 of τ as switches and denote the set of all switches by Sw(τ ). Then τ − Sw(τ ) consists of a disjoint collection of smooth open arcs. These components will be refered to as the branches of τ and we will denote set of all branches by Br(τ ). Then τ is a train track in S if 1. For each switch v there is an open neighborhood U of v and a well dened tangent line L ∈ Tv (S) so that τ ∩U is the union of a nite collection of open arcs, each tangent to L at v. 2. The components of S−τ are either once-punctured k -gons with k≥1 or unpunctured k -gons with k ≥ 3. Example 2.2.2. See Figure 2.2. On the left is a train track in the 4-punctured sphere. On the right is a 1-complex in the 3-punctured sphere which is not a train track (S0,3 −τ contains a punctured disk with no cusps). Figure 2.2: A train track in S0,4 (left) and a 1-complex that is not a train track in S0,3 (right) Denition 2.2.3. Let C be a family of smooth simple closed curves disjointly embedded in a surface S so that no component belonging to C is homotopic to a point or a puncture. We 7 say that a train track τ carries C if there is a smooth map ϕ ∶ S → Ÿ, called the supporting map, so that 1. ϕ(C) ⊆ τ 2. ϕ is homotopic to the identity map 3. The restriction of the dierential dϕp to the tangent line to C at p is nonzero for every p ∈ C. Similarly, we say a train track τ′ is carried by τ is there is a supporting map ϕ∶S →S meeting the conditions given above. Denition 2.2.4. A mapping class [f ] ∶ S → S is carried by a train track τ ⊂S if there is a representative f ∈ [f ] so that f (τ ) is carried by τ. Note that the supporting map ϕ for a map f carried by τ can always be chosen so that switches are sent to swiches and edges are sent to edge-paths. See [2] for details. Denition 2.2.5. Suppose β is carried by τ with supporting map ϕ chosen so that edges are sent to edge-paths and vertices are sent to vertices. Then viewing τ as a graph, the map β∗ = ϕ ○ β ∶ ∣τ ∶ τ → τ is the train track map induced by β . Example 2.2.6. Let β = σ1 σ2 −1 . Let τ ∈ D3 be the train track depicted in Figure 2.3 (left) with branches labeled as indicated. A representation of β(τ ) is shown in Figure 2.3 (below). The induced train track map β∗ ∶ τ → τ is dened by e1 ↦ e3 e4 ↦ e5 e2 e4 e2 ↦ e1 e5 ↦ e4 e2 e5 e3 e5 e3 ↦ e2 Denition 2.2.7. Let g∶G→G be a graph map sending vertices to vertices and edges to edge-paths. Suppose G has edges e1 , . . . , ek . Then the transition matrix for g∶G→G is 8 e2 e4 e5 e1 e1 e3 β(e5 ) β(e4 ) Figure 2.3: τ ⊂ D3 (left) and β(τ ) (right) where β = σ1 σ2−1 dened as M = (aij )1≤i,j≤k where aij is the number of times β(ej ) passes over ei (ignoring orientation). Example 2.2.8. Consider again the braid and train track used in Example 2.2.6 and shown in Figure 2.3. Using the denition above, we have transition matrix ⎛0 1 0 0 0⎞ ⎜ ⎟ ⎜ ⎟ ⎜0 0 1 1 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 0 0 0 1⎟ . ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜0 0 0 1 1⎟ ⎜ ⎟ ⎜ ⎟ ⎝0 0 0 1 2⎠ 2.2.1 Measured train tracks Let τ be a train track in a surface S and let v∈τ be a switch. By denition every branch incident to v approaches the switch along a well dened tangent line Lv . This allows us to partition the branches incident to a switch into two sides. Let e1 , . . . , ek be the branches of τ and let w = (w1 , ..., wk ) be an assignment of real-valued weights to the edges of τ with w(ei ) = wi . We say that w satises the switch conditions if at each switch the sum of the weights of the branches on each side are equal. 9 Denition 2.2.9. Let τ be a train track with edges e1 , . . . , ek and let w be an assignment of real-valued weights to the branches of τ. If w satisies the switch conditions, then the pair (τ, w) is called a measured train track. 2.3 The Bestvina-Handel Algorithm In [2] Bestvina and Handel gave an algorithmic proof of the Nielsen-Thurston classication by using train tracks to encode the properties of (F u , µu ) and (F s , µs ). Bestvina and Handel's proof showed that given we can always construct a graph G⊂S and a graph map g∶G→G induced by [f ] so that g sends vertices to vertices and for any branch b the map g∣b is an immersion. An example of a graph map induced by a braid is given in Example 2.2.6. What follows is a brief overview of how G is constructed. Denition 2.3.1. A bered surface is a compact surface F decomposed into arcs and poly- gons modelled after k -junctions as shown in Figure 2.4. The components of F − {junctions} are called strips. We will be interested in bered surfaces which are subsurfaces of S0 associated to f. We say a bered surface F ⊂ S0 carries f if 1. F ↪ S0 is a homotopy equivalence, 2. f sends decomposition elements to decomposition elements 3. The junctions of F are sent to junctions by f In particular, each arc belonging to a strip in F is sent to an arc or into a junction of F, but junctions must be sent to junctions. Every bered surface F is associated to a graph G obtained by crushing each decompo- sition elements of F to a point. The edges (resp. vertices) of G correspond to the strips (resp. junctions) of F. If F carries f, then there is a map g∶G→G induced by f which sends vertices to vertices and edges to edge-paths dened in the obvious way. 10 Figure 2.4: k -junctions for k = 1, 2, 3 Recall that the link of a vertex v ∈ G is a graph lk(v, G) with vertices corresponding to the edges in G emanating from v. If ei and ej emanate from v then the corresponding vertices in lk(v, G) are connected by an edge if ei and ej are incident to a common 2-cell. For convenience we will refer to vertex of Lk(v, G) corresponding to e as e ∈ Lk(v, G) when there is no risk of confusion. With that in mind, suppose v and w are vertices in G with g(v) = w and e is an edge emanating from v. Then g(v) is an edge-path in G with initial edge e′ emanating from w. The derivative of g∶G→G is the map Dg ∶ Lk(v, G) → Lk(w, G) dened by e ↦ e′ Denition 2.3.2. We say ei and ej in Lk(v, G) belong to the same gate if there is some k>0 such that D(g k )(ei ) = D(g k )(ej ). In other words, edges ei and ej belong to the same gate if there is some power of g that sends both edges to an edge path with the same initial edge segment in G. Before proceeding we review some matrix theory. Let A = (ai,j ) and B = (bi,j ) be n×n matrices with non-negative integer entries. We will write A≥B or A>B to mean that ai,j ≥ bi,j for all i, j . By aki,j we mean the i, j -th entry of Ak . We say A = {aij } is irreducible if for each ai,j there is a k>0 so that aki,j > 0. If there is a k for which A k > 0, we say A is primitive. 11 We can associate to A a directed graph GA . The graph consists of n vertices v1 , . . . , vn and an edge oriented from vj to vi whenever ai,j is non-zero. Lemma 2.3.3. 1. A is irreducible if and only if for every pair of vertices vi , vj in GA , there is an oriented edge-path connecting vj to vi . 2. A is primitive if and only if there is an integer n such that for every vi , vj in GA , there is an edge path of length n connecting them. A primitive matrix A is Perron-Frobenius if it has integer entries. Theorem 2.3.4 (Perron-Frobenius theorem for primitive matrices). Let A be a non-negative n × n matrix. If A is primitive, then there is a eigenvalue λ > 0 of A such that given any other eigenvalue λ′ of A we have ∣λ′ ∣ < λ. Note that in the case that A is primitive and has integer entries, then λ > 1. By denition the transition matrix M (see Denition 2.2.7) is a square matrix with non-negative integer entries. Therefore if M is irreducible there is a unique positive unit eigenvector with positive eigenvalue λ which is the spectral radius of M and is called the growth rate of M. We will also occasionally refer to this value as λ = λ(F, f ) = λ(G, g) when it is convenient. Example 2.3.5. Let β = σ1 σ2−1 ∈ B3 . Let G be the graph depicted in Figure 2.5. A visual representation of the map induced by σ1 σ2−1 is also shown. However, the actual map sends edges to edge paths along the edges of G. The edges represented as circles are peripheral to punctures and thus do not contribute to the transition matrix for the real edges of the induced train track τ. Comparing σ1 σ2−1 (G) to G we see that the edgepath σ1 σ2−1 (e1 ) passes once through e1 and e2 . The edgepath σ1 σ2−1 (e2 ) passes through e1 once and e2 twice. Then 12 the transition matrix for the real edges is e1 e2 e1 ⎛ 1 1 ⎞ M= ⎜ ⎟ ⎜ ⎟ e2 ⎝ 1 2 ⎠ As we will see in later sections, the stretch factor for σ1 σ2 −1 is the largest real root of the characteristic polynomial of M which is h(x) = x2 − 3x + 1. It will also turn out that h(x) is the homology polynomial for β G e1 e2 σ1(G) σ-12 σ1σ-12(G) σ1σ-12(e2) σ1σ-12(e1) Figure 2.5: A graph map induced by σ1 σ2−1 The following is a consequence of [2]: 13 Proposition 2.3.6. A mapping class [f ] is pseudo-Anosov if and only if there is a train track τ invariant under [f ] and the transition matrix for the real edges of τ under the train track map is Perron-Frobenius. 2.3.1 From ecient graph maps to train tracks The graph map g∶G→G can be used to recover a train track τ that carries β along with a train track map representing β. Suppose v is a vertex of G and that there are k gates at v , g1 , . . . , gk . Replace v with a small circle C and identify k points q1 , . . . , q k ∈ C . We assume the gates are labeled to match the ordering of their associated gates when traveling counterclockwise around C. For each gate gi , we arrange all edges belonging to the gate to intersect C orthogonally at pi . Suppose there is an edge e0 ∈ G so that the edgepath g(e0 ) passes through v . Then there is some subpath of g(e0 ) of the form ei ej with ei ∩ ej = {v} Suppose ei and ej belong to gates gi and gj respectively. Then we add an edge ϵij connecting pi to pj within the region bounded by C. We assume ϵij intersects C orthogonally. After performing this operation at each vertex of G we have a train track τ. The edges in τ that come from edges in G are called real edges. The edges connecting gates are called inntesimal edges. At a vertex with k gates, the resulting inntesimal edges form a k -gon (possibly with one edge missing). Denition 2.3.7. A vertex of G is odd (even respectively) if the corresponding inntesimal edges form a polygon with an odd (even respectively) number of sides. If v corresponds to a polygon that is missing a side, we say v is partial. Dene ϕ∶τ →τ as follows: If e is a real edge in τ , then g(e) = ei ej ⋯ek is an edge-path in G. Suppose ei ej enters and exits the vertex v ∈ G. By the operation described above, there is an inntesimal edge η∈τ 14 connecting ei to ej . Repeating this at each vertex g(e) passes through, we get the edge path ϕ(e) = ei ηej ⋯. If η is an inntesimal edge connecting real edges ei and ej . Then there is an edge e∈G for which g(e) contains ei ej as a subpath. The map g 2 (e) contains the subpath g(ei )g(ej ) and determines ϕ(η). See Figure 4.7 and Figure 4.7 for an example of this process for the braid σ1 σ2 σ3−1 . After constructing τ, Bestvina and Handel use the train track and map to recover the invariant measured foliations for the pseudo-Anosov mapping class. The following is a con- sequence: Proposition 2.3.8. Let β ∈ Bn be pseudo-Anosov carried by a train track τ . There is a 1-to-1 correspondance between the components of S4 − τ and the singularities of F u and F s . In particular, a non-punctured disk with k corners corresponds to an order-k singularity in the interior of S4 and a punctured disk with k corners corresponds to an order k singularity occurring at a puncture. 2.4 The homology polynomial Recall the homology polynomial discussed in the introduction. Let [f ] ∈ Mod(S). The main result of [3] is the following. Theorem 2.4.1 ([3], Theorem 1.1). Let [f ] be a pseudo-Anosov mapping class in a closed, orientable surface S with possibly nitely many punctures. Let f ∶ S → S be the pseudo- Anosov representative of [f ] with the Bestvina-Handel graph and graph map g ∶ G → G and transition matrix M . Then 1. The characteristic polynomial of M , ∣xI − M ∣, has a divisor h(x) which is an invariant of [f ]. The dilatation of [f ] is the largest real root of h(x). It is associated to the induced action of f∗ on H1 (X, R) where X = S when τ is orientable and X is the orientation cover of τ when is not orientable. 15 2. The homology polynomial decomposes as a product p(x) ⋅ s(x) of two polynomials, each a topological invariant of [f ]. a) p(x) is the puncture polynomial and records the action of g∗ on the radical of a skew-symmetric form on W (G, g). It is related to the way f permutes the punctures of S . b) s(x) is the symplectic polynomial and records the action of g∗ on the non-degenerate symplectic space W (G, g)/Z and contains the dilitation of f as its largest real root. In Chapter 3 we will wish to compare homology polynomials for distinct mapping classes. If [f ], [g] ∈ Mod(S), we write their homology polynomials as h([f ], x) and h([g], x) respec- tively. In what follows we shall restrict our attention to braids. As we see, every train track that carries a braid is non orientable. First we recall the Euler-Poincarè-Hopf formula (see [5], Exposè 5, Section 1.6) Theorem 2.4.2. Let S be a genus g surface, possibly punctured, with a singular foliation F and singular points x1 , . . . , xk . For 1 ≤ i ≤ k let Pi denote the order of xi . Then k 4 − 4g = ∑(2 − Pi ). i=1 Lemma 2.4.3. Let β ∈ Bn be a pseudo-Anosov braid and let τ be a train track that carries β . Then τ is not orientable. Proof. It suces to show that β must produce at least one odd-ordered singularity. By Theorem 2.4.2, k 4 = ∑(2 − Pi ) i=1 If each Pi > 1 the above equality fails. Therefore at least one singularity is odd-ordered and τ is not orientable. 16 a1 b a2 Figure 2.6: The branches a1 and a2 form a corner. The branch b does not form a corner with either ai . 2.4.1 The orientation cover When τ is not orientable we lift to a special branched double of S determined by τ. Denition 2.4.4. Let b1 and b2 be branches in τ that meet at a switch v. Recall that b1 and b2 intersect v along a well dened tangent which allows us to partition the branches meeting v into two sides. If b1 and b2 approach v from the same side, we say the angle between them is 0. Otherwise, the angle between them is π. In the latter case we say the branches b1 and b2 form a corner (see Figure 2.6. Denition 2.4.5. Let τ ⊂S be a non-orientable train track and x some basepoint x ∈ τ. If S is not homotopic to τ , then let S0 be the surface obtained by puncturing each unpunctured disk component of S − τ. Then S0 is homotopic to τ and we may identify π1 (S0 , x) with π1 (τ, x). Dene ϵ ∶ π1 (τ, x) → Z/2Z ≃ {−1, 1} by γ ↦ (−1)#corners in γ Then the kernel of ϵ is all loops in τ with an even number of corners. The covering space cooresponding to the kernel of ϵ, after lling in any added punctures, is called the orientation cover for τ. It is a two-fold branched cover of S and the ber of τ is an orientable train track in the cover. 17 The following is result of Theorem 2.4.1. Theorem 2.4.6. Let β ∈ Bn be a pseudo-Anosov mapping class with train track τ . Let D̃ denote the orientation double cover for τ and denote its involution by ι. Then ι∗ ∶ H1 (D̃, R) → H1 (D̃, R) has two eigenspaces E + and E − corresponding to eigenvalues 1 and −1 respectively. The homology polynomial of β is the characteristic polynomial of β∗ ∣E − . 2.4.2 W (G, g) Given a train track τ constructed from g ∶ G → G, there is a natural surjection π∶τ →G sending real edges to real edges and collapsing all inntesimal polygons to a point. Let V (τ ) be the R-vector space of real weights on the branches of τ. Dene V (G) similarly. Let W (τ ) ⊂ V (τ ) be the subspace of assignments that satisfy the switch conditions. The surjection π∶τ →G induces a surjection π∗ ∶ V (τ ) → V (G). Denition 2.4.7. We dene W (G, g) = π∗ (W (τ )). It is the subspace of W (G, g) consisting of weight assignments that extend to an assignments of weights on τ that satisfy the switch conditions. There is a convenient way to determine if an element in V (G) is in W (G, g). Lemma 2.4.8 ([3], Lemma 2.9). An element η ∈ V (G) belongs to W (G, g) if and only if for each non-odd vertex the alternating sum of the weights at the incident gates is zero. Lemma 2.4.9 ([3], Lemma 2.11). If τ is orientable, then dim W (G, g) = #(edges of G) − #(vertices of G) + 1 otherwise dim W (G, g) = #(edges of G) − #(non-odd vertices of G) 18 2.4.3 The decomposition h(x) = p(x)s(x) Theorem 2.4.10 ([3], Theorem 3.8) . Let p(x) and s(x) be the characteristic polynomials of g∗ ∣Z and g∗ ∣W (G,g)/Z respectively. The map g∗ preserves the direct sum decomposition W (G, g) ≈ Z ⊕ (W (G, g)/Z) so that h(x) = p(x)s(x). Moreover we have 1. The polynomial p(x) is an invariant of the pseudo-Anosov mapping class [f ] ∈ Mod(S). The restriction g∗ ∣Z encodes how [f ] permutes the puntures whose projections to τ have even numbers of corners. In particular, g∗ ∣Z is a periodic map, so that all the roots of p(x) are roots of unity and the polynomial p(x) is palindromic or anti-palindromic. 2. The polynomial s(x) is an invariant of [f ]. The skew-symmetric form ⟨⋅, ⋅⟩W (G,g) nat- urally induces a symplectic form on W (G, g)/Z . The map g∗ induces a symplectomor- phism of W (G, g)/Z . Hence ks(x) is palindromic. 3. The homology polynomial h(x) is either palindromic or anti-palindromic. 2.5 The Burau representation The Burau representation for braids will play a crucial role in what follows. In particular, there is an equivalent twisted homological version of the Burau representation which allows us to represent braids as acting on the rst homology group of an innite cyclic cover Dn∞ → Dn with deck group ⟨t⟩ ≃ Z. In Chapter 3 we will see that the quotient Dn∞ / ⟨t2 ⟩ is equivalent to the orientation cover for a train track that carries a braid. For more information on this topic see [6] sections 3.1-3.3 and [1]. 19 Denition 2.5.1. Let Z[t, 1/t] denote the ring of Laurent polynomials and let n ≥ 3. Dene V1 , . . . , Vn−1 ∈ GLn−1 (Z[t, 1/t]) as ⎛−t 0 0 ⎞ ⎛I 0 0⎞ ⎜ ⎟ ⎜ n−3 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ V1 = ⎜ 1 1 0 ⎟ ⎜ Vn−1 = ⎜ 0 1 t ⎟ ⎟ and ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 0 In−3 ⎠ ⎝ 0 0 −t⎠ and for 1 1, it is well known that sup{sr(ψ[β, η] ∣ η a root of unity }≤λ (3.1) where ψ[β, η] is the (reduced) Burau matrix for β and with the substitution t = η. and sr(ψ[β, η] is its spectral radius. The left side of Equation (3.1) is called the Burau estimate for the stretch factor of a pseudo-Anosov braid at η. Let ϕ ∶ π1 (Dn , d) → Z denote the map sending elements to their winding numbers, and for 23 any k≥2 let ϵk ∶ Z → Zk denote the standard quotient mapping. Clearly ker(ϵk ○ ϕ) ⊂ ker ϕ. Denition 3.1.1. For each k≥2 let pk ∶ Dn → Dn (k) denote the covering space of associated to the kernel of the map ϵk ○ ϕ ∶ π1 (Dn , d) → Zk with Dn = Dn /tk (k) (∞) qk ∶ Dn → Dn (∞) (k) Let denote the corresponding covering map. The main result of [1] that we build upon is Theorem 3.1.2 ([1], Theorem 5.1) . Let β be a pseudo-Anosov braid with stretch factor λ and (reduced) Burau matrix ψ[β, t]. Then the following are equivalent (1) sr(ψ[β, η] = λ for some root of unity η . (2) sr(ψ[β, −1]) = λ and −1 is the only root of unity for which this equality is true. (3) The invariant foliations F u and F s have odd-ordered singularities at each puncture and all singularities in the interior of D are even-ordered. (4) D(2) is the orientation double-cover for F u and F s . We state two additional results of [1] which we will wish to use in later sections. Lemma 3.1.3 ([1], Lemma 3.2) . Let T be the generator of the deck group for the covering p(k) ∶ X (k) → X and let h(k) and h(∞) be the lifts of h to X (k) and X (∞) . The eigenvalues of T∗ restricted to SC(k) are 1, ηk , ηk2 , . . . , ηkk−1 where ηk = e2πi/k . Denote by E0 , . . . , Ek−1 the corresponding eigenspaces in SC(k) . Then each subspace Em is h(k) ∗ -invariant, and the action of h(k) ∗ on Em is given by the matrix M (ηkm ), obtained by substituting ηkm into the matrix M (t) of h∗ . (∞) 24 Theorem 3.1.4 ([1], Theorem 3.4). Let h ∶ X → X be a homeomorphism of the locally path- connected, semi-locally simply connected topological space X , whose rst homology group we assume to be free and of nite rank. Suppose ρ ∶ H1 (X) → Z is a homomorphism which satises ρh∗ = ρ, and let X (∞) and X (k) = X (∞) /T k denote the covering spaces over X corresponding to ρ and ξk ○ ρ, with covering projection q (k) ∶ X (∞) → X (k) . Let h(∞) and h(k) denote lifts of h to these covering spaces. If M = M (t) ∈ GL(r, R) denotes the matrix of ∶ H1 (X (∞) ) → H1 (X (∞) ) as an R-module isomorphism, then the action of h∗ on the (∞) (k) h∗ invariant subspace SC(k) = q∗(k) (H1 (X (∞) , C)) is given by the direct sum h∗ = M (1) ⊕ M (ηk ) ⊕ ⋯ ⊕ M (ηkk−1 ) (k) where M (ηkj ) denotes the complex matrix obtained by substituting ηkj = e2πij/k into M . Fur- thermore, any eigenvector of h(k) ∗ not lying in S (k) has eigenvalue which is a root of unity. 3.2 The homology polynomial from the burau representation We now build upon Theorem 3.1.2 and show that under the same singularity conditions given in Theorem 3.1.2(3) the characteristic polynomial of the Burau matrix of a pseudo-Anosov braid is the homology polynomial. Fix some β ∈ Bn be a pseudo-Anosov braid with stretch factor λ and homology polynomial h(x). Suppose g∶G→G is an ecient graph map corresponding to β that induces train track τ ⊂ Dn . Let χ(β) = χ (ψ[β, −1]) = ∣xI − ψ[β, −1]∣. That is, χ(β) is the characteristic polynomial of ψ[β, −1]. Lemma 3.2.1. The stretch factor of β is the largest real root of χ(β) if and only if χ(β) = h(x). Proof. If χ(β) = h(x) then λ is the largest real root of χ(β) since h(x) always contains λ as its largest real root. (2) Conversely, suppose that λ is the largest real root of χ(β). Then by Theorem 3.1.2 Dn ψ[β, −1] (2) is the orientation cover for τ and by Lemma 3.1.3 represents the action of β∗ on 25 H1 (Dn ; R) −1. (2) the eigenspace of corresponding to the eigenvalue Then by Theorem 2.4.6 χ(β) is equal to the homology polynomial. Using Lemma 3.2.1 and Proposition 2.3.8. we are able to restate Theorem 3.1.2 in terms of the homology polynomial, graph maps, and train tracks. Proposition 3.2.2. Suppose β ∈ Bn is pseudo-Anosov with dilatation λ and homology poly- nomial h(x). Then the following are equivalent: (1) χ(β) is equal to the homology polynomial for β ; (2) The spectral radius of ψ[β, e2πij/k ] = λ for some 0 ≤ j < k ; (3) The spectral radius of ψ[β, −1] = λ and −1 is the only root of unity at which this occurs; (4) The vertices of G occuring at the punctures of Dn and in the interior of Dn are odd and even respectively. (5) Dn is the orientation double-cover of τ (after attacking a punctured disk to the bound- (2) ary of Dn ). Proof. By Theorem 3.1.2 conditions (2), (3), and (5) are equivalent. By Lemma 3.2.1 and Theorem 3.1.2 (2) is equivalent to (1). Finally, by Proposition 2.3.8 (4) is equivalent to the third statement of Theorem 3.1.2 which implies (4) is equivalent to (3). 3.3 Proof of Theorem 1.0.2 3.3.1 Overview The goal of this section is to prove Theorem 1.0.2. Unless stated otherwise all braids are assumed to be pseudo-Anosov. Denition 3.3.1. We say that β produces a k -ordered singularity if the invariant foliations associated to β have a k -ordered singularity. 26 By Proposition 3.2.2 h(β, x) = χ(β) if and only if β does not produce certain types of bad singularities. Denition 3.3.2. A singularity produced by β is bad if it is odd-ordered and occurs at an interior point of Dn or even-ordered and occurs at a puncture of Dn . By Theorem 1.0.1 if β does not produce bad singularities h(β, x) = ∣xI −ψ beta (−1)∣. In the case that β does produce bad singularities Theorem 1.0.2 says that we can algorithmically construct some new braid β ′ which produces no bad singularities so that h(β, x) is recoverable from h(β ′ , x). We construct β′ from β using two operations. The rst involves puncturing Dn at each odd-ordered singularity in the interior of Dn . This can be thought of as "inserting a strand" into β. The second operation involves lling in any punctures of Dn at which β produces an even-ordered singularity. This can be thought of as "forgetting a strand". If βi+1 is constructed from βi using one of these two operations we see that βi+1 produces exactly one less bad singularity than βi . 3.3.2 Odd-ordered singularities in the interior We now prove assume that a pseudo-Anosov braid β ∈ Bn produces and xes an odd-ordered singularity at a point s in the interior of Dn . We show that declaring s a new puncture results in a pseudo-Anosov braid β ′ ∈ Bn+1 with the same homology polynomial as β. Lemma 3.3.3. Let β ∈ Bn be a pseudo-Anosov braid that produces an odd-ordered singularity at a point s in the interior of Dn . Suppose β xes s and let Dn+1 = Dn −{s}. Dene β ′ ∈ Bn+1 by β ′ = β∣Dn −{s} . Then 1. β ′ is pseudo-Anosov 27 2. β ′ produces one less bad singularity than β 3. We have p(β ′ , x) = p(β, x) s(β ′ , x) = s(β, x) h(β ′ , x) = h(β, x) Proof. We rst prove that β′ is pseudo-Anosov. Consider β as an element of Mod(Dn ) where Dn = (D, {p1 , . . . , pn }) is the disk with marked points p1 , . . . , p n . Let Dn+1 = (D, {p1 , . . . , pn , S}). Let T denote the transition matrix for a train track τ ⊂ Dn that carries β. The singular point s is in Dn − τ so we may embed a copy of τ in Dn+1 . Since β acts as the identity on some neighborhood of s β′ is also carried by the image of τ embedded in Dn+1 . Then the transition matrix for β′ is also represented by T and the submatrix representing the transition matrix corresponding to the real edges of τ is also Perron-Frobenius. Then by Theorem 2.3.4 β is pseudo-Anosov. The singularities of β′ are the same as those of β except we have replaced an odd-ordered singularity in the interior with an odd-ordered singularity at a puncture. Thus β′ produces one less bad singularity than β. It remains to show (c). If the singularity at s is order k then s cooresponds to a vertex vs of G with k gates. Dene G′ as G with the vertex vs replaced by a k -gon, with k partial vertices v1 , . . . , vk cooresponding to the gates of vs and k edges e1 , . . . , e k with ei connecting vi to vi+1 for i2 then Dn−1 − τ is still a train track so g∗′ ∶ τ ′ → τ ′ carries β′ with no modication from g∗ ∶ τ → τ . Again, p(β ′ , x) = p(β, x)/(x + 1) by the same reasoning as above. Furthermore, the transition matrix for β′ acting on τ is the same as the transition matrix for β which means β′ is pseudo-Anosov (again see Theorem 2.3.4.) Suppose W (G, g) ≈ Z ⊕ W (G, g)/Z is of dimension M. Let η1 be the generator of Z associated to p. Let {η1 , . . . , ηr , γ1 , . . . , γ M −r } be a basis for W (G, g) so that the {ηi } generate Z. Since g∗′ ∶ W (G′ , g ′ ) → W (G′ , g ′ ) is unmodied from g∗ ∶ W (G, g) → W (G, g) {η1 , . . . , ηr , γ1 , . . . , γM −r } is still a basis for W (G′ , g ′ ). The generator η1 is not longer an element of Z ′. This leads to p(β ′ , x) = p(β, x)/(x + 1). The element represented by η is now a generator of W (G′ , g ′ )/Z ′ xed by β ′. It follows that s(β ′ , x) = (x + 1)s(β, x) and h(β ′ , x) = p(β ′ , x)s(β ′ , x) = (p(β, x)/(x + 1)) ((x + 1) ⋅ s(β, x)) = h(β, x) as desired. 3.3.4 Conclusion and proof The following result from [3] gives a corollary that will be used in the proof of Theorem 1.0.2. 32 vj+1 ei vj+1 p e*i ei+1 vj vj Figure 3.3: τ (left) and τ′ (right) after lling in an order-2 singularity Lemma 3.3.5 ([3], Corollary 4.5) . Let m > 0. If g ∶ G → G is a graph map representing a pseudo-Anosov mapping class β then g m ∶ G → G represents β m . Suppose that the homology polynomial for β hβ (x) = sβ (x)pβ (x) and that sβ (x) = ∏ (x − zi ) and pβ (x) = ∏ (x − wj ) , zi , wj ∈ C. i j Then hβ m (x) = sβ m (x)pβ m (x) with sβ m (x) = ∏ (x − zim ) i pβ m (x) = ∏ (x − wjm ) j hβ m (x) = ∏ (x − zim ) ∏ (x − wjm ) i j See [3], Corollary 4.5 for a proof. We now prove Theorem 1.0.2 Proof of Theorem 1.0.2. Suppose β0 produces q odd-ordered singularities at the interior points s1 , . . . , s q and r even-ordered singularities at punctures p1 , . . . , p r and that these q+r are the only bad singularities produces by β. Suppose ϵ ≥ 0 of the r even-ordered singularities at the punctures of Dn are order-2. Choose k so that β = β0k xes all q+r bad singularities. 33 Let β̂ be the braid obtained by applying Lemma 3.3.3 at each odd-ordered singularity in the interior of Dn . Then ̂ x) = h(β, x) and whβ h(β, produces no odd-ordered singularities in the interior of Dn . Now let β′ be the braid obtained from β̂ after applying Lemma 3.3.4 at each even-ordered singularity at a puncture. Then β′ produces no bad singularities and χ(β ′ ) = h(β ′ , x) = h(β,̂ x)/(x + 1)ϵ = h(β, x)/(x + 1)ϵ By Lemma 3.3.5 if h(β0 , x) = ∏i (x − zi ) then χ(β ′ ) = h(β ′ , x) h(β, x) ∏i (x − zik ) = = (x + 1)ϵ (x + 1)ϵ as desired. 3.4 An algorithm for constructing β ′ from β What follows is an algorithm for constructing β′ for an arbitrary pseudo-Anosov β0 . Let β0 be pseudo-Anosov. If β0 produces no bad singularities then χ(β0 ) = h(β0 , x). If β0 produces at least one bad singularity, we apply the following steps to obtain β ′. (1) Choose k≥1 so that β = β0k xes all bad singularities. (2) If β produces an odd-ordered singularity at an interior point s of Dn puncture Dn at s and dene β̂0 as the image of β in Dn+1 = Dn − {s}. Repeat until every interior point with an odd-ordered singularity is punctured. Let β̂ be the resulting braid in Dn+q where q is the number of interior points punctured. (3) If β̂ produces an even-ordered singularity at a puncture P of Dn+q then ll in P and let β̂′ be the resulting braid in Dn+q−1 . Repeat until every puncture with an even-ordered singularity is lled in. Let β′ be the resulting braid. By Theorem 1.0.2 χ(β ′ ) = h(β ′ , x) and h(β ′ , x) is related to h(β, x) as described in Theorem 1.0.2. 34 CHAPTER 4 EXAMPLES AND APPLICATIONS In this chapter we give examples and applications of Theorem 1.0.2. This result is only useful if we can use it to avoid nding train tracks and singularity types. Otherwise we may as well compute h(x) from denition. With this in mind we rst introduce a large family of braids for which Theorem 1.0.2 can be used to compute h(x) from the braid word alone. As a simple example we rst show that h(β, x) = ∣xI − ψβ (−1)∣ for any pseudo-Anosov β ∈ B3 . When applying the Nielsen-Thurston Classication to a mapping class on a surface with boundary we rst attack punctured disks to each boundary component. In particular, we attach a punctured disk to the boundary component of Dn to consider β ∈ Bn as an element of Mod(Sn+1 ) where Sn+1 is the sphere with n+1 points removed. Proposition 4.0.1. If β ∈ B3 is a pseudo-Anosov braid, then h(β, x) = ∣xI − ψβ (−1)]∣ Proof. Suppose β ∈ B3 is pseudo-Anosov carried by a singular foliation F on S4 . Let x1 , . . . , x k be the singular points (possibly occurring at a puncture) of F. Let Pi ≥ 1 de- note the order of xi . Recall that Pi ≥ 3 if xi is in the interior and Pi ≥ 1 if xi occurs at a puncture. According to Theorem 2.4.2, k 4 = ∑ (2 − Pi ) i=1 For convenience assume x1 , x2 , x, 3, x4 occur at the punctures of S4 . Then Pi ≥ 1 for 1≤i≤4 and Pi ≥ 3 for i > 4. Therefore 4 k 4 = ∑ (2 − Pi ) + ∑ (2 − Pi ) i=1 i=5 k ≥ 4 + ∑ (−1) i=5 35 Therefore there can be no singularities in the interior of D3 . The above application of the Euler-Poincarè-Hopf formula also implies that Pi = 1 for i = 1, 2, 3, 4. Let β ∈ B3 be pseudo-Anosov. By the above the foliations associated to β have exactly four singularities. Each is odd-ordered and each occurs at a puncture. By Theorem 1.0.1 h(β, x) = ∣xI − ψβ (−1)∣. 4.1 Application of Theorem 1.0.2 to a large family of braids The family of braids presented in this section and the methods for studying them are an extension of the methods of [7]. We rst dene two building blocks for constructing the elements. Denition 4.1.1. For any integers m, p ≥ 1 we dene two elements of Bm β(m,p) = (σ1 σ2 . . . σm−1 )p and β(−m,p) = (σ1−1 σ2−1 . . . σm−1 −1 )p n ··· ··· β α m Figure 4.1: β⋆α An illustration of β(3,2) is given in Figure 4.3 (left). Elements of B are constructed from the above with a modied form of concatination. Denition 4.1.2. Let β ∈ Bn and α ∈ Bm . Let β ′ ∈ Bn+m−1 be the image of β under the usual inclusion map σi ↦ σi and let α′ be the shifted image of α under the map σi ↦ σn+i−1 . Then β ⋆ α = β ′ α′ ∈ Bn+m−1 . See Figure 4.1. 36 m1 mk ··· ··· ··· β(m1 ,p1 ) β(m2 ,p2 ) ··· β(mk ,pk ) m2 Figure 4.2: β(m1 ,p1 ),...,(mk ,pk ) Denition 4.1.3. A sequence of ordered pairs {(mi , pi )}ki=1 is a pA-sequence if 1. ∣mi ∣, pi > 0 for all i 2. ∣mi ∣ and pi are relatively prime for all i 3. The sequence m1 , . . . , mk is alternating. We dene B = {β{(mi ,pi )}ki=1 ∣ {(mi , pi )}ki=1 is a pA-sequence}. where β{(mi ,pi )}ki=1 = β(m1 ,p1 ) ⋆ β(m2 ,p2 ) ⋆ ⋯ ⋆ β(mk ,pk ) . An illustration of β(m,p) is given in Figure 4.3 (left). Denition 4.1.4. Dene ⎧ ⎪ ⎪ ⎪ ⎪ β(m,p) if m even γ(m,p) = ⎨ ⎪ ⎪ ⎪ p ⎪ (σ2 ⋅ β(m+1,1) ) if m is odd ⎩ and γ{(mi ,pi )}ki=1 = γ(m1 ,p1 ) ⋆ γ(m2 ,p2 ) ⋆ ⋯ ⋆ γ(mk ,pk ) β∈B β = β{(mi ,pi )}ki=1 (mi , pi )i=1 . k If then for some pA-sequence In this case we dene γ(β) = γ(β{(mi ,pi )} ) = γ{(mi ,pi )} ) 37 β3,2 γ3,2 γ(4,3) = β(4,3) Figure 4.3: Comparison of βm,p and γm,p Note that γ(m,p) is always a braid on an even number of strands. A comparison of β(m,p) and γ(m,p) is given in Figure 4.3. We will show the following: Theorem 4.1.5. Let β ∈ B. Then 1. β and γ(β) are pseudo-Anosov 2. h(β, x) = h(γ(β), x) 3. h(γ(β)), x) = ∣xI − ψγ(β) (−1)∣ To prove these braids are pseudo-Anosov we use combined tree maps [7]. Denition 4.1.6. For any m≥1 let Tm+ and Tm− be trees of star type shown in Figure 4.4. Each has m valence-1 vertices and 1 valence-m vertex. See Figure 4.4. T0 is the trivial tree consisting of exactly one vertex. Given a sequence S = {(m1 , p1 ), . . . , (mk , pk )} Dene k (−1)i+1 TS = (⋃ Tmi ) /(ri ∼ li+1 ) i=1 38 Label the edges of Tm+1 e1 , . . . , em1 in the order indicated in Figure 4.4. Then the edges of Tm−2 are em1 +1 , . . . , em1 +m2 and so forth. For 1 ≤ j ≤ k, dene gj ∶ TS → TS by ei ↦ ei i < m1 + ⋯ + mj−1 − 1 ei ↦ e Then gS ∶ TS → TS is given by gS = gkpk ○ ⋯ ○ g1p1 . w0 ··· w1 wm v2 w2 ··· v1 v0 vn Tn,+ Tm,− Figure 4.4: Trees of star type Example 4.1.7. Let S = {(3, 2), (4, 1)}. The combined tree map gS ∶ TS → TS is shown in Figure 4.5. The following is a consequence of [7] (section 3): Proposition 4.1.8. Let gS ∶ TS → TS be a combined tree map for a pA-sequence S and let MS be the transition matrix of gS . Then M is Perron-Frobenius. Given TS as above, we can produce a train track in the punctured disk: 1. Replace each valence 1 vertex with a 1-gon bounding a punctured disk. 2. Each valence-2 vertex shared by two trees is replaced by a 1-gon bounding a punctured disk as indicated in Figure 4.6. 39 3. The valence-m vertex in each Tm± is replaced with an m-gon bounding a disk. Finally, we extend gS to ϕS ∶ τS → τS by permuting the inntensimal edges to match the rotation of Tm± . The following is a result of [7]. Proposition 4.1.9. Let S be a pA-sequence and let gS ∶ TS → TS be the induced combined tree map. 1. The transition matrix MS is Perron-Frobenius 2. The induced train track τS carries βS . 3. MS is the transition matrix for the real edges of τS . See [7] Section 4. The assumption that each pair (m, p) are relatively prime is needed for MS to be Perron-Frobenius. Proof of Theorem 4.1.5. By Proposition 4.1.9 the transition matrix for the real edges of βS is Perron-Frobenius. Then by Theorem 2.3.4, βS is pseudo-Anosov. If βS ∈ Bn then γS ∈ Bn+n′ where n′ is the number of pairs (mi , pi ) in S with mi odd. However, by construction, γS is carried by a copy of τS embedded in Dn+n′ with the same train track map representing γS . Therefore the transition matrix for the real edges of a train track invariant under γS is Perron-Frobenius and γS is pseudo-Anosov. Using combined tree maps we can predict the singularity types produced by βS . Specif- ically, if S = {(mi , pi )}ki=1 , βS produces an order-1 singularity at each puncture and an order-mi singularity in the interior of S4 for each mi ≥ 3. The braid γS is βS after applying Theorem 1.0.2 at each odd-ordered singularity in the interior. Then by Theorem 1.0.2 we have h(βS , x) = h(γS , x) = ∣xI − ψγS (−1)∣ as desired. 40 Figure 4.5: The tree map gS ∶ TS → TS for S = {(3, 2), (4, 1))}. Figure 4.6: Constructing τ from TS Proposition 4.1.10. Let S = {(mi , pi )}ki=1 be a pA-sequence. Then β = βS = β(m1 ,p1 ) ⋆ ⋯ ⋆ β(mk ,pk ) is pseudo-Anosov. 41 Proposition 4.1.11. Let {(mi , pi )}ki=1 be a pA-sequence. Let β⋆ = β{(mi ,pi )}ki=1 and γ⋆ = γ{(mi ,pi )}ki=1 Then 1. γ⋆ is pseudo-Anosov 2. The characteristic polynomial of the Burau matrix for γ⋆ is the homology polynomial: h (γ⋆ , x) = ∣xI − ψ[γ⋆ , −1]∣ 3. The homology polynomials for β⋆ and γ⋆ are equal h (β⋆ , x) = h (γ⋆ , x) 4.2 An example comparing the computation of h(x) from denition and computing h(x) using Theorem 1.0.2 In this section we will compute the homology polynomial for a pseudo-Anosov braid from denition and then using Theorem 1.0.2 The ecient graphs used in the following examples were determined with the help of [10]. 4.2.1 An odd ordered singularity in the interior of the disk Let β = σ1 σ2 σ3−1 denote the braid in B4 represented as a mapping class in the 4-punctured sphere. We will nd an ecient graph and graph map that carries β and construct the corresponding train track. After this we will nd the homology polynomial for β using both the denition given in Section 2.4 and by using Theorem 1.0.2. 42 The graph map, transition matrix, and train track Let G be the graph as depicted in Figure 4.7. The edges and vertices are labeled and will be refered to throughout this example. Also depicted is the graph maph g ∶ G → G. An orientation is given for convenience. Recall that if G has k edges, the transition matrix of g ∶G→G is the k×k matrix with ij -th entry equal to the number of times the edgepath g(ej ) passes through ej in either direction. The transition matrix for the map constructed above is e1 e2 e3 e4 e5 e6 e7 e8 e1 ⎛ 0 1 0 0 0 0 0 0 ⎞ ⎜ ⎟ ⎜ ⎟ e2 ⎜⎜ 0 0 1 1 0 0 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ e3 ⎜⎜ 1 0 0 1 0 0 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ e4 ⎜⎜ 1 0 0 2 0 0 0 0 ⎟ ⎟ T= ⎜ ⎟ ⎜ ⎟ e5 ⎜⎜ 0 0 0 0 0 1 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ e6 ⎜⎜ 0 0 0 0 0 0 1 0 ⎟ ⎟ ⎜ ⎟ e7 ⎜⎜ 1 0 0 1 0 0 0 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ e8 ⎝ 0 0 0 2 1 0 0 0 ⎠ To construct a train track τ that carries β we need to determine the gates at each vertex. Consider edges e3 and e4 emanating from v4 . As is shown the edgepaths g(e3 ) and g(e4 ) have the same initial segment. Therefore they belong to the same gate. The peripheral edges are permuted, and the two ends are never sent to the same initial segment, so at v4 we have three distinct gates. See Figure 4.8 for a visual representation of τ. The enlarged dashed circles represent the gates and inntesimal edges that replace the vertices of G. The vertex v2 is odd. To see this, let U be a neighborhood of v2 so that U ∩ G consists of three open-ended arcs emanating from v2 . Since v2 is xed by g and the arcs are permuted, we see that for any p>0 the edgepaths g p (e1 ), g p (e2 ), and g p (e3 ) never coincide and belong to distinct gates. 43 e1 e2 v1 v3 e7 v2 e6 v4 v5 e8 e5 e3 e4 Figure 4.7: From top to bottom: G, σ1 (G), σ1 σ2 (G), and σ1 σ2 σ3 −1 (G) Figure 4.8: The train track induced by the graph map shown in Figure 4.7. The homology polynomial from W (G, g) As seen above v2 is an odd vertex and all others are non-odd. For w = (w1 , w2 , w3 , w4 , w5 , w6 , w7 , w8 ) ∈ W (G, g), wi denotes the weight assigned to ei , 44 Figure 4.9: The basis element η1 . All other edges are assigned a weight of 0. i = 1, . . . , 8. Let B = {η1 , η2 , η3 , η4 } where η1 = (2, 0, 0, 0, 1, 0, 0, 0), η2 = (0, 2, 0, 0, 0, 1, 0, 0), η3 = (0, 0, 2, 0, 0, 0, 1, 0), η4 = (0, 0, −2, 2, 0, 0, 0, 1) Recall that an element w ∈ V (G) belongs to W (G, g) if at each non-odd vertex vi the alternating sum of the weights at incident gates is zero (see Lemma 2.4.8). Thus B ⊂ W (G, g). According to Lemma 2.4.9, the dimension of W (G, g) is #(Edges) − #(Non-odd vertices) =8−4=4 which implies B is a basis for W (G, g). See Figure 4.9 for a depiction of η1 . Let T be the transition matrix given above and let P denote the matrix for the map R8 → R4 which projects onto the rst four coordinates. Let Q = (η1T η2T η3T η4T ) denote the 8×4 matrix with column vectors equal to the elements of B. 45 Then the homology polynomial for β is h(x) = ∣xI − P T Q∣ = x4 − 2x3 − 2x + 1 The homology polynomial from the Burau representation The image of β under the Burau representation is ⎛ 0 t 0⎞ ⎜ ⎟ ⎜ ⎟ Ψβ (t) = ⎜ ⎟ ⎜1 − t 1 − t 1 ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎝ −t −t −t⎠ 1 1 By Proposition 3.2.2 if χ(Ψβ (ν)) = ∣xI − Ψβ (ν)∣ is equal to the homology polynomial then ν = −1. However this will not hold for β because of the bad singularity occuring at v2 above. In fact χ(Ψβ (−1)) = (1 − x)3 . Following the strategy outlined in Section 3.3.2 we will "add a strand" by declaring the bad singularity a new puncture. The resulting braid is β = σ2 σ1 σ2 σ3 σ4 −1 ∈ B5 which is shown in Figure 4.10. We now have ⎛0 −1 0 0⎞ ⎜ ⎟ ⎜ ⎟ ⎜0 −1 −1 0⎟ ⎜ ⎟ Ψβ (−1) = ⎜ ⎟ ⎜ ⎟ ⎜2 2 2 1⎟ ⎜ ⎟ ⎜ ⎟ ⎝1 1 1 1⎠ and ∣xI − Ψβ (−1)∣ = x4 − 2x3 − 2x + 1 = h(x) as expected. 46 Figure 4.10: The braid σ1 σ2 σ3 −1 . 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