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arXiv:math/0301250v1 [math.GT] 22 Jan 2003

Flowlines transverse to ?bred knots and links

Robert Ghrist Eiko Kin

Department of Mathematics, University of Illinois, Urbana IL, 61801 USA Department of Mathematics, Kyoto University, Oiwake-cho Kitashirakawa Sakyo-ku Kyoto-shi, Kyoto 606-8502 Japan Email: ghrist@math.uiuc.edu kin@kusum.kyoto-u.ac.jp

Abstract Let K be a knot or link in S 3 which is ?bred — the complement ?bres over S 1 with ?bres spanning surfaces. We focus on those ?bred knots and links which have the following property: every vector ?eld transverse to the ?bres possesses closed ?ow lines of all possible knot and link types in S 3 . Our main result is that a large class of ?bred knots and links has this property, including all ?bred nontorus 2-bridge knots. In general, su?cient conditions include a pseudo-Anosov type monodromy map and a su?ciently high degree of symmetry. AMS Classi?cation numbers Primary: 57M25,37C27 Secondary: 37E30 Keywords: train tracks, templates, ?bred knots, braids, vector ?elds

1

1

Introduction

We consider ?brations of knot or links complements in S 3 from the point of view of transverse vector ?elds. Periodic ?owlines of a vector ?eld on S 3 possess knotting and linking information which is intimately related to the dynamical properties of the ?ow. Examples abound of classes of systems for which simple dynamics implicate simple knotted orbits (those whose complement cannot possess a hyperbolic structure) and vice versa: see [21] [Morse-Smale ?ows], [10] [Smale ?ows], [7, 9] [integrable Hamiltonian ?ows], [4] [the Lorenz system], and [8] [?ows tangent to plane ?elds]. In this paper, we demonstrate that complicated ?brations typically force complicated knot and link types in any transverse vector ?eld — indeed, as complicated as can be imagined. Consider a knot or link K in S 3 which is ?bred; that is, there is a ?bration π : S 3 ?K → S 1 with ?bre a spanning surface Σ for K . Choose any vector ?eld X transverse to the ?bres of π (in particular, X must be nonvanishing). Birman and Williams originally asked the question, “Which knot types are forced to exist as periodic orbits of X ?” The question splits into cases based on Thurston’s classi?cation theorem for surface maps applied to the monodromy Φ : Σ → Σ of π . If the complement of K is geometrically “simple” (e.g., the unknot, torus knots, iterated torus knots), then only a ?nite set of knot types are forced to exist for all X (again, typically unknots, torus knots, and iterations). On the other hand, if the complement of K is “hyperbolic” (in the sense that the monodromy map is of pseudo-Anosov type), then there are always an in?nite number of distinct knot types as periodic orbits in any vector ?eld X transverse to π . The paper [5] carefully constructed a template (a branched surface with semi?ow) in the complement of the ?gure eight knot which captures all knot and link types forced by a transverse vector ?eld. It was there noted that, although many knot types were present, others were seemingly impossible to locate. To the contrary, the paper [14] showed that this template is universal — every knot and link type can be found on this branched surface. It follows that any vector ?eld transverse to a ?bration of the ?gure eight knot complement must possess all knot and link types as orbits. In this paper, we explore to what extent this property holds for other ?bred knots and links. De?nition 1.1 A ?bred knot/link K ? S 3 is said to be universally ?bred if each vector ?eld X transverse to the ?bration possesses closed orbits of all knot and link types. 2

An obvious prerequisite for being universally ?bred is that the monodromy be su?ciently complex: pseudo-Anosov type. For in the case of periodic type, the forced links may have only a ?nite number of knot types represented (by suspending the periodic monodromy). Our results are phrased in the language of braids: see §2 for details and Fig. 1 for examples of braids. We summarize our results in a simpli?ed form; see §5 for stronger results. Main Theorem: Let b be a braid of the form

?N ? ? b = σ11 σ22 · · · σN ,

(1)

where ?i = +1 or ?1 for each i (see Fig. 1[right]) and not all the ?i are the same. Then for every m > 1, the closure of bm is a ?bred knot/link in S 3 of pseudo-Anosov type. Furthermore: (a) The closure of bm has the universal ?bration property for all but ?nitely many m. (b) The closure of b2 always has the universal ?bration property.

?1 ?1 ?1 ?1 (c) Whenever b contains the subword σi?1 σi σi+1 σi+2 (or σi?2 σi?1 σi σi+1 ), m is universally ?bred for all m > 1. then the closure of b

Figure 1: [left] an example of a braid on seven strands; [right] an example of a braid ?1 ?1 ?1 represented by Eqn. (1), b = σ1 σ2 σ3 σ4 σ5 σ6

From result (b) above, we obtain a complete classi?cation in the case of 2-bridge knots: Corollary: Every ?bred non-torus 2-bridge knot has the universal ?bration property. 3

The proof of the Main Theorems utilizes a construction initiated in [5], based on work of [15] (and, ultimately, the work of Alexander). This involves taking a branched cover of S 3 over an unknot and lifting a branched surface constructed from a train track. The outline of the paper is as follows. In §2, we review basic de?nitions and tools needed for the proofs. In §3, we introduce a family of pseudo-Anosov braid types and present their train tracks. We proceed with the general template constructions in §4, and identify universal subtemplates to prove the Main Theorem in §5. The ?nal section, §6, gives a summary of to what extent we can classify the universally ?bred knots and links. We propose a class of examples of pseudo-Anosov ?brations which we believe do not induce all knot and link types in the transverse ?ow.

2

Background

Several of the topics below require entire books to do the material justice: we limit ourselves to the minimal set of ideas necessary to follow the remaining sections.

2.1

Fibred knots and links

A knot/link K is said to be ?bred if there is a ?bration π : S 3 ? K → S 1 with ?bre a spanning surface Σ for K . The complement of a neighborhood of K in S 3 is homeomorphic to the mapping torus of a homeomorphism Φ : Σ → Σ: i.e., S 3 ? N (K) ? (Σ × [0, 1])/(x, 0) = (Φ(x), 1). The spanning surface Σ is = called the ?bre surface, and the homeomorphism Φ which is well-de?ned up to isotopy is called the monodromy. The union of the closed orbits of the suspension ?ow by Φ on S 3 ? N (K) is called a planetary link of Φ. It follows from the Thurston classi?cation theorem for surface homeomorphisms [20] that the monodromy of a ?bred knot/link is isotopic to Φ′ , one of the following three types of maps: (1) periodic: (Φ′ )n = Id for some n; (2) pseudo-Anosov: Φ′ possesses a pair of transverse (singular) foliations λu and λs along which Φ′ is uniformly expanding and contracting respectively; 4

(3) reducible: there exists a collection of disjoint simple closed curves C ? Σ such that Φ′ (C) = C , and each connected component of Σ?C has negative Euler characteristic. We say that a ?bred knot/link K is of periodic, pseudo-Anosov, or reducible type respectively if its monodromy map is isotopic to a periodic, pseudo-Anosov, or reducible map. We will be most interested exclusively in those ?bred knots/links of pseudoAnosov type (e.g., the ?gure-eight knot) since these alone may permit universal ?brations. We may reduce the problem to the case where the monodromy is the pseudo-Anosov representative thanks to a theorem of Asimov and Franks [1] which guarantees that any map isotopic to a pseudo-Anosov map (and hence any vector ?eld transverse to the ?bration) may not remove any of these periodic orbits — isotoping the monodromy may generate more, but not less. Hence, the set of planetary links of the pseudo-Anosov map is “minimal” with regards to changing the monodromy map (or the vector ?eld). Henceforth, by the planetary link of a knot/link is meant the planetary link of its pseudo-Anosov monodromy map.

2.2

Braids and branched covers

We use the language of braids throughout the remainder of the paper. Recall that braids are isotopy classes of disjointly embedded arcs monotonically connecting ?xed endpoints (as in Fig. 1). Braids on n strands form a group Bn under concatenation, with standard generators σi denoting an elementary crossing of the ith over the (i + 1)st strand. Inverses correspond to reversing the crossing of the sign.1 A closed or geometric braid is obtained by joining the two sets of endpoints around some ?xed braid axis: algebraically, this corresponds to taking the conjugacy class of the braid in the braid group. The classi?cation of monodromies into periodic, pseudo-Anosov, or reducible type has its analogue for braids. Let β denote the closure of b, an n-braid with braid axis α. Since α is unknotted, its complement ?bres over S 1 with ?bre a disc A meeting β at n points. The monodromy map on A is ?xed on the boundary and, by removing the intersection with β , becomes a map on an n-punctured disc. The Thurston Classi?cation theorem applied to this monodromy map implies that one may classify a braid b as being of periodic

There is a tradition of dynamicists and topologists using opposite sign conventions for braids: we employ the dynamicists’ convention and apologize for the inevitable annoyance.

1

5

type, pseudo-Anosov type, or reducible type. The pseudo-Anosov type braids will be of central importance in analyzing ?brations. While ?brations of knot complements are di?cult to visualize (the explanations of the trefoil, in [19, 10I], and the ?gure-eight knot, in [5] are not short!), the branched covering construction of Goldsmith [15] and Birman [3] provides a concrete mechanism for analysis. We use the method as presented in [5, pp. 26-30], keeping similar notation. Lemma 2.1 [15, Lemma 1] Let β be a single-component unknotted geometric braid with braid axis α and meridional disc A spanning α. Let p : S 3 → S 3 be a branched covering space projection whose branch set is β . Then α = p?1 (α) is a non-trivial ?bred knot/link in S 3 with ?bre A = p?1 (A). A result of Birman [3] shows that every ?bred knot may be obtained as p?1 (α) for some β and some (irregular) cover p. Moreover, the ?bration of the complement of α is itself a lift of the ?bration of the complement of α. For the remainder of this paper, we work with pairs of curves (α, β) which are exchangeable, i.e., there is an isotopy of S 3 which presents α as a geometric n-braid with braid axis β and vice versa [17]. Let a (resp. b) denote the braid in Bn whose closure is α, with β as its axis (resp. the braid whose closure is β with axis α). In this case, α is the closure of am if p is m-fold.

2.3

Train tracks and templates

The problem of how to understand the complex dynamics associated to a pseudo-Anosov surface homeomorphism was resolved with the theory of train tracks. The central idea, going back to Williams [22], is to project the foliation down to a branched manifold. We call attention to [18, 2] for treatments of this theory from di?erent perspectives. Recall that a pseudo-Anosov surface map is characterized by the pair of transverse singular foliations, λs and λu , along which the map is uniformly contracting and expanding respectively. Roughly speaking, a train track for a pseduo-Anosov map is a branched 1-manifold obtained by cutting open the singular foliation λs in a certain manner and collapsing each leaf to a point: see [12] for examples relevant to this paper. The dynamics of the pseduo-Anosov homeomorphism is then easily represented by an induced self-immersion of the train track. We note that there are several presentations available for train tracks. For the remainder of this paper, we follow e.g., [2, 6] and represent the 6

train track as a graph. The induced map is then easily represented as a map taking each edge of the graph to a sequence of edges. Two simple examples of train track graphs and associated graph maps are illustrated in Fig. 10. The problem of how to capture all of the knot and link data of a dynamically complex 3-dimensional ?ow was successfully tackled by Birman and Williams in the early 1980s [4, 5] in the theory of templates (a.k.a. “knotholders”). De?nition 2.2 A template is an embedded branched 2-manifold with boundary, out?tted with an expansive semi?ow. Templates have a natural decomposition into a ?nite number of branchline charts (Fig. 2[left]), each containing a number (> 1) of incoming and outgoing strips whose ends are joined respecting the direction of the semi?ow. The term “semi?ow” means that the ?ow is well-de?ned in forward time, but not in backward time (at the branch lines in particular — there is a loss of uniqueness). By “expansive,” it is meant that the semi?ow expands volume everywhere.

x1

x4 x3

x2

Figure 2: A branchline chart for a template [left]; an example of a template with an unknotted periodic orbit on it [right]. In the alphabet {x1 , x2 , x3 , x4 }, this knot corresponds to the word x2 (x2 x4 )2 = (x1 x1 x2 x4 x2 x4 ). 1

Every template carries an in?nite link of periodic orbits of the semi?ow. There is a well-de?ned symbolic language for describing these orbits. Upon labeling each strip of a template T with a symbol xi , it is clear that to each forward orbit of the semi?ow on T is associated a semi-in?nite word in the alphabet {xi }. Furthermore, from the expansivity of the semi?ow, it can be shown that periodic ?owlines on T are in fact in bijective correspondence with (admissible) periodic words in this alphabet, up to cyclic permutations of the words (see [5] or [13, Lem. 2.4.1] for details). 7

The Template Theorem [5, Theorem 2.1] of Birman and Williams gives very general conditions under which the dynamics of a three-dimensional ?ow is accurately captured by a template: see [13] for a comprehensive treatment. For our purposes, we note that for a ?bred knot/link K with pseudo-Anosov monodromy, there is a template T ? S 3 ? K whose knot and link types are in bijective correspondence with those of the ?ow except for a ?nite number of orbits on the boundary of the template. The template is obtained by suspending the train track graph map for the monodromy, and cutting along the vertices of the graph as necessary. These cuts may produce “extra” orbits on the templates not in the original ?ow. To minimize confusion, we note that for the remainder of this paper, any such exceptional orbits which we “lose” are unknots, and are easily found elsewhere in the interior of the template. De?nition 2.3 A universal template in S 3 is one which possesses all knot and link types as closed orbits of the semi?ow. The paper [14] established that universal templates exist in abundance. From this work and [13] comes a practical criterion for determining when a given template is universal. One additional de?nition is needed. Given any closed orbit κ on a template T ? S 3 , the twist, τ (κ), is de?ned to be the twist number of the normal bundle to T along κ (which is either an annulus or M¨bius band having κ as its core). For the remainder of this paper, we consider o twists for unknotted curves only, avoiding any ambiguity in how to count twist: an unknot with zero twist is one whose normal bundle can be isotoped to be a plane annulus. The following criterion is a very slight modi?cation of [13, Cor. 3.2.17], with a near-identical proof. Since the method of proof relies heavily on terminology and techniques of [13], we suppress the details. Theorem 2.4 Let T be a template in S 3 . Suppose that there exist three disjoint closed orbits on T , κ, κ′ , κ′′ such that (1) they are separable unlinked unknots; (2) τ (κ) = 0, τ (κ′ ) > 0, τ (κ′′ ) < 0; and (3) these three unknots intersect some branchline of T as in Fig. 3[left] with the speci?ed adjacencies and strip crossings. Then T is a universal template. 8

κ′

κ

κ′′

?

?

+

Figure 3: κ′ , κ, κ′′ from left to right [left]; a template and its cartoon representation — top and bottom are identi?ed [center,right].

The proofs of all the main theorems in this paper consist of constructing and simplifying templates for ?bred knots and links, and then searching for the three unknots {κ, κ′ , κ′′ } to which one may apply Theorem 2.4. Because these templates are quite complex, we introduce a “cartoon” notation (cf. [16]). (1) Strips in the template are collapsed along the direction transverse to the semi?ow to become braided curves, except in a neighborhood of the branch lines; (2) Each half-twist of each strip is represented by a label + or ? within a box, depending on the sign of the crossing; (3) Templates will often be “cut open” with the top and bottom to be identi?ed. For example, the template in Fig. 3[center] is expressed in cartoon form in Fig. 3[right].

2.4

Templates for planetary links

The branched cover construction allows one to construct a template for the planetary link of certain ?bred knots/links. Recall the notation that β , the closure of b, is an unknotted geometric braid with braid axis α; p is an m-fold branched cover over β ; and α = p?1 (α) is the lift of the braid axis, which, in the case of exchangeable braids is the closure of the braid am . We summarize the procedure of [5, pp. 26-30] in the case of an exchangeable braid. If b is a braid of pseudo-Anosov type, then α is ?bred and also of 9

pseudo-Anosov type, since the monodromy map for α projects under p to the monodromy map of α represented by the pseudo-Anosov braid type b. The following procedure yields a template for the ?bration of the closure of am : (1) Construct a train track graph for b along with its induced map; (2) Suspend the train track graph map and cut as necessary to obtain a template T in the solid torus S 3 ? α; (3) Peel o? a copy of the unknot β from the boundary of T and ?nd a spanning disc D for this unknot; (4) Cut T along D and glue m copies of this cut template together end-toend cyclically to obtain U m , the template for the ?bration of the complement of α. This procedure was carried out carefully in [5, pp. 28-30] for the ?gure-eight ?1 knot, which is the closure of the braid (σ1 σ2 )2 . Note that in this case a = ?1 b = σ1 σ2 is the simplest pseudo-Anosov braid type. The train track graph for this braid is an interval, and the graph map is represented in Fig. 10[left]. We reproduce this example in cartoon notation in Fig. 4 in preparation for the proof of Main Theorem (c). Familiarity with this example will reveal patterns in the general case.

3

Families of ?bred knots and links

We focus on the following families of braids along which to perform the branched cover construction. De?nition 3.1 Given integers n1 , n2 , · · · , nk ≥ 1 (with k > 1), denote by b := b(n1 ,··· ,nk ) the braid on (1 + k nr ) strands given by the braid word r=1

?1 ?1 σ1 σ2 · · · σn1 σn1 +1 · · · σn1 +n2 · · · σ n1 n2 (?1)(k+1) 1+ k?1 nr r=1

···σ

nk

(?1)(k+1) k r=1 nr

.

(2)

Thus, b is equivalent to the form given in (1). For example, the braid in Fig. 1 is b(2,2,1,1) . Clearly, one can also use the mirror image of these braids, and all the theorems we prove hold here as well. Lemma 3.2 The closure of b(n1 ,··· ,nk ) is self-exchangeable. Speci?cally, given β the closure of b(n1 ,··· ,nk ) with braid axis α, there exists an isotopy of S 3 taking α to the closure of b(n1 ,··· ,nk ) with braid axis β . 10

+

?

+ ? +

? + ?

Figure 4: The branched cover construction for the ?gure-eight knot complement. The ?1 suspension of the train track for the braid b = σ1 σ2 is a template [upper left]. By peeling o? a copy of β and cutting along its spanning disc [lower left], one has a fundamental domain for the template of the m-fold cover [right].

Proof: We induct on the length of the braid word b. Assume as an induction hypothesis that the N -strand closed braid β exchanges with α as in Fig. 5[left]: speci?cally, that there is an isotopy which is ?xed on a solid ball as indicated. ?1 Now, for a braid of the form bσN (in Fig. 5[right]) or bσN , ?t the ?nal strand within a larger ?xed solid ball and perform the isotopy given by induction. Fig. 6 shows that this ?nal strand of β may be exchanged relative to a yet smaller ?xed ball. Note that in the diagram, the set of crossings in the braid word b is ?ipped and rotated, but is kept together rigidly. Lemma 3.3 The braid b = b(n1 ,··· ,nk ) is of pseudo-Anosov type. A proof of this can be explicitly given by one of the [several] algorithms available.2 We note, however, that the braid b is precisely that considered by Gabai and Kazez [12], who show the pseudo-Anosov braid type. By Lemma 3.3, we have the following;

We have used the Bestvina-Handel algorithm [2] for this family: though straightforward, the algorithm is lengthy and of little consequence for the remainder of this article.

2

11

b b

b

Figure 5: The induction hypothesis: the closure of the braid b can be exchanged with its axis rel a ?xed ball [left]; the setup for the induction step [right].

Corollary 3.4 For each integer m > 1, the closure of (b(n1 ,··· ,nk ) )m is a ?bred knot/link of pseudo-Anosov type. The braid b(n1 ,··· ,nk ) induces a very simple map on the closed disc consisting of a collection of k rotations which alternate direction, and the associated train track is very straightforward to compute. Let G = G(n1 ,··· ,nk ) be a chain of k radial “stars” S1 , · · · , Sk , each star having valence (ni + 1) vertex if ni > 1 (Fig. 7). The radial stars alternate their up-down orientation, and small loops (corresponding to the periodic orbit) are attached at the ends of the radial oriented edges as in Fig. 7. The set of non-loop edges in Si equals {e0 } if i ni = 1 (resp. {e0 , · · · , eni } if ni > 1). For example, see Fig. 8. i i Lemma 3.5 An e?cient 3 graph map g = g(n1 ,··· ,nk ) : G → G for the train track graph of b(n1 ,··· ,nk ) is given as gk ? · · · ? g1 , where gi rotates the i-th star Si in the clockwise (resp. counterclockwise) direction if i is even (resp. odd), see Fig. 9. The algebraic presentation of the train track map is simple to derive knowing that it is composed of rotations. For simplicity, we ignore all loop-edges in the train track. Given an oriented edge E , we denote the same edge oriented in the opposite direction by E . The following lemma is used to derive a normal form of the template T(p,q) in Section 4.1.

For the de?nition of the e?cient graph maps, see [2, pp. 114]. In general, the e?cient graph map is not unique for the braid b.

3

12

twist

?ip

Figure 6: The induction step: the closure of bσN can be exchanged with its axis rel a ?xed ball.

e0 i e1 i eni?1 i eni i e0 i eni i eni?1 i

e0 i e0 i

e1 i

Figure 7: i odd, ni = 1 [upper left]; i even, ni = 1 [upper right]; i odd, ni > 1 [lower left]; i even, ni > 1 [lower right].

Lemma 3.6 The images of the edges under g = g(p,q) are as follows; For p = q = 1, g(e1 ) = e0 e0 , 2 1 g(e2 ) = e0 e0 e0 . 1 2 2 For p, q > 1, g(ep ) = e0 eq e0 , 1 2 2 1 g(eq ) = e0 e0 eq e0 e1 , 2 2 2 1 1 2 g(ei ) = ei+1 (i = 0, · · · , p ? 1), 1 1 g(ei ) = ei+1 (i = 0, · · · , q ? 1), 2 2 13

b

b b

slide

b

b

b

e1 1 e0 1 e2 2 e1 2 e0 2 e2 1 e0 1 e0 2

e2 3

e1 3

e3 3

e0 3

PSfrag replacemen

Figure 8: G(1,2) [left]; G(2,1,3) [right].

e0 i ? gi(e0) i eni i eni?1 i e0 i ? gi(eni?1) i gi(e0) i gi(eni ) i

Figure 9: i even, ni = 1 [left]; i even, ni > 1 [right].

see Fig. 10.

?

?

Figure 10: g(1,1) [left]; g(2,2) [right].

4

Subtemplates in branched covers

The various cases of the Main Theorem all possess the same strategy of proof. Given the braid b = b(n1 ,··· ,nk ) , we suspend the train track G = G(n1 ,...,nk ) by the train track map g = g(n1 ,...,nk ) to obtain a template T(n1 ,...,nk ) . We then use m the branched covering method to obtain a template U(n1 ,...,nk ) which captures m . Since, in general, U m the monodromy of the closure of the braid b (n1 ,...,nk ) is much too complex to visualize, we focus on a particular subtemplate which we may show to be universal. In this section, we prove the technical results for restricting attention to appropriately simple subtemplates.

4.1

A normal form

We now derive the branched surface T(p,q) obtained from G(p,q) by suspending g(p,q) from the previous section. In what follows, we assume p, q > 1. (Other 14

cases are much simpler and left to readers.) Let q? and qr are the valence (p + 1) vertex and (q + 1) vertex of S1 and S2 respectively. These suspend to give the knots k? and kr . Notice that the set of loop-edges in S1 ∪ S2 is a periodic orbit, say A(p,q) , corresponding to the closure of b(p,q) . To visualize T(p,q) as a template, we need to “cut open” q? , qr and A(p,q) , and ?atten out the strips in some canonical way. This is illustrated in Fig. 11.

e2 1 ep?1 1 e1 1 eq 2 ep 1 e0 1 eq?1 2 e1 2 e2 2 e2 1 e1 1 e0 1 eq 2 eq?1 2 e2 2 e1 2 e0 2 e0 2 ep?1 1 ep 1 e2 1 e1 1 eq 2 e0 1 e0 2 e1 2

?

e2 2 eq?1 2

ep 1

ep?1 1

Figure 11: Flattening a pair of stars into strips.

Recall that g(ep ) = e0 eq e0 . We split the edge ep into E and E ′ so that 1 2 2 1 1 g(E) = e0 eq and g(E ′ ) = e0 . Split the edge eq into F, F ′ and F ′′ so that 1 2 2 2 g(F ) = e0 , g(F ′ ) = e0 and g(F ′′ ) = eq e0 e1 . For the template T(p,q) , we can 2 2 2 1 1 use the symbols xi=0..p+1 , and y,zj=0...q+1 for the ?rst star, and second star respectively. The edge e0 goes to x0 , and the edge e0 goes to z0 , etc., and ep is 2 1 1 split into xp and xp+1 corresponding to E and E ′ , and eq split into zq , zq+1 , y 2 corresponding to F, F ′ and F ′′ . We denote the left end point and right end point of the branch segment of the strip x by p? (x) and pr (x) respectively. Since the initial vertex of e0 , e1 , · · · , ep is the same vertex q? , we must abstractly identify 1 1 1 (1) pr (x1 ), pr (x2 ) ,· · · , pr (xp ) and p? (x0 ). In the same manner, we need to identify (2) pr (x0 ) and p? (y), (3) pr (y) and p? (zq+1 ), (4) pr (zq+1 ) and p? (zq ), (5) p? (z0 ), p? (z1 ), · · · , p? (zq?1 ) and pr (zq ), and (6) p? (xp ) and pr (xp+1 ). For the remainder of the paper, we conveniently leave these boundary curves as distinct. When constructing subtemplates of this branched surface, we are careful never to use these edges within the subtemplates. In this way, we do not need to worry about the changes to the periodic orbits set we have made. The normal form of T(p,q) appears in cartoon form as in Fig. 12.

m In order to construct U(p,q) , the template for the m-fold branched cover of T(p,q) , we must “peel o?” a parallel copy of β and cut T(p,q) by a spanning disc

15

+ PSfrag

+

+

?

?

xp+1 xp xp?1

x3

x2

x1

x0

y

zq+1

zq zq?1

z3

z2

z1

z0

Figure 12: The normal form for the template T(p,q) .

for this unknot: cf. the ?gure-eight knot case in Fig. 4. Thus, the fundamental m domain of U(p,q) depends on the linking properties of β with T(p,q) . Lemma 4.1 Let β be the closure of b(p,q) and β ′ a parallel copy. Then β ′ links T(p,q) as in Fig. 13. Proof: Note that g(n1 ,n2 ) has three ?xed points q? , qc , qr , where q? and qr are as above and qc ∈ eq (see Lemma 3.6). These suspend to give the three unknots 2 k? , kc , and kr . It is clear from the image of the train track (see Figs. 10 and 14) that the ?xed point qc is “in back” of the image of all the other edges, and that consequently the suspension of this ?xed point, kc , is an unknot which is of twist zero and separable from all other orbits on the template. Hence, ?k(kc , β) = 0. Since the train track map rotates each star about its central vertex with opposite orientations, it is easy to see that ?k(k? , β) = +1 and ?k(kr , β) = ?1 (see Fig. 14[left]). Since the strip y contains kc and ?k(kc , β) = 0, y and β are linked as in Fig. 14. Because of the way we unfolded the star in Fig. 11, the way that zq+1 and β are linked is same as that of y and β . Notice that the edges e0 , · · · , ep were all “split o?” of q? , and ?k(k? , β) = 1 1 +1. Hence x0 , · · · , xp , xp+1 and β are linked as in Fig. 14. Since the edges q?1 e0 , · · · , e2 , F (? eq ) were also “split o?” of qr and ?k(kr , β) = ?1, the strips 2 2 z0 , · · · , zq?1 , zq and β are linked as in Fig. 14. This completes the proof. 16

+

+

+

?

?

xp+1 xp xp?1

x3

x2

x1

x0

y

zq+1

zq zq?1

z3

z2

z1

z0

Figure 13: A parallel copy of the closed braid β links certain strips of T(p,q) .

k?

kc

kr

′ k?

′ kc

′ kr

β

Figure 14: p = q = 2 [left]; k = 4 and n1 = n2 = n3 = n4 = 2 [right].

17

m By Lemma 4.1, the fundamental domain of U(p,q) is as in Fig. 15, where we m have cut along a spanning disc of β . To form U(p,q) , take m copies of this fundamental domain joined cyclically end-to-end: see, e.g., Fig. 17.

+

+

+

?

?

m Figure 15: The fundamental domain of U(p,q) , cf. Fig. 4.

4.2

Subtemplates

m The (very complicated) normal form U(ni ,ni+1 ) is tame compared to the full m template U(n1 ,...,nk ) . Fortunately, we do not need to consider the most general case.

Lemma 4.2 Suppose i is odd [resp. even]. Then T(ni ,ni+1 ) [resp. its mirror image] is a subtemplate of T(n1 ,...,nk ) . 18

Proof: We prove this result on the level of the train track (by subdividing and removing extraneous portions), and then suspend this to obtain the result for subtemplates. We only prove the case ni , ni+1 > 1, for other cases are similar. Let G = G(n1 ,··· ,nk ) and g = g(n1 ,··· ,nk ) . For the proof of the lemma, it is ni+1 enough to show that there exist subdivisions I2 I1 = eni , J3 J2 J1 = ei+1 of G i that satisfy the following: Remove I2 and J2 from Si and Si+1 respectively. Then (Si ∪Si+1 )′ denotes the graph obtained from (Si ∪Si+1 ) by identifying the right end point of J3 and the left end point of J1 . Then g((Si ∪ Si+1 )′ ) = gi+1 (gi (Si ∪ Si+1 )), see Fig. 16. We demonstrate the case i = 1, the proof for other cases being similar. Denote by ci the following edges of the train track: ci = e0 if ni = 1 and ci = eni e0 i i i if ni > 1. By induction on k , one demonstrates that g(en1 ) = e0 c2 · · · ck , 1 1 g(en2 ) = e0 c3 · · · ck?1 ck ck ck?1 · · · c2 e0 e1 , and g(e) = g2 (g1 (e)) for e ∈ S1 ∪ 2 2 1 1 S2 \ {en1 , en2 }. Take subdivisions I2 I1 = en1 and J3 J2 J1 = en2 such that 1 2 1 2 g(I1 ) = e0 c2 , g(J3 ) = e0 , and g(J1 ) = c2 e0 e1 . Since g(I1 ) = g2 (g1 (en1 )), 1 2 1 1 1 n g(J3 )g(J1 ) = g2 (g1 (en2 )) and g(e) = g2 (g1 (e)) for e ∈ S1 ∪ S2 \ {en1 , e2 2 }, we 2 1 have g((S1 ∪ S2 )′ ) = g2 (g1 (S1 ∪ S2 )).

J2 I2 I1 J3 J1 ?

g(I2)

g(J2)

I1

J3 J1

Figure 16: g = g(2,2,1) [top]; (S1 ∪S2 )′ [lower left]; g((S1 ∪S2 )′ ), which equals g2 (g1 (S1 ∪ S2 )) [lower right].

m Lemma 4.3 Suppose i is odd [resp. even]. Then U(ni ,ni+1 ) [resp. its mirror m image] is a subtemplate of U(n1 ,...,nk ) .

Proof: We consider the case of i odd and ni , ni+1 > 1; the other cases share identical proofs. The key step in this lemma is to determine the linking properties of β with the subtemplate obtained from Lemma 4.2. 19

′ ′ ′ ′ Note that g(n1 ,··· ,nk ) has three corresponding ?xed points q? , qc , qr , where q? and ′ are (n + 1) vertex and (n qr i i+1 + 1) vertex of Si and Si+1 respectively, and ni+1 ′ ′ ′ ′ qc ∈ ei+1 . Denote by k? , kc , kr the three unknots given by suspending these points. Let β and β ′ be the closures of b(n1 ,n2 ) and b(n1 ,··· ,nk ) respectively. ′ As in the proof of Lemma 4.1, one shows that ?k(k? , β ′ ) = ?k(k? , β) = +1, ′ ′ ?k(kc , β ′ ) = ?k(kc , β) = 0, and ?k(kr , β ′ ) = ?k(kr , β) = ?1, see Fig. 14(b). This together with Lemma 4.2 implies Lemma 4.3.

5

The Main Theorems

We now have assembled all the ingredients to prove the Main Theorems.

5.1

Main Theorem (c)

We begin with the easiest and least general of the three Main Theorems. A proof can be obtained implicitly by a combination of Lemma 4.3 with a geometric argument from [14, pp. 444-5]. However, to set a pattern for the proofs of Main Theorems (a) and (b), we give a di?erent proof of (c). Theorem 5.1 Let b = b(n1 ,··· ,nk ) where nj = nj+1 = 1 for some j . Then the closure of bm is a knot/link in S 3 with the universal ?bration property for every m ≥ 2.

m Proof: From Lemma 4.3, we may restrict attention to the subtemplate U(1,1) m within U(n1 ,··· ,nk ) . The template T(1,1) is given in cartoon form in Fig. 4, along m with a representation of the fundamental domain of U(1,1) . Our strategy is to m identify three unknots on U(1,1) — κ, κ′ , and κ′′ — to which we can apply Theorem 2.4.

Speci?cally, using the symbolic representation of the orbits as periodic words, let κ = (y), κ′ = (xm ), and κ′′ = (zm ). Recall that (xm ) means to follow the periodic orbit which traverses the x-strip m times. While there is an ambiguity in the symbolic representation (in terms of which fundamental domain the symbol is on), the only choice is for (y), and by cyclic symmetry, the result is independent of the choice. It is trivial to see from Fig. 17[left] that (1) these three curves are separable unknots; and (2) κ has zero twist, κ′ has positive twist, and κ′′ has negative twist. The only other ingredient needed to invoke Theorem 2.4 is that the three 20

orbits meet at a branchline as in Fig. 3. This is not the case, since the strip z does not fully cover the branchline. To remedy this, we replace (xm ) with (yxm ). This orbit is still a separable unknot (using [13, Prop. 3.1.19]) whose (already positive) twist has increased by two and which now passes through the triple branchline in precisely the manner of Fig. 3. Hence, by Theorem 2.4, m U(1,1) is universal.

+ + ? ? + ? ?

+ ? + ? ? + +

m Figure 17: A cartoon representation of the fundamental domain of U(1,1) [left]; the ′ ′′ unknots κ, κ , and κ in the case m = 6 [center]; all three orbits pass through the same branch line [right].

5.2

Main Theorem (b)

The following theorem will allow us to completely classify the ?brations of 2bridge knot complements in the next section. Theorem 5.2 The closure of (b(n1 ,··· ,nk ) )2 has the universal ?bration property.

2 Proof: To prove Theorem 5.2, it su?ces to show that U(p,q) is universal for all p and q . We assume p, q > 1, the other cases being both simpler and similar to Main Theorem (c). The strategy, as in the previous theorem, is to ?nd three 2 well-chosen unknots κ, κ′ , κ′′ on U(p,q) .

As before, choose κ := (y)∞ the untwisted unknot separable from all other orbits on the template. For p odd (resp. even), de?ne κ′ := (yx0 x1 x2 · · · xp?1 xp ) κ′′ := (yzq z0 z1 · · · zq?1 ) resp. κ′ := (yx1 x2 · · · xp?1 xp ) , resp. κ′′ := (yzq+1 z0 z1 · · · zq?1 ) , 21

see Fig. 18. For q odd (resp. even), de?ne

see Fig. 19. Note that there is an ambiguity as to which fundamental domain the orbit segment y of κ′ and κ′′ passes through. We specify as follows: take κ′ and κ′′ such that the orbit segment y passes through the fundamental domain where κ lies. The ?gures imply that these are separable unlinked unknots with τ (κ) = 0, τ (κ′ ) > 0, and τ (κ′′ ) < 0. These unknots κ, κ′ and κ′′ pass 2 through the triple branchline as in Fig. 20. Hence, by Theorem 2.4, U(p,q) is universal.

+ +

+

? ?

+ +

+

? ?

+ +

+

? ?

+ +

+

? ?

Figure 18: The orbit κ′ in Main Theorem (b); p is odd [left]; even [right].

5.3

Main Theorem (a)

For part (a) of the Main Theorem — that the universal ?bration property holds for m su?ciently large — we give a more precise formulation. 22

+ +

+

? ?

+ +

+

? ?

+ +

+

? ?

+ +

+

? ?

Figure 19: The orbit κ′′ in Main Theorem (b): q is odd [left]; even [right].

Theorem 5.3 Let N := mini (max{ni , ni+1 }). The closure of (b(n1 ,··· ,nk ) )m has the universal ?bration property for all m ≥ N (N ? 1).

m Proof: By Lemma 4.3, it is enough to show that U(p,q) is universal for each m ≥ N (N ? 1), where N = max{p, q}. As before, we assume p and q greater than 1.

Let m = pt′ + s′ = qt′′ + s′′ be Euclidean integer factorizations of m; hence 0 ≤ s′ < p and 0 ≤ s′′ < q . As m ≥ N (N ? 1), it follows that t′ ≥ s′ and m t′′ ≥ s′′ . Consider the following three periodic orbits on U(p,q) : κ := (y) κ′ := κ′′ := (yx1 · · · xp )t ?s (x0 x1 · · · xp )s (zq+1 z0 z1 · · · zq?1 ) 23

t′′ ?s′′

′ ′ ′

.

s′′

(3)

(zq z0 z1 · · · zq?1 )

κ′

κ

κ′′

κ′

κ

κ′′

Figure 20: A closeup of the branch line in the proof of Main Theorem (b); p and q even [left]; p and q odd [right].

The orbit κ is, as before, an untwisted unknot which lies within one fundamental m domain of U(p,q) . The orbit κ′ passes through p(t′ ?s′ )+(p+1)s′ = pt′ +s′ = m m fundamental domains of U(p,q) . Except for the occasional excursions about the y orbit (which do not change the knot type), the orbit κ′ can be thought of as a 1m strand braid about the braid axis of U(p,q) : it is therefore an unknot. All twisting in the x-strips is of a positive nature, so that κ′ is an unknot with positive twist. A similar argument shows that κ′′ passes through q(t′′ ? s′′ ) + (q + 1)s′′ = m fundamental domains and is an unknot with negative twist separable from κ′ . These orbits are a triple of separable unknots which satisfy the conditions of m Theorem 2.4, as illustrated in Fig. 21. Hence U(p,q) is universal.

κ′ κ κ′′

Figure 21: A closeup of the branch line in the proof of Main Theorem (a).

24

6

Concluding remarks

We have displayed a large family of ?bred knots/links whose ?brations force all knot and link types in the complement. It remains an open question how large this family can be: Question 6.1 Is every ?bred knot/link of pseduo-Anosov type universally ?bred ? We conclude this paper with a few remarks on this question.

6.1

2-bridge knots

Recall that a 2-bridge knot is one which can be isotoped so as to have precisely four critical points (two minima and two maxima) with respect to the standard height function in R3 . We can a?rmatively answer Question 6.1 in this category. Corollary 6.2 Every ?bred non-torus 2-bridge knot has the universal ?bration property. Proof: The work of Gabai and Kazez [12] explicitly shows that the class of ?bred 2-bridge knots is precisely that class of knots whose ?bres are a chain of plumbed Hopf bands, and that the monodromy is of pseudo-Anosov type if and only if the Hopf bands are not all of the same sign, which is to say, if and only if it is not a torus knot. We claim that such a chain of plumbed Hopf bands has a closed braid form of (b(n1 ,··· ,nk ) )2 . The proof is inductive on the number of Hopf bands and illustrated in Fig. 22. The key step is noting that the ?bre for the closure of (b(n1 ,··· ,nk ) )2 is a stack of discs attached by a pairs of half-twisted bands: this is precisely a chain of plumbed Hopf bands. From Theorem 5.2, we have that this knot must be universally ?bred.

6.2

Connected sums

We can enlarge the class of universally ?bred knots considerably by taking connected sums. Proposition 6.3 The connected sums of any ?bred knot/link with any universally ?bred knot/link is universally ?bred. 25

Figure 22: Plumbings of Hopf bands [right] can be arranged as a closed braid of the form (bn1 ,··· ,nk )2 [left].

Proof: In [11, Corollary 1.4.], it is shown that the Murasugi sum of ?bre surfaces is “natural”. Speci?cally, if Ki is a ?bred knot/link (i = 1, 2) with ?bre surface Σi and monodromy Φi satisfying Φi |?Σi = id, then the Murasugi sum of Σ1 and Σ2 along a 2k -gon D common to Σ1 and Σ2 is ?bre surface and has ? ? ? monodromy Φ1 ? Φ2 , where Φi is equal to Φi on Σi and the identity map elsewhere. Even more importantly, the proof of this result uses vector ?elds transverse to the ?bration to achieve the result. Thus, we actually have that a transverse vector ?eld of the Murasugi sum decomposes naturally. The simplest kind of Murasugi sum (where the attachment is along a 2-gon in the spanning surfaces) is precisely the connected sum. Choose the monodromy Φ1 of the universally ?bred knot to be the pseudo-Anosov representative, isotoped so as to be the identity map on the the boundary. Likewise, choose the monodromy Φ2 to be the Thurston canonical form isotoped to be the identity on the boundary. The composition monodromy on the connected (in this case, Murasugi) sum is easily isotoped to have an invariant curve separating the universal pseudo-Anosov component from the Thurston form of Φ2 . Thus, the theorem of Asimov and Franks implies that any transverse vector ?eld must have at least this set of periodic orbits — all knots and links. As a corollary of the proof, we see that in certain instances, more general Murasugi sums (e.g., plumbing) can preserve the universal ?bration property as well. One would need to control (1) the dynamics on the attaching region, to preserve the pseudo-Anosov property of the monodromy, and (2) the location of the attaching region, so that it does not interfere with the universal subtemplate. 26

6.3

Too-twisted monodromies

We believe that there are pseudo-Anosov ?bred knots or links whose monodromies are too “twisted” to admit a universal template in the suspension. Our proposed example stems from the fact that the proof of Main Theorem (c) requires a su?ciently high order branched cover. Conjecture 6.4 For p and q su?ciently large, the closure of (b(p,q) )3 is not universally ?bred. Our reason for conjecturing the non-universality stems from the di?culty in ?nding enough unknotted orbits on which to build a universal subtemplate.

References

[1] D. Asimov and J. Franks, Unremovable closed orbits, in Geometric Dynamics, Ed. J. Palis, Lect. Notes in Math. 1007, Springer-Verlag, (1983) 22-29. [2] M. Bestivina and M.Handel, Train-Tracks for surface homeomorphisms, Topology 34 (1) (1994) 109-140. [3] J. Birman, A representation theorem for ?bered knots and their monodromy maps, in Topology of Low-dimensional Manifolds, Lecture Notes in Math. 722, Springer-Verlag (1977) 1-8. [4] J. Birman and R. Williams Knotted periodic orbits in dynamical systems – I: Lorenz’s equations, Topology 22(1), (1983) 47-82. [5] J. Birman and R. Williams, Knotted periodic orbits in dynamical systems – II: Knot holders for ?bered knots, Contemp. Math. 20, (1983) 1-60. [6] A. de Carvalho and T. Hall, Braid forcing and star-shaped train tracks, preprint ArXiv math.DS/0204115, 2002. [7] J. Casasayas, J. Martinez Alfaro, and A. Nunes, Knots and links in integrable Hamiltonian systems, J. Knot Theory & Rami?cations 7(2) (1998) 123-153. [8] J. Etnyre and R. Ghrist, Gradient ?ows within plane ?elds, Commun. Math. Helv. 74 (1999) 507-529. [9] A. Fomenko and T. Z. Nguyen, Topological classi?cation of integrable nondegenerate Hamiltonians on isoenergy three-dimensional spheres, in Adv. Soviet Mathematics Vol. 6, A. Fomenko ed., pp. 267-296, AMS, 1991. [10] J. Franks, Knots, links, and symbolic dynamics, Ann. Math. 113 (1981) 529-552. [11] D. Gabai, The Murasugi sum is a natural geometric operation. II, Contemp. Math. 44, (1985) 93-100.

27

[12] D. Gabai and W. Kazez Pseudo-Anosov maps and surgery on ?bred 2-braidge knots, Top. & Its Applications 37 (1990) 93-100. [13] R. Ghrist, P. Holmes and M. Sullivan, Knots and Links in Three-Dimensional Flows, Lect. Notes in Math. 1654, Springer-Verlag (1997). [14] R. Ghrist, Branched two-manifolds supporting all links, Topology 36 (2) (1997) 423-448. [15] D. Goldsmith, Symmetric ?bered links, in Knots, Groups, and 3-Manifolds, Ann. of Math. Studies 84, Princeton Univ. Press, Princeton, (1975) 3-23. [16] P. Melvin and N. Tu?llaro, Templates and framed braids Phys. Rev. A, 44(6) (1991) 3419-3422. [17] H. Morton and M. Rampichini, Mutual braiding and the band presentation of braid groups, in Knots in Hellas ’98 Delphi, 335-346. [18] R. Penner and J. Harer, Combinatorics of Train Tracks, Ann. of Math. Studies 125, Princeton Univ. Press, Princeton (1992) [19] D. Rolfsen, Knots and Links, Publish or Perish Press, 1976. [20] W. Thurston, On the geometry and dynamics of di?eomorphims of surfaces, Bull. Am. Math. Soc. 19 (2) (1988) 417-431. [21] M. Wada, Closed orbits of nonsingular Morse-Smale ?ows on S 3 , J. Math. Soc. Japan 41(3) (1989), 405-413. [22] R. Williams, One dimensional nonwandering sets, Topology, 6 (1967), 473-487.

28

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