lain Roderick Aitchison- Isotoping and Twisting Knots and Ribbons

8/3/2019 lain Roderick Aitchison- Isotoping and Twisting Knots and Ribbons

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Isotoping and Twisting Knots and Ribbons. .

Bylain Roderick AitchisonB.Sc. (Hon.) (University of Melbourne, Australia) 1979M.Sc. (University of Melbourne, Australia) 1980C.Phil. (University 0' California) 1983

DISSERTATIONSubmitted in partial satisfaction of the requirements for the degree of

OOCTOR OF PHIlDSOPHYin

Mathematics

in the .GRADUATE DIVISION

OF 1HEUNIVERSITY OF CALIFORNIA, BERKELEY

.. '

Approved:

.................... . . . . . .. . . .. . . . . . . . . . . . . . . . . . .


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1

Isotoping and Twisting Knots and Ribbons

ABSTRACT

In higher dimensions the Seifer t form determines up to isotopy thegeometr ically defined "simple" knots. Every class ical knot i s simple.However, there are numerous examples fo r which the Seifer t form f a i l sto provide sufficient distinguishing invariants. I t was conjecturedthe Alexander polynomial would determine classical fibered knots towithin f ini te ly many possib i l i t ies .

This paper addresses the queetd.om "Is there a geometrically definedclass of class ical knots fo r which the Seifer t form acts as a completeinvariant?"

Geometric conditions known to place rest r ic t ions on the Seifer t formare the properties of being fibered, sl ice and doubly sl ice . For fiberedknots Thurston' s classification of diffeomorphisms o f sur fa ce s leads tofurther subclasses. For fibered ribbon knots, Casson and Gordon haveshown such a knot bounds a ribbon in some homotopy bal l whose complementf ibers over the ci rc le with f iber a handlebody. That th is bal l be stan-da.rd i s .a t present knowledge an additional requirement. Doubly sl iceknots are known in higher dimensions to in teract with codimension oneembedding phenomena.

Our main resul t i s the const ruct ion of infini tely many dist inct knotsKi in 83 which are prime, fibered, of genus 2, ribbon, bounding ribbonsin B4 whose complements f iber over S1 with f iber a handlebody, have monodromy which i s pseudo-Anosov, and thus have complements in S.3 admittinga complete hyperbolic structure, are symmetric sl ices of the O-spun f ig-ure 8 knot, are doubly sl ice , being cons tructed by ambient iso topy of


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a closed surface i n S3. and y et have id en tical S e i f e r t forms. Thus thec l a s s i c a l dimension is even fu rt h e r removed from the higher ones thanpreviously expected.

Along the way we construct fibered knots by ambient isotopy of submanifolds o f spheres, give a shor t proof t ha t every closed orlentable3-manifold has a fibered knot and inf initely many hyperbolic fiberedlin k s and characterise geometrically the simple fibered doubly s l i c eknots i n higher odd dimensions.

Finally we consider the twisting constuction of Stallin g s and re l a t et h i s on the one hand to the Gluck construction on 2-knots and to theconstruction of d if f er en t ribbons by isotopy on th e other.


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CONTENTS

Introduction

Definit ions, Notation and Conventions

Fibered Knots

Diffeomorphisms of SurfacesSlice Knots and Ribbons

Constructing Mapping Tori

Doubly Slice Knots

Further Aspects in Dimension 4Bibliography

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Isotoping and Twisting Knots arid Ribbonsor

Some Simple Samples of Simple Simple Knots

o INTRODUCTIONThe fundamental problem of knot theory i s to find sufficient comp-

utable means of distinguishing examples. In higher dimensions the Seifert form distinguishes a ll odd dimensional simple knots (31), and everyclassical knot is simple. However there are many examples in the classical dimension of knots which are not so distinguished. Simplicity is ageometric condition, and the natural question is whether there i s a geometrically defined class of classical knots for which the abelian invariants arising from the Seifert form form a complete set . Geometric cond-i t ions manifest in the form the Seifert form may take, and again forhigher dimensions, algebraic propert ie s a re realised geometrically.]

The fundamental difference between the higher and lower dimen-sional cases i s the availabil i ty of the techniques of the s-cobordismtheorem and the Whitney tr ick. Failure of the Whitney tr ic k in dimension4 leads to much of the richness and pathology of the classical dimension.For example, in higher dimensions an algebraically slice simple knot iss l ice, whereas t he re a re infinitely many examples of dis t inct classicalknots with t r iv ia l abelian invariants, those with Alexander polynomial1. The recent work of Freedman (16) and Donaldson (57) shows that on theone hand these are a ll topologieally s l ice, but tttat there a re many whichcannot bound smooth slic es, fo r reasons relat ing to possible smoothstructures on 4-dimensional manifolds.

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For f 'ibered knots, the Alexander polynomial cannot be t r iv ia l exceptf'or the unknot. I t has been long known that the only possible f iberedknots of genus 1 are the t refoi l and figure 8 knots. Based on the fewexamples known at the time, Burde and Neuwirth conjectured that therewere only finitely many distinct examples of f ibered knots with a givenAlexander polynomial. Morton (36) showed this to be false at the sametime as Stal l ings (49), the la t te r introducing the notion of " tw is ti ng"for the purpose of generat ing more examples of fibered knots. These ex-amples were distinguishable by their Alexander modules, from which theAlexander polynomial arises, and thus the obvious question was whetherthe Alexander module determined fibered knots to within finitely manypossibili t ies. Quach (44) showed that there were abstractly infinitelymany distinct examples with the same Alexander module structure, andKanenobu (26) gave the f i r s t specific examples of this phenomena. Subse!i'fuently Morton (37) showed that there are infinitely many distinct fiberedknots for any given possible Alexander module.

Quach and Weber (45) rediscovered the examples of Stallings, andmade the further observation that they are a l l r ibbon knots . Recently,Bonahon has produced infinitely many dist inct examples of fibered knotswith the same Seifert form, with pseudo-Anosov monodromy, but which can-not be ribbon. In (6 ), he appeals to a recent result of Casson and Gordon (58) who show that a necessary condition for a fibered knot to beribbon is that the monodromy extend over a handlebody. There are no knownexamples of f'ibered ribbon kno ts not distinguished by their abelian in-variants.

The condition of being doubly slice places further restrict ions onthe Seifert form, and as before, these algeb raic conditions are always

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geometrically realised f'or higher odd dimensional simple knots.The main result of' this paper is the f'ollowing

Theorem: There are imini te ly many dist inct knots K in S3 suCh thatna) K is primen

i i ) K i s f'ibered of' genus 2ni i i ) K i s ribbon, bounding a ribbon R in B4n niV) B4 - R f'ibers over S1 with f 'iber a handlebodynv) K has pseudo-Anosov monodromy, and thus S3 - K admits an n

complete Riemannian metric of' constant curvature -1vi) K is doubly slicen

vii) Kn i s a SYmmetric slice of' the a-spun f'igure 8 knotvii i) K i s constructible by isotopyn

iX) the Seif'ert f'orm of' K i s independent of' nn

A number of concepts arise in the s ta tement of the theorem, andaccordingly the paper i s divided into sections The proof' i s construct-ive, and thus we reach the f'inal result only at the end after a numberof' brief diversions along the way.

In the f i r s t section, we give some definitions, and describe brieflythe f'undamental notion of twisting knots, arising from the calculus of'f'ramed l inks (30). This i s the basic tool of' Stallings (49), Morton (37)and Bonahon ( 7 ) and implicit in Kanenobu (26) and Quach and Weber (4.5).One of' the aims of this work i s to relate this concept to the work ofCasson and Gordon (.58) and to put the results in the broader context of'four and higher dimensions.

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4Section 2 i s devoted to f ibrations, and we discuss in some detai l

the t refoi l and figure 8 knots, crucial to our subsequent constructions.The notion of constructing diffeomorphisms of surfaces and handlebodiesby isotopy in S3 is discussed in this context. We briefly mention theconst ruct ions of fibered knots due to Stallings (49) and apply this togive a new proof that every closed orientable 3-manifold has a fiberedknot. We also proveTheorem 2.2 Every closed orientable 3-manifold has infinitely manyfibered l inks whose complements admit a hyperbolic structure.An interesting phenomena relating fibered l inks and hyperbolic structuresin the 3-sphere i s introduced.

The monodromy of a fibered knot i s a surface diffeomorphism, and sowe discuss Thurston's work in section 3. The techniques of train tracksi s described in the context of distinguishing diffeomorphisms. We proveTheorem 3.7 Infinitely many distinct pseudo-Anosov maps may be obtained by ambient isotopy of a genus 2 surface in S3.

Section 4 briefly descr ibes the relationship between the Seifertform and r ibbon knots , .. and the work of Casson and Gordon.

In section 5 we use the construction of section 3 and apply i t toproveTheorem 5.2 The knots K of Kanenobu are genus 2 fibered ribbon knotsm,nbounding ribbons in B4 whose complements f iber with f iber a handlebody.Theorem 5.4 The knots S of Stallings, Quach and Weber are genus 2mfibered ribbon knots, bounding ribbons as in Theo!em 5.2

We also introduce the examples for our main theorem, and the underlying technique for generating many more,


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5The techniques involved are handlebody decompositions OI 4-maniIolds

and the calculus OI Iramed l inks.Doubly sl ice knots are discussed in section 6, and we prove

Proposition 6.1 The monodromy Ior a simple Iibered doubly sl ice knotin S2n+1 determines the SeiIert Iorm.

We then give a new prooI OI a special case OI Zeeman's theoremTheorem 6.2 (56) I IK is a Iibered knot, then K # -K i s doubly slice.

The point OI this prOOI i s to introduce a new idea into the studyOI Iibered doubly sl ice knots, the notion OI "construction by isotopy":Theorem 6.4knots in S3.

There exi st in in it ely many prime Iibered doubly slice

Our construction gives r ise to the generation OI zill ions OI candi-dates Ior algebraically doubly sl ice Iibered classical knots which areribbon, but probably not geometrically doubly slice. We conjectureConjecture: Every doubly slice Iibered knot i s constructable by isotopy.As evidence Ior this we proveTheorem 6.5 Every simple, Iibered doubly sl ice knot in S2n+1, I J f ~ 3,may be constructed by isotopy.

The construction gives a Iurther maniIestatfuon OI codimension 1embedding phenomena in the study OI doubly sl ice knots. We also observea phenomena we cal l synchronicityTheorem 6.7 Ininitely many doubly sl ice knots arise synchronouslyas I ini te dimensional slices OI ininite dimensional doubly slice knots.

,11Finally in section 7 we discuss remaining 4-dimensional aspects OIthe main theorem, the Gluck construction and the implications OI "motion"in the study OI knots and ribbons.


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1 DEFINITIONS, NOI'ATION AND CONVENTIONSWe work in the smooth category unless otherwise stated.

As general references, we use Gordon (22), Kervaire (29) and Rolfsen et6).n. . n n+2An n-knot K as the amage of an embedded n-sphere S in S

A Seifer t manifold Wn+1 for Kn i s any connected, orientable submanifoldof Sn+2 with boundary Kn Every knot has infini tely many Seifer t mani-

( ) n+2 )folds. The Qomplement C K of K i s S - K, and the exterior X(K isthe complement of an open tubular neighbourhood of K.

In odd dimensions, the Seifer t form of a knot K2n-1 i s the integralbil inear form defined o H (W2n; Z)/ Torsion X H (W2n; Z)/ Torsion byn nthe l inking matrix V ::: (V..) with respect to the Seifer t manifold W2n~ and choice of basis (a . ) fo r H (W2n; Z)/Torsion. Here V.. ::: l k ( a . , a ~ )n ~ Ji s th e lin kin g number in S2n+l determined by taking representativecycles ai and a . fo r the classes (a.) and (a . ) , pushing a . off W2n,J J Jusing the orientation to choose a direction, to obtain ai:, and calculJating the l inking number. The matrix V i s the Seifer t matrix fo r K withrespect to the Seifer t manifold and basis chosen.

Examples: In f ig . i , we give the knots cons truc ted by Stall ings (49),and rediscovered in Quach and Weber (45). These knots wil l be construct-ed as representatives of a much larger class l a te r in th i s paper. Weshal l denote them by S In fig.2, we give a similar class, K whichm m,nwere originally discovered in Kanenobe (26), and rediscovered by theauthor, and these too p laya ro le in what follows.

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fig. 1

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The knots of Stal l ings,Quach and Weber, S m

m fu l l twists

a+ b+ c+a 0 0 1 0b 0 0 1 1 = V Seifer t matrixc 1 0 1 0d 1 1 0 m-1

Under row and column operations, we have:tAlexander module presentation tv - V equivalent tor- t + 1 -m t Jt 2 _ t +


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b

+ b+ + d+a ca 0 0 -1 0b 0 0 -1 -1 Seifertc -1 0 n-1 -n matrixd 1 -1 -n m+n-1

Alexander [ t 2 _ : + 1 t(n-m) ]module t 2 - 3t + 1

fig. 2


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For these examples, we see that there i s a genus 2 punctured surfaceserving as Seifert manifold, and for each we have chosen the basisrepresented by the embedded circles a,b,c,d.

Since W2n is not unique, V i s not well defined. However, the Thom-Pontryagin construction shows that every Seifert manifold is cobordantto any other. This leads to the notion of S-equivalence of Seifert mat-r ices:

where a and b are respect ively integer row and column vectors. An elem-entary reduction is the reverse operation .

Two Seifert matrices V and V' are S-equivalent i f each 'can be obt-ained from the other by a sequence of unimodular integral congruences,elementary enlargements and reductions.

From V we obtain a presentation matrix tv - vt for the Alexandermodule over z[ t , t -1] for the knot K. This depends only on the S-equiv-alence class of V, and the p resentat ion may often be simplified by rowand column operations. For figures 1 and 2 we obtain the simpler matrices shown. From the Alexander module, the elementary ideals and i t hAlexander polynomials may be defined, and are invariants of the knottype. These are so-called abelian invariants of the knot. In particularDefinition: A(t) = det (tv - Vt ) i s the Alexander polynomial.

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For the examples of 1 and 2, only two Alexander polynomials arise .10

In the f i r s t case, the knots are distinguished by thei r Alexander modulebut fo r the second class, when m = n we see that the knots have the sameabelian invariants , and must be distinguished by other means. We shal lsee l a te r that a l l these examples a re f ib er ed , the f i r s t class havingbeen discovered as counterexamples "btl the con jecture of Burde andNeuwirth tha t there were a t most f in i te ly many dist inct ftbered knotswith the same Alexander polynomial. Other examples of t ~ i s phenomenahave been constructed by Morton


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We say that K' (resp. F) i s obtained from K (resp. ') by twistingonce along the unknotted c ir cle J . In general, i f wetwist k times aboutan unknotted circle J" we rem ov e J and take as result ing object that ob-tained by giving k fu l l twists to everything passing through the discbounded by J . For more detai ls we refer to Rolfsen and Kirby (30). I tshou ld be clear that every kno t may be "unknot ted" in this way.

A boundary l ink i s any l ink a ris ing a s the boundary of adis jo int embedding of once-punctured orientable surfaces in S3.

There i s the obvious generalisation to higher dimensions, but inthe classica l dimension there i s additional signifigance in the casewhere there are two components, one of which i s unknotted in S3;

Proposition:!.1 Let K, J be components of a boundary l ink, with J theunknot. Then any knot K I obtained from K by twisting about J has thesame Seifer t form as K, and thus the same abelian invariants .

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Proof: Choose a Seifer t surface for K, dis jo int from that of J , withembedded circles on F representing a b as is fo r H1(F; Z). Push off eachcircle ai to obtain an embedded circle a in a neighbourhood o.f F. Wethus have an object given by the data (F, K, (a., a:), transformed by]. ] .twisting about J to (F u, K', (a!, a! ' ) ) . We claim tha t lk( a . , a ~ i s the]. ]. ]. Jsame as lk( a! ,a I ). This i s because each of a i ' a l inks J algebraically]. J Jzero times, as J bounds a surface dis jo int from F, and so twisting doesnot change the linking numbers. See f ig . 5 $

a:a

f ig ; 5


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Examples: This construction immediately gives r ise to infinitely many12

distinct examples with the same abelian invariants as the unknot r Merelytwist about ei ther component of a boundary l ink, both of whose component sare unknotted, as in the following f ig .6 .Such a l ink is a special caseof a Brunnian l ink. one for


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2 FIBERED KNarS

The role of fibrations over the circle in 3-manifold theory wasbrought to the fore in the fundamental paper of Stal l ings (48). Forknot complements in S3, i t was soon realised that the only examples forwhich the fiber i s of genus one are the t refoi l and th e figure 8 knot.This was the motivation for the conjecture of Burde and Neuwirth, coupled with lack of examples for higher genus. Techniques for constructingfibered knots were subsequently given by Stall ings in (49), and Harer(23) showed that every fibered knot or l ink stably arises using two ofthe techniques, twisting and plumbing, the lattercnotion having beenused by Murasugi (38) ..previously.Fibrations over S1: We begin with a more general cont.ext, than knotcomplements.Definition: A fibration over S1 with f iber F i s any manifoldarising as the quo ti en t o f F X [-1, 1) under the equivalence relat ion(x, -1)"" (f(x) , 1) where f: F......:;.. F i s some diffeomorphism.

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Note: When F i s a manifold with boundary, we will require f to rest -r ic t to the identi ty in a neighbourhood of the boundary. We shal l onlybe concerned in what follows with f o,n;ientation preserving. We shal ldenote the f ibrat ion by F Xf S1 or Mf when the context makes interpret"a tion c lear .Definition: A fibered knot K in Sn+2i s a knot such that X(K) i s asmooth f ibrat ion over S1.

Note that the unknot i s a fibered knot. More generally, the f iberF is a natural Seifert manifold for K, and the diffeomorphism f - themonodromy - i s defined only up to isotopy and conjugation. The induced


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maps : H*(F; z ) H*(F; Z) will also be ref'erred to as "the aonosdromy" when the context excludes ambiguity.

In the classical dimension, themonodromy has characteristic poly-

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nomial which is exactly the Alexander polynomial f 'or the knot, modulof'actors t S The degree of' the polynomial i s twice the genus of' the f'iber.Moreover the monodromy may be recovered f'rom the Seif'ert f'orm V, corres-ponding to the Seif'ert surf'ace the f'iber of' the f 'ibration, by taking

-1 tM=V V . Up to sign, the same f'ormula holds f'or the monodromy of' anyfibered s imple knot in higher dimensions. We also note that in the caseof fibered knots in S3, the fiber surf'ace i s the unique Seifert surfaceof minimal genus. Furthermore, note that in the cases just considered,the Seifert form i s in fact unimodular.

I t i s an unsolved problem to determine the extent to which a Seifert f'orm can be recovered from a given monodromy. See Durfee (13).

We wil l be constructing fibrations by hand, in the following sense:I f asked to describe a simple example of a f ibrat ion, one might respondby taking some physical object in R3, moving i t around a circle whiledeforming i t in some way, and returning i t to i t s original position insome twisted f'ashion. For example, the following figures we see thet r fo i l and figure 8 knots braided inside a solid torus S1 X D2:

7In each case, the disc D2 intersects the knot in 3 points. I f we

remove t he kno t, and push the thrice punctured disc around the circle,i t undergoes a non-trivial dif'feomorphism, giving a non-trivial bundle.


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In th is way we construct non-trivial diffeomorphisms of a boundedsurface in 33 by isotopy. Less t r iv ia l examples arise by consideringthe f ibrat ion associated to the complements of the t re fo i l and figure 8knots in 33. In th is case, the genus one f iber, when pushed around thef ibrat ion, undergoes a diffeomorphism which plays a signifigant role inthe construction of t he knots S a n d K of f igs. 1 and 2. Abstractlym m,nwe know the monodromy, but we will need to see the effect on a chosenset of embedded circles representing a b asis fo r H

1(T2; Z) where T2 iso 0the punctured torus.

Case 1: The t refoi l .A convenient picture for the monodromy may be found in Rolfsen. The

monodromy is periodic, of period 6, as i l lustrated in t he f igure:

f ig.8We wish to have the knot fixed, and also to see the image of the

i-spine indicated. Isotoping the O-spine q back to i t s o rigina l pos it -ion, we see that the arc joining the fixed po,;int p on the boundary to qmay be presumed fixed by the monodromy. The configuration below is isot-

1.5

opic to that above.

f ig.9


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16I f; 'we v iew T2 as obtained by identifying opposite sides of a square,

2considered as the fundamental domain in the universal covering space R ,the monodromy corresponds to the map i nduced by the matrixThe puncture corresponds to removing a neighbourhoodof the vert ices a s i nd ic at ed :

f ig.10

The diffeomorphism may be constructed by Dehn twist ing to the l e f t

-e

along the circles C1 and C2 as shown: fig.11

The key observation i s that the f ibrat ion of 83 - t re fo i l leads toa diffeomorphism of a c lo sed sur face of genus 2. We thicken T2 in 83 to, 0obtain a neighbourhood of the f iber. This i s a genus 2 handlebody, onthe boundary of which we see the original t r e f o i l ~ . Pushing T2 around theOi' 0f ibration, the handlebody undergoes a dif'feomorphism, obtained by cutt ingalong the a n ~ ~ i C1C1 and C2C2 , twist ing to the lef ' t and reglueing. Theinduced map on the boundary i s thus the composition of Dehn twis ts

-1 -1to to to t c ' where t means a r ight handed Dehn twis t about c.221 1 c


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fig.12

Carrying out an ambient isotopy in 83 we obtain a more convenientpicture for la ter use:

f ig . i3

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The fibration of the :figure 8 knot complement does not lend i t s e l fto such a convenient descript ion. Fortunately, Birman and Williams ( 5)have given e:lpl lc it pictures which resUlt in the diffeomorphism shown inf ig.14, when we push the punctured torus around the :f ibration. That thearc pq i s preserved up to isotopy - a fact not explici t in (5 ) - canbe easily deduced by considering as before the l i f t to the universalcover R2 This corresponds to the matrix r2 -1] Using the same modelfo r T; as with the t re fo i l , l4 1it i s clear tha t the arc pq from th e fixe d point on the boundary to theO-spine i s preserved under the isotopy of T2 from the l inearly inducedform to tha t corresponding to the f ibrat ion in f ig . 14. This i s shown inf ig .15.

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fig.14

>

f ig.15


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-1This may be achieved. by the composition of Dehn twists t c t c1 2

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f ig.16As with the t re fo i l , we obtain an induced diffeomorphism of the

genus 2 handlebody, obtained by isotopy in 83, which on boundary maybe described. by tB t ; l t a l t c as in f ig .17.1 122

i s clear tha t the same investigatfuon may be carried out fo r anarbitrary fibered l ink in 83

fig.17


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20Constructing fibered knots in 3-manifolds

Stall ings (49) has given three ways of construct ing fibered knotsin the 3-sphere, We briefly describe sbme'conclusions of two of theseand give some further applications.A The Twisting Construction. This i s the main technique implicit inthe results of this paper. Stal l ings shows that the twisting of knots,as described briefly in the f i r s t sec ti on , le ads in special cases tothe construction of new fibered knots or l inks from old.

Specifically, and we quote l o o s e l y ~" Suppose F i s a fiber surface for a fibered l ink L in S3, and that C i san embedded circle on T, unknotted in S3 and with Ik(C , C+) = O. Then anyl ink L' obtained from L by twist ing a longC+is f ibered , with f iber T Yobtained from T by twli.stlng along C+. Moreover, the monodromy is that ofL composed with some number of Dehn twists along the circle C on T."

This i s the technique used by Bonahon, Morton (37) and Stal l ings(49) to construct examples of fibered knots with the same Alexanderpolynomial or module.Example: The knots S a l l arise from S , the square knot, by twist-m 0ing along the circle b. Thus they are a l l fibered knots. See f ig . i .

Later in this paper we shal l see how this phenomena may be i n t e r ~I

preted in a 4-dimensional context.We shal l refer to th is construction as Stall ings twists.

Another famous "twist" i s the so-cal led "Lickorish twist" , used byLickorish (33) to show that every closed orientable 3-manifold M3 hasa framed l ink description - M3 arises by removing some embedded solidto r i in S3 and replacing them by a twisted diffeomorphism of the boundary.


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Wallace (55) has obtained the same decomposition via different methods.For further details , we refer to Rolfsen and Kirby (30). The importantpoint we need i s that we may assume that sucq a surgery d e s c r ~ p t i o n i sintegraL.

We now put this in the context of Stall ings ' generalisation of atechnique due originally to Murasugi (38):B Homogeneous Braids. Every l ink in SJ may be arranged by isotopyas a closed braid on n str ings, for some n. The la t te r may be descr ibedby any of a number of open braids on n str ings, each of which in turnmay be described by a word in the symbols t 1 , , t n_1, where t i denotesa right handed crossing of the i t h strand over the i+1s t , as we move up

-1the page and with the s tr ands numbered from le f t to rigfut. t . i s then1a l e f t handed crossing of the i+1s t strand over the i t h, as in fig. 18:

\ -1 )t . .: \+1 t i : Ji fig.18 i+1We use the convention that the braid i s constructed from the word

by moving up the page as we read the word from lef t to right.

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Example

fig.19


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This is an example of a homogeneous braid - one for which everyoccurrence of each t . is with the same sign.1

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Theorem (Stallings) Any link arising.as the closure of an open homo-geneous braid in which every t . occurs at least once is a fibered l ink.1Moreover, the fiber surface is that arising from the closed braid byapplying SeifertUs algorithm.

An application:Myers 09) and Gonzales-Acuna have proved that every closed orient

able 3-manifold M3 has a fibered knot. We will give a simple version ofa proof of th is , based on homogeneous braids:Theorem; 2.1 Every closed orientable 3-manifold has a fibered knot.Proof: Take any integral framed l ink description for M3, and denoteby L the link in S3 on which we perform surgery to obtain M3

Arrange L in S3 as a closed braid on n+1 str ings, corresponding sayto the closure of the open braid corresponding to the word w on symbolst . , i = 1, . . . , n. Now introduce n unknotted and unlinked circles C.,1 1woven into the origina l b ra id as follows: Each C. runs once around the1braid between the i t h and i+1 st strands, and every t ime we reach acrossing, corresponding to the symbol t i or t ~ we embed Cilocal ly asfig.20:

-1t . =1)( C.1

fig.20


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2 )

fig .2 1

We apply t h i s t o th e b ra idf i g . 1 9 t o o b tain f i g . 21 ;Example;

... , In ge ne ra l, th e braid corresponding t o the word w on th e t . , i = 1,n becomes th e b ra id corresponding t o th e word won s ym bo ls s.,. Jj = 1 , 0 0 00 ' 2n v ia the replacements

-1 s- 1t i - -+ s2 i-1 2 i s 2 i - l-1 -1t i --+ s 2 i s 2 i - l s2 i

The r e s u l t i n g braid i s homogeneous, and i t i s easy t o arrange t h a tf o r th e o ri g i n al word w every t . occurs. I t follows t h a t th e new b rai daobtained i s f'ibered. Wea knot;

connect th e C. together a s i n f'ig.22 t o obtain" l Jf'ig.22


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The fiber surface determines a framing of the normal bundle foreach of the knots K. making up our original link from the surgery desJcription of JYP. I f we do surgery on each K. corresponding to this framingJwe can extend the fibration over the glued-in solid torus. Thus to obtaina fibered knot in M , we must ensure that we induce the right framingsby the fibration. By further twisting of the new component about the K.Jas in fig.23, we can arrange that the introduced knot l inks each K. anyJdesired number of times. Thus a ll possible framings on the K. may beJinduced by fiber surfaces, which completes the proof.

24

+1: -1 :

fig.23

c.a

Myers has shown further that every closed orientable 3-manifold" IIadmits a simEle knot - one whose complement has a complete hyperbolic

structure. I t i s cer ta in ly t rue that every such M has a fibered simpleknot, although no proof of this has a ~ p e a r e d . On the other hand, we caneasily concludeTheorem; 2.2 Every closed orientable J-manifold has infinitely manydistinct fibered links whose complements are hyperRolic.Proof: We observe that the links we constructed above are not onlyhomogeneous, but in fact al ternat ing. Menasco (34) has shown that everyprime alternating link which i s not a (2 , q)-torus link has complementadmitting a complete hyperbolic structure, generalising the results of


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Thurston (53) on the complements of t he f igure 8 knot and the Whiteheadl ink. As we may introduce arbitrarily many crossings by f.mrther twistingof the introduced component, or by connect-summing with fibered hyperbolic knots in S3, i t i s clear t ha t i nf in it el y many different fiberedl inks may be obtained.

A Further A p p l i c a t i o n ~We observe the following phenomena:

Given a word w on symbols t . as above. there corresponds an open braid.J.wthus corresponds to an open al ternat ing braid , to which we may assoc-iate any invariants of the corresponding closed braid. In particular,there i s naturally associated the hyperbolic volume and the stretchingfactor of the monodromy of the f ibration. We will discuss this la t te rconcept in the next section. noting here that a f ibration over S1 ofdimension 3 admits a complete hyperbolic structure i f and only i f themonodromy i s "pseudo-Anosov", from which the stretching factor may bedefined.

The volumes of hyperbolic 3-manifolds are at present diff icult tocompute. The most succesful technique has been to employ decompositionsby ideal polyhedra, in particular tetrahedra, of open 3-manifolds with

I

cuspidal ends. Details may be found in Thurston (53), Menasco (34) andNeumann and Zagier (41). For l ink complements in s3, no systematictechnique is known, except for links which are a lt erna ting . In Par ti cu l-ar, there is at present no known procedure for determining a decomposition by ideal tetrahedra for bundles over S1.

The point of this discussion is that by the procedure given abovei t i s possible to generate many examples of knots and l inks which are on

25


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th e one hand a l t e r na t i ng, an d on th e o t h e r fibered. From these examplesi t may be possible to determine a rel at i o n s h i p between imformationd eriv ab le from th e IJionodromy, and th e hyperbolic volume. We see f o r ex aample t h a t i f we s tar t with th e figure 8 knot, and carry out th e i terat-iv e procedure w4 W4 ' W--7 .. . e t c . we o b tain a sequence of determinablemonodromes as well a s a geometric p i ct u re a r i s i n g from th e a l t e r na t i ngst r uc t ur e o f th e l i n k s .

I t is d i f f i c u l t t o discern a ny fu nd amen tal signifigance to th e i n f -i n i t e sequences of volumes a r i si ng from an a r b i t r a r y word under i terat-ion. There are obviously i n f i n i t e l y many p o ssib le l oc a l modificationswe could have used instead o f th e simple one above, t o convert an a r b -i t r a r y b rai d i n t o an a l t e r na t i ng one.

I t should also be remarked t h a t we obtain d i f f e r e n t b rai d s i f wef r e e l y reduce any given word. For example, i f we start with the twocomponent unlink as a b rai d on two s t ri n g s , described by th e word t t:l,1 1we obtain th e Borromeen r i n ~ s .

26


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3 DIFFEOMORPHISMS OF SURFACESTo dis tinguish the knots we are going to construct, we need to

use some invariants arising in Thurston's description of diffeomorphismsof closed orientable surfaces.

We motivate his results by considering a very special case, thatof the monodromy of the figure 8 knot. We saw this ariises from thel inear map on R2 corresponding to the matrix [2 -1] This mat rix issymmetric, and thus th ere are two - i 1orthogonal eigenvectors corresponding to the two reciprocal irrat ionaleigenvalues. I f we uniformly foliate R2 by l ines par al le l to the eigenvectors, we obtain a pair of t ransverse non-singular fol iat ions of R2invariant under the diffeomorphism. I f we assign a positive uniformtransverse measure to these fol iat ions, we see that one is contracted

27

by v, the other expanded by l /v , where vvalues. This i s indicated in, fig. 2.5:

the smaller of the two eigen-

fig. 2.5

These invariant object!? p ro je ct to the torus. Consider now thediffeomorphism of the I t a : . t u s ~ ' , a s described by Dehn twists, and observe theeffect of assigni:t;l.g some positive weights a and to the circles of thei-spine as in f ig26. Notice i f we modify the neighbourhood of the 0-spine as in fig.27, we obtain i t s image may be smoothly collapsed backonto the original. This gives the new weights 2a+b and a+b.


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ba

>

28

b a

fig. 26

b::l '--,

fig.27I f we choose weights which correspond to largest eigenvalue of

the l inear map (a, b) ...-.-+. (2a + b , a + b), the resul t ing weightedgeometric object is 'invarian'f ' in some sense under the diffeomorphism,up to scale. We notice now that the eigenvalue for th is map and for the

2measured foliat ion arising on T by projecting the invariant foliat ionby geodesics on R2 are in fact the same. We can nowdefinit ions and Thurston's results .

more formal

,;I,For an arbi trary closed orientable surface, Euler characteristicconsiderations preclude the existencefol iat ions. Hence we allow k-prongsingulari t ies, k ) 3, as shown: fig.28

of non-singular codimension one

~ 1 6 J 1 ~ ~ ~ r


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29

Definition: A measured :foliation ( ~ m) on a closed sur:face F is a:foliation "3- with isolated prong singulari t ies, and an invar iant t rans-verse measure m ( one :for which the measure assigned an arc transverseto the foliat ion remains constant as we push the arc along a lea:f).De:finition: A homeomorphism :f of the surface F i s pseudo-Anosov i fthere i s a pair o:f transverse measured :foliations ('1-u, mu) and ( ~ S , mS)with the same singulari t ies, and a real number a> 1 such that

:f( u, mU) = ('"'", amu) ( ~ s) ( ,-


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30T h e o r e m ~ ( T h u r s t o n ) " ~ Given a diffeomorphism f of the closed orientablesurface F, one of the following i s t r u e ~

t.) f i s isotopic to a periodic diffeomorphism f 'i i ) f i s topologically isotopic to a pseudo-Anosov diffeomorphism f '

i i i ) f i s reducible, ie . there i s a collection of disjoint embeddedhomotopically non-trivial circles Ci ' i = 1, , n such that the rest""rict ion of f to the components of F cut along the Ci i s of type i ) or i i ) .

Case i i ) excludes i ) and i i i ) , and in this case ft i s unique up toconjugation by a diffeomorphism isotopic to the identi ty.

We saw that for the figure 8 knot, the stretching factor also arosewhen we considered an invariant-up-to-scale weighted "train track".Definition: A train track on a surface F i s an embedded graph such that

i ) a t every vertex, a l l edges are parallel - f ig . 30i i ) there are a t least 3 edges at each vertex - fig.J1

i i i ) no vertex i s a cusp - f i g ~ 2iV) every contractible component of the complement has a t leas t 3

horns .J3v) no component of the complement i s an annulus with smooth boundary.Given a train track, there are many ways in which a positive measure,

may be assigned to each edge so that the sum of the weights entering a

fig.30 fig.34

vertex from one side equals the sum on the other side tl'ig'34)

L 'f: '< b8' : f . ~


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fig.35

31

For the fig ure 8 knot, the invariant foliation arose :from a :foliation o:f the universal cover R2 by invariant geodesics. For sur:faces of

2genus greater than one, t he unive rsal cover may be taken to be H , hyper-bolic Z-space, since a ll such surfaces admit metrics of constant curva*ture -1. A l i :f t of the di:ffeomorphism f to HZ extends to a continuousmap on the closed disc, with :fixed points on the sphere at ro, 81 Pairsroof dist inct points in 81 determine geodesics in HZ, and thus there isroa set of invariant geodesics under the l i f t o:f the map. After isotopy,we obtain under projection to F a map fV which preserves a set of embed-ded geodesics on F. We cal l such an object an invariant geodesic lamin-ation on F, and there is naturally associated, as was the case of theJ..-'-i",UJ..I:J 8 knot, two transverse measured foliations.

If ' we take a suf':ficiently small neighbourhood of the lamination,we see an object that appears as a thickened t rain track. The traintrack we seek is obtained from this by smoothly squeezing it down ontoi t s 1-spine, as in the f'ollowing :figure:


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32The invar iance of the lamination implies, a s with th e figure 8 knot,

that the image of the train track under f t may be isotoped into the -neighbourhood of the lamination in such a way that i t smoothlyyollapsesback onto the original. I t i s easy to arrange that the vert ices are sentto vertices, in which case each edge of the train track i s wrapped aroundevery other edge with some non-negative multipl ici ty. We will shortlyi l lustrate this by examples of subsequent relevance. Label the edges byei , and le t r i j be the number of times the edge fl(e i ) runs around theedge e . af ter collapsing.J

The natural correspondence between the geodesic lamination and meas-ured foliation leads to the natural assignment of a weight to each edgeof the t rain track. Given the weight w. assigned to the edge e ., the map1 1f ' induces via the matrix R = ( r..) a l inear t ransformation of the weightlJvector w = (Wi)' defined by f ; (r!) = Rr! This has already been i l l -ustrated for the f igure 8 knot. The result we observed there genera li ses:Theorem: J.3 When f is pseudo-Anosov, the stretching factor is the uniquelargest eigenvalue of R, i f Rn i s positive for some n.

This i s discussed in more detail in Bonahon ts paper ( ? ) . We makeno attempt to provide details , but employ the technique as way of i l lus-trat ion. The difficulty l ies in general in determining an invariant t raintrack.Examples: Pseudo-Anosov maps obtained by isotopy in S3

We will show that a diffeomorphism of pseudo-Anosov type can beconstructed by ambient isotopy in S3 of a closed, orientable surface.The examples arise from the f ibrat ion of th e figure 8 knot complement, 'since as start ing point we take the diffeomorphism of fig.1?, redrawnhere with an additional "waist" curve for the solidhandlebody.


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TheoremlltJ-

33

I t i s clear that we may Iftwisttf the handlebody about the waist circle Wby ambient isotopy in 83 to induce on the boundary F a dif'feomor-phism corresponding to a Dehn twist twabout W. Hence by composing withthe induced map :from the figure 8 monodromy we obtain the maps f , wherenn -1 -1f = tw t g t c to tcn 1 122

There ex is t in f' in i te ly many distinct diffeomorphisms ofa genus 2 closed orientable surface which are pseudo-Anosov and obtainedby ambient isotopy of the standardly embedded surface ,in 83.Corollary 3. 5 83 has a Heegard decomposition with pseudo-Anosov glueingmap.

This i s reminiscent of the result of Fathi and Laudenbach (14),who show that there i s a non-contractible loop in the space of' Heegardspli t t ings of #k82 X 81 for every k ) 2.

I

proof of' Theorem: We only have to prove that the maps f' , for n 1,na re i so topic to pseudo-Anosov maps. We will do th is by constructing aninvariant t rain track, from which we then extract the s tretching factors

. (i,

to show the f are distinct .nF:tmn,;the description of' f in terms of Dehn twists we observe thatn

the circles on which we twist right are mutually disjoint , as i s the casefor those on which we twist lef ' t . This has a fortunate consequence:


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Suppose we are g i v e n ~ a ' C o l l e c t i o n of homotopi.cally non-tr ivialembedded circles on a surface F, which fa l l into two c la ss es , th e mem-

34

bers of each of which are mutually disjoint . I f the complement of a ll thecircles on F i s a collection of discs, many pseudo-Anosov maps may beconstructed by assigning + to the members of one class and - to the

-/members of the other. Any diffeomorphism obtained by Dehn twisting alonga l l of the circles, in any order, is pseudo-Anosov provided we alwaystwist according to the assigned sign. This result i s due to Bob Penner(43), and i s a generalisation of a technique found in (15). The proof i sconstructive - the circumstances allow the direct construction of aninvariant t rain track.

We demonstrate the technique for the maps f , which satisfy thencondition above when n is positive. Note that we have already seen anexample - when we constructed the t rain track for the figure 8 monodromy.

In f ig .)6 we show how to modify the intersec tions of the circles onF to obtain a candidate for the train track we seek.

r, +

Ifig.36

This results in the following object on F:

fig.3?


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36After further collapsing we obtain fig.38 with edges labelled 1, . .

00,12. Assigning weights consistent ly requires the satisfaction of comp-at ibi l i ty conditions at each of the vert ices, and thus the tota l weightsystem i s determined by the weights assigned to a proper subset of theedges, a fact that emerges naturally as we try to determine the stretch-ing factor. This i s discussed in more d etail in Bonahon ( 7 ) .

fig.38The beauty of the diffeomorphisms in Penner's class i s that the

trMn track constructed i s invariant under each of the consecutive Dehntwists. To see th is , we isotope a l l of the twisting c ir cl es i nt o a neigh-bourhood of the train track, as in the following figure 40.

I f for example we twist along the circle C2' the image of the traintrack naturally collapses back onto t he o ri ginal :

fig.39


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37


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Hence to see the automorphism of the weights, we do not need to seethe image of the t rain track under the entire diffeomorphism. Instead,we compute the e ffe ct fo r each Dehn twist , and obtain the f inal result

38

by composition. All of the necessary imf'ormation contained in the wayt he twi sti ng circles l ie in the neighbourhhod of the train track, andcan be encoded in the following table:

Wraps IntersectsC2 e7 e92 e9 e5 e1 e2e10e4 e1 e3C e8 e1 e4 e3e11e5 e1 e2C e6 e8W e1e5e12e4

e11e11e12e10e6e2e1

This records the edges we t ravel along as we go along any givencircle 9 and the edges we cross t ransversely as we do so. I f we twistaround a given circle, and collapse back the image of the train track,the weights assigned to the edges intersected by the circle are addedto the edges along which the circle wraps. The resulting l inear map forf acting on the weights i sn

where each of the matrices R* i s the induced map on weights determinedby the Dehn twist on C*. We have given the results explici t ly, in themost convenient fashion for matrix multiplication:


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J 2 0 0 0 2 0 0 0 2 2 2 1 0 0 0 0 0 0 0 0 0 0 01 2 0 0 0 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 01 1 1 0 0 1 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 01 1 0 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 01 1 0 0 1 1 0 0 0 1 1 1 0 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

RzR2= 0 0 0 0 0 0 1 0 0 0 0 0 O o 00 o 0 i o 6b1 0o 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 01 1 0 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 1 01 1 0 0 0 1 0 0 021 . 1 0 0 0 0 0 0 0 0 0 1 0 00 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0o 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1

J 0 1 0 0 0 1 0 0 1 1 1 J 2 0 0 0 2 0 0 0 2 2 21 1 1 0 0 0 1 0 0 1 1 1 1 2 0 00 1 0 0 0 1 1 11 0 2 0 0 0 1 0 0 1 1 1 1 1 1 0 0 1 0 0 0 1 1 11 0 1 1 0 0 1 0 0 1 1 1 1 1 0 1 0 1 0 0 01 1 11010101 0 0 1 1 1 1 1 0 0 1 1 0 0 0 1 1 1o 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0R R-R :=1 2 2 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 01 0 1 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 1 0 0 0 0o 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 2 1o 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 2 1 11 0 1 0 0 0 1 0 0 1 2 1 0 0 0 0 0 0 0 0 0 0 1 0o 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1

I1 0 0 0 0 0 0 0 0 q 0 0 1 8 1 0 0 8 1 0 0 9 10 9o 1 0 0 0 0 0 0 0 0 0 0 6 6 1 0 0 5 1 0 0 6 7 60 0 1 0 0 0 0 0 0 0 0 Q 6 5 2 0 0 5 1 0 0 6 7 6o 0 0 1 0 0 0 0 0 0 0 0 6 5 1 1 0 5 1 0 0 6 7 6o 0 0 0 1 0 0 0 0 0 0 0 6 5 1 0 1 5 1 0 0 6 6 6o 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0RrR1R2R2 = o 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 O ~ 1 0 0 0 1 000000 0 0 1 0 1 0 0 5 4 1 0 0 4 1 1 0 5 6 5o 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 1 1o 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 0 0 2 1 1o 0 0 0 0 0 0 0 0 0 1 0 5 4 1 0 0 5 1 0 0 5 7 5o 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1

39


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J'"

40

n+1 0 0 0 0 0 0 0 0 0 0orro.o 0 0 0 0 0 0 0 0001 0 0 0 0 0 0 0 0 0n 0 0 1 000 0 0 0 0 nn 0 0 0 1 0 0 000 0 n

pIlp-R R-R = 0 0 O ; C O : ~ 0 1 0 0 0 0 0 0~ v r ' 1 1 2 2 ,0 0 0 0 0 0 1 0 0 0 0 0

000 0 0 0 0 1 0 0 0 0o 0 0 0 0 0 0 0 1 000o 0 0 0 0 0 0 0 0 1 0 0o 0 0 0 0 0 0 0 0 010n 0 0 0 0 0 0 0 0 0 0 n+1

1 8 1 0 0 8 1 0 0 9 10 9661 005 1 006 7 6652 0 051 006 7 6651 1 051 006 7 66510151 0 0 6 61 1 0 0 0 2 0 0 0 2 1 1o 0 0 0 0 0 1 0 0 0 C o ~ o__ ~ ._ - , - " . _. ,." . , - ~ " , ~ . ~ ._, .", ""- - ' ' ' ' ' - ' . " ' " ' r ""5 4 1 0 0 4 1 1 0 5 6 51 1 0 0 0 1 ,0 0 1 1 2 11 1 0 0 0 1 0 002 1 15 4 1 0 0 5 1 0 0 5 7 5'000 0 0 0 0 0 0 0 0 1

o 0 8n n+8 +1W't! t (1,9n e #+9

66

oo 9n 10n+9 +10

6 76 7o 9n 10n 9n+6 +7 +6

o 9n 10n 9n+6 +6 +6021 1o 0 67 0

~ i : I - - " ' - ' ~ ' ' ' ' -o 5 6 51 121q f ,1 ",{le""---o 9n 10n 10i:!+1

o

oooo

oo1oo\o

oo1ooIn

11

55

o

o

o oo 8n n+5 +11 8n n+5 +1

0020010040010,_ 0) 1"o 0 8n

o1

11n 8n n+11 +8 +1661652

l1n 8n n+6 +5 +111n 8n n+6 +5 +11100005411 1 01 1 0t)Ll.n 8n n

= A =n

/

Any weightings wi fo r the edges ei are subJect to the re la t ions

Wi = w3 + w5 w9 -:I- wl1 = w3 + w7 w4 t w6 = w8 + w10 + w12These determine a subspace of R12 which in tersects the posit ive

" d t i l R12 .ua ran + ln a subset E. E i s thus invariant under the act ion of An.


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41Observe now that our t ra in track t i s "minimal" in the sense that

we cannot remove any edges and s t i l l have a train track invariant underf This is because given any edge e ., there i s some other edge which isn 1wrapped around i t by f , which may easily be, seen by referring to thentable of wrappings and intersections. Furthermore, our t ra in track tsat isf ies each component of F - t is a disc with three "hornsn Thefollowing i s taken direct ly from Bonahonus Theorem 4.3 and note addedin proof:Theorem: 3.6 Let f be a diffeomorphism of the cillosed surface F admitt-ing a; minimal train track t such that each component of F - t is a discwith three horns, and le t A denote the matrix of the induced l inear mapon E, the space of allowable weights. Then f is topologically isotopicto a pseudo-Anosov diffeomorphism f ' i f and only ifAX::j X for everynon-trivial X in E. I f so, moreover, the s tretch ing factor a of f ' isthe unique eigenvalue of A that admits an eigenvector X in E.aRemark: The weighting of t corresponding to Xa is projectively thatinduced from the invariant transverse measured fol iat ions on F. NeceSB--L ty of the requirement that t be minimal follows immediately fromour examples f ; t is invariant even for n = 0, in which case the diffeonmorphism i s reducible, with reducing circle the figure S knot on theboundary of the handlebody. On the other handProposition: 3.7 I f n I , A X = X for some X in E has only the t r iv ia lnso'Iutd.on c-Lhus f i s peeudo-Anoeov ,nProof: From the f i r s t row of A we see that A X =X = (Wi) requiresn na l l Wi zero except perhaps w4 ' w5' ws' or w9 Hence X is 0 or does notl ie in E, as follows from the vertex relat ions; eg 0 =Wi =Wz + w4


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42necessitates w4 =0, and similarly for w5' ws and w9.

We see in fact that any non-zero vector in E has every Wi , O.Proposition: 3.8fnnction of n.

The stretching factor a i s a str ict ly increasingn

Proof: For matrices with non-negat ive entries, the theory of Perron-Frobenius shows that the eigenvalue we seek is given by (4)

an = sup f v i Y in E, AnY ~ v Jwhere, by AnY ~ v we mean each entry of AnY i s greater than or equal tothe corresponding entry of vY. The entries of A Y are either constantsnor polynomials in n, for fixed Y. However, i f we i terate three times, wefind that every entry of AJy involves n. Hence we apply the characterisnation of a3 as given above: the entries of AJy are str ict ly increasingn nfnnctions of n, and so i f A3y vY, A3+1 Y '> AJy vY, showing that then n na are str ict ly increasing with n, and we conclude f in al ly t ha t thendiffeomorphisms f are mutually non-conjugate and non-isotopic.n


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434 SLICE KNDrS AND RIBBONS

When Quach and Weber rediscovered the knotsS in f ig . l , they mademthe addit iona l observation that they are a l l ribbon knot.s,Definition: A knot Kn in Sn+2 i s ribbon i f i t bounds an immersedn+l-ball Bn+l a l l of whose self-intersections are t ransverse doublepoints, the connected components of which are embedded copies of BnThe preimage of each such Bn i s two copies , one of which i s properlyembedded in Bn+l , the other lying entirely in the interior.

Fig.l provides a configuration in which the immersed discs are app-arant. We will shortly see that the knots K are also ribbon, a factm,mthe reader may easily discern directly"

The interest in r ibbon knots is that they are the boundaries ofparticularly nicely properly embedded balls in Bn+3 , To see thfus, merelypush the self-intersections of the immersed Bn+l into a collar of Sn+2in Bn+3 , and use general position, Ribbon knots are the nicest examplesof slice knots:Definition; A knot Kn i s slice i f the pair (Sn+2, Kn) i s the boundaryof some disc pair (Bn+3, Bn+l ) .

In even dimensions , every knot i s slice, though not a l l such areribbon. In odd dimensions, an obstruction for a knot to be slice arisesfrom the Seifert form:JDef'furr::kbion: Kn i s an algebraically slice knot i f i t has a Seifertform S-equivalent to one of form

A necessary condition for a knot to be slice is that i t i s a l g e b ~raical ly sl ice.


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In higher odd dimensions, Levine (31) has proved the remarkableTheorem:4.1 A knot K2n-1 in S2n+1 is slice i f and only i f it i s algebraically slice,. , for n ~ and K2n-1 simple.

Returning to the classiealdimension, i t i s a lorgstanding conject-ure that a knot is .slice i f and only i f i t i s ribbon. Casson and Gordonhave made two signifigant contributions to this problem. On the one hand(11), they have introduced invariants whose vanishing is necessary fora knot to be r ibbon, a lthough the possibi l i ty of the knot nonethelessbeing slice i s not ruled out. These are finer obstructions than thealgebraically slice condition, but we shal l not have cause to calculatethem here. The obstructions for our examples will vanish, as they are infact a ll going to be ribbon.

Their other contr ibut ion (58) has as start ing point the observationthat ribbon knots enjoy a nice property: there is a surjection of thefundamental group of the knot complement onto that of the ribbon compl-ement in the ball . The la t te r complement having a handle decompositionwith handles of index &2, as may easily be seen by looking at the cri t-ical points of the embedded ribbon in the collar of Sn+2.

44

ball vn+3 i f i t boundsDefinition: A knot Kn . Sn+2. h t .bbon ti h 1n lS omo ow r l on In some omo ogy

1 Rn+1 t t t t .s lce such ha he na ural incluSlon

TI (Sn+2 - K),i s a surjection.Casson and Gordon (58) then prove

Theorem:4.2 A fibered knot in a homology 3-sphere i s homotopy ribboni f and only i f the capped off monodromy extends over a handlebody.


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Thus in part icular, a necessary condition for a fibered knot to beribbon is that i t s monodromy be null-cobordant (ie i t must extend oversome orientable .3-manifold). In his thesis , Bonahon ( 6) calculated theoriented cobordism group of diffeomorphisms of closed orientable surfacesand in (7 ), he applied th is to produce infinitely many examples offibered knots in S.3 which are not ribbon, but which nonetheless have thesame algebraically .slice Seifert form.

On the other hand, the knots K of f ig.2 a l l have the same Alex-n,nander module, and thus the same abelian invariants. That these are dis-t inct i s proved in Kanenobe (26).

We will now show how to construct the knots of figs. 1 and 2, andin so doing prove the stronger statement than that implied by Casson andGordon's work, namely tha.t the complement in B4 of the r ibbons theybound f'Lber-eoven thEIj3iele with f iber a genus two handlebody. Cassonand Gordon's work shows that this property is s ati sf ied i n some homotopy4-ball , but unt i l the Poincare conjecture i s se ttle d in the smooth catBegory we may not conclude that our resul t is universally t rue.

We should remark at this stage that the only class of fibered ribbonknots previously known to the author are the knots Sm' the 89 knot andother related examples arising from the analysis in Akbulut and Kirby(2 ) and Aitchisonand Rubinstein (1 ) of the homotopy 4-spheres constructed by Cappell and Shaneson (8 ,9 ). In fact , the technique forconstructing such knots there leads to the construction of S a n d K m m,n


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.5 CONSTRUCTING MAPPING TORIWe will show how the various knots Sand K arise, by reversingm m,n

the following observation: I fK i s a fibered ribbon knot in S3, boundinga ribbon R in B4 such that B4 - R fibers, over the circle with fiber a

4handlebody, we can reconstruct B by f i r s t explici t ly constructing themapping torus Mf of the diffeomorphism f corresponding to the fibration,and then adding a 2-handle to ki l l off the S1 factor, corresponding toglueing the ribbon back into the ball .

I f we were to use an arbi trary diffeomorphism of a handlebody, weof course do not in general expect to obtain the 4-ballo On the otherhand, we will now show that the diffeomorphisms of the genus two handlebody constructed in section 2 a l l lead to B4.:

The following technique was f i r s t used by Akbulut and Kirby ( 2 ),generalised by Montesinos (3.5) and also used in Aitchison and Rubinstein.

46

For the diffeomorphism f: #s S1 X D2 - - ~ > #s S1 X D2,

struct the mapping torus in stages. Denote # S1 X D2 by H s swe con-o..

fig.41

1. Since Mf arises by iden ti fy ing the ends of Hs X I by f , we mayisotope f so that i t rest r ic ts to the identi ty in a neighbourhood N ofsome point p on the boundary. As O-spine forH we take some other pointsq in the interior of N. A 1-spine for H i s then an embedded bouquet ofscircles with common point q, each circle intersecting H - N in an arcsl i running once around one of the handlesas in f i g / a . We will need toof the annuli A., as shown, under fo1Note: 1. l ies on A. .1 1


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47This gives a handle decomposition fo r Hs with a O-handle and s 1

handles b . , i = 1, , s , Taking H X I with induced handle decomposition1 . . . . . .we obtain Mf by identifying H X r - 1 ~ w i t h H X {1} via f :s s2. Identify N Xf-1} with N X (11 by adding a 1-handle b*. The ends ofthe arcs 1. Xf-1} may be ident i f ied with the i r images under f by pairs of1arcs running over the 1-handle b*, and taking the union of th ese, theI i Xf-1\ and thei r images in Hs X fq we obtain a dis joint se t of embedded ci rc les a i The mapping torus construction i s completed by adding di,s 2-handles with attaching circles the a . , and with f ramings determined1by the A. and thei r images. The situation i s depicted in fig.l42..1

:::fI

..

.- ..-......

....

..... ' ....", ...... ....... "--. \; ,10..,..,,

I....'"-, ............ ,...

t.... ....... .. . ..'... . ...X Isf ig.42


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48

We wish to apply this to the diffeomorphisms f Hence we need ton"coordinatise" our description:

For O-handle, take B4 viewed as B3 X 1. The boundary 3-sphere thusnaturally inherits a structure as the union of the two 3-balls B3 X[-1and B3 X f 1 , and the S2 X I running between them.. I t is convenient ttl:>take S3 as R3 U 00 , with B3 X f 1T the unit bal l , and B3 Xf-1} the exteerior of say the bal l of radius 2.

....e..f '". -".. .. ..

... ' -," ...- .-, .....n. 11'".... . .."Ill .. .. -... .. ,... .. ..'62,,.

..d e . ,tI_. .. ' . 1If.. ..._ .. . . . . I l ...... ..-.- ..-.. . '.... .... .., -.... , : q X (1l \ . \.. J .. ../ .

b Pi O b fig.43

As fn

i s a diffeomorphism of H2 ' Hg X I in th.is case has 2 1-handles b1 and b2, thickened from a1 and a2 These are attached to the unitball , serving as O-handle for H2, with a1 Us ends at the "East ll and "West"poles, and a2 to the North and South. Thus in our model for H2 X I theends of b1 and b2 are attached to th3-balls in the neighbourhoods ofthe "po'Lea" and poles, as in fig . 43:

We have also drawn in t he a tt aching bal ls for the 1-handle b*oFor the moment, we consider only the case n = O. In this case, we have


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a very convenient picture for the construction. The diffeomorphismmfOis exactly th e f igur e 8 knot monodromy, thickened. Hence the attachingcircles a1 and aZ appear as in fig.44. The reader may wish to refer tofig.16. Another consequence i s that the framings are untwisted, beingdetermined by annuli lying in the planes of the representation, for theidentif icat ion by adding the Z-handles d1 and dZ along a1 andaZ'

fig.44

Inside Mf we see a neighbourhood DZ X S1 of the circle p X I / ~ .oThis appears as a neighbourhood of the arc z in the f igure. We are goingto attach another Z-handle d* along this circle, and show that the resul ti s B4 First ly, we sl ide the Z-handles d1 and dZ off b* using d*, and thencancel these la t te r handles The result i s f i g . 4 5 ~ I t will be convenientto keep track of the genus Z surface which appears as the boundary of theunit bal l , surgered by tubes running over b1 and bZ' I f we remove a discneighbourhood of the point p we obtain a punctured surface, with the knot


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KO i t s boundary. When we f i n a l l y a r r i v e at the t r iv ia l handle decompos4i t i o n fo r B4 , KO w i l l appear a s .a knot. We w i l l see t h a t th e image of Wunder t he c an ce ll in g of handles remains unchanged.

f i g . 4 5

I n f ig.46 , we have isotoped KO over b1 so t h a t th e 2-handle d1 ca nbe isotoped to reduce th e number o f times i t runs over b1 geometricallyto one.

f i g . 4 6

50


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Now s l i d e d1 o f f b1 using d2, and cancel b1 with d2 to obtain f i g .47- We must a l s o s l i d e KO a s i n d i c a t e d . After fu rt h er isotopy, it is

c l e a r tha tb2 i s cancelled by d1, bu t i n doing so, KO must be isotopedal s o . We f i n a l l y obtain B4 , i n th e boundary of which we see KO and W,as well as th e S e i f e r t surface f o r KO on which Wl ies . We have also l a b -e l l e d two o th er unknotted c i r c l e s and Q2 which l ie on t he s ur fa ce .

51

fig.47

fig.48


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:fig50 fig.51

fig.49

52


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53

In fig.51 we isotopecthe knot KO to a more recognisable form - weobtain figure 8 # -figure 8. Furthermore, the circles Q

1and Q

2satisfy

the conditions of Stallings, and so twisting on each of them we obtainthe knots K of Kanenobe ,m,n

That KO i s a fibered knot may be deduced from the const ruct ion.Let f be the restr iction of f to ~ H 2 ' Thus aMf is the mapping torus Mf ,1 0a genus 2 surface bundle over S Adding the 2-handle ~ has the effectof surgering Mf along the circle swept out by p, and thus S3 i s obtain-2 1 1 2ed by removing D X S and replacing by S X D Hence KO i s isotopic toS1 X toh proving that the complement of KOfibers.

A similar conclusion holds whenever we obtain B4 by adding a 2-hand.Le to H X S1, for any diffeomorphism g: H ---J)o H fixing a neighbour-s g s shood of a point p on ~ H s ' along the circle swept out by p. The boundaryK of a disc neighbourhood of p on ~ will always be a fibered ribbonsknot in S3, bounding a ribbon R.

Suppose now that C is an embedded circle on aH ,bounding a discsappears unknotted in S3=in H , and that after handle cancellation Cs

d(M U d*) as constructed above.gProposition:5.1 Any knot K' obtained from K by twisting along C isagenus 2 fibered ribbon knot, bounding a ribbon RV whOse complement in B4f ibers over 81 with fiber H sProof; Let g* be any diffeomorphism of H obtained from g by followsing with a cut along D, twisting a number of times and reglueing. Weclaim this is the monodromy of B4 - Rv, for some ribbon R V To see this ,observe that C in S3 bounds D in H c..M c: B4 . (We can in fact "see" thiss gdisc in the representation of the mapping torus construction).


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Twisting along C changes the mondromy of K by a number of Dehn twistsalong C on the f iber surface. Considered as 3H , this can be achievedsby cutting along D and twisting the handlebody. Hence in the 4-dimensional perspective, a neighbourhood of the disc D in B4 is a 2-handleattached to the complementc the twisting is achieved by removing th is2-handle and replacing i t with a new 2-handle with the same attachingcircle but with different framing. We are "blowing down" the ribbon D

Hence R' i s exactly R, twisted along C. This is because the pairK, C bounds a ribbon l in k a ris ing naturally from the construction of themapping torus - in our ini t ia l handlebody picture for M U d* both RandgD appear as "ribbon" discs, and under isotopy and handle sliding, onlyribbon intersections are introduced. Twisting along C thus twists R to

va ribbon R Applying th is to the cases above, w.e have

Theorem: 5.2 The knots K of Kanenobe are genus 2 fibered r ibbon knotsm,nwhich bound ribbons R such that B4 - R fibers over the circle withm,n m,nf iber the ~ e n u s 2 handlebody.Proof; I t i s clear that the circles Q1and Q2 sat isfy the conditionsof the discussion above.

Now observe we may do the same -t-lith W. In th is case, denoting theknot obtained by twisting n times by K , we see that the monodromy fornK is exactly f , and hence the Kn are a l l distinct for nz O. Moreovern n I'Wand KO form a boundary l ink ,and so a l l of the1


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Theorem: 5.3 There exist infinitely many dist inct knots K in SJ suchn

55

that i ) K i s fibered, of genus 2ni i ) K i s ribbon, bounding a ribbon R in B4n n

i i i ) B4 - R fibers over S1 with f iber a handlebodyniV) SJ - K has pseudo-Anosov monodromy, and thus admits a comn

plete Riemannian metric of constant curvature -1v) the Seifert form of K i s independent of n, and thus the Kn n

cannot be distinguished by abelian invariants.Remark: We of course expec t that the knots K also have pseudo-Anosovm,nmonodromy, but we have not carried ou t a thorough i nves tiga tion , a s themonodromy i s not obviously of Penner type. On the other hand, ~ v e r y diff -eomorphism of a genus 2 surface commutes with the hypoelliptic involutionand so the kind of train track analysis used by Bonahon (7 ) can be app-l ied.

The Trefoil: We carry out the same construction for the monodromy ofH2 arising from the t refoil . The only difference i s t he a tt aching circlesfor d1 and d2 are now determined by fig.10, and so we obtain the diagramsin f ig .52. As above, we slide and cancel handles in the obvious fashionto obtain B4, in which we see So = t refo i l # - t refoi l and i t s genus 2Seifert surface, as in fig.53

In this case, however, we see that the circles Q1 and Q2 are in factisotopic, and thus by twisting we only obtain a 1-parameter family ofknots Sm' ;I'hese a re exactl y the knots of Stallings, Quach and Weber, andso we have provedTheorem:.s4.The knots S bound ribbons in B4 whose complements fiber overmS1 with fiber a genus 2 handlebody.


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fig.52

56


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57

f ig. 53


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586 DOUBLY SLICE KNQrS

n n+2 n+)Suppose K i s a slice knot in S ,bounding a ribbon R in B I f we double the pair (Bn+) , R) we obtain a new knot DR in sn+) , whichintersects the equatorial Sn+2 in the original knot Kn.Kn i s thus asymmetric slic.e of the knot DR. Usually DR will be non-trivial in Sn+).A natural question i s then "Which knots Kn arise as sl ices of an unknotted Sn+1i A' Sn+)?" Of course, in this situation we do not expect thesl ice to be symmetric.Definition: A knot Kn in Sn+2 i s doubly slice i f i t i s the intersect-. . Sn+) n+1lon ln of the equatorial n+2-sphere with an unknotted S

In the classical dimension, the f i r s t non-trivial example i s dueto S t a l ~ i n g s (unpublished), who showed that the square knot So i s doublyslice. We will give a proof below. Techniques for recognising certaindoubly sl ice knots in the classical dimension were given by Hosokawa (25)but the most signifigant breakthrough came via the work of Zeeman ontwist-spinning knots (56), who .showed that for any knot Kn, Kn#_Kn i sdoubly slice. Hence both So and Ko are doub.ly s l ice.

The square knot is fibered, but non-prime, and the only doubly sl iceprime knot with less than 10 crossings i s 946, as shown by Sumners (51).This i s not fibered. We will give infinitely many examples of prime,fibered doubly slice knots in the classical dimension, a ll of genus 2.

The condition of being doubly slice places algebraic restrict ionson the possible forms of the Seifert form for such a knot.Definition: A knot is algebraically doubly sl ice i f i t admits a Sei-fe r t form of form [: :1 with A and B square.


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In higher odd dimensions, a simple algebraically dOUbly sl:lceknotis doubly slice, but unlike the case of sl ice knots, there are .addit-

ional obstructions when the knot is either non-simple or even d:lmension-al. These additional conditions arise from obstructions in codimens:i:on 1embedding phenomena. We wfull observe a further manifes ta tion of thisshortly. The interested reader may refer to the work of Sumners (51),Kearton (28), Stolzfus (50), Hitt (59), Ruberman (47) and Levine (32)for more details in the general higher dimensfuonal case.

We noted earlier that i t is in general unknown how to recover theSeifert form, given the monodromy on H* for a fibered simple knot. Wewill show there is a solution when the knot is doubly sl ice:Proposition:6.1 The monodromy M for a fibered simple doubly sl ice knotK in S2n+1 determines the Seifert form.

59

Proof: Since K is doubly sl ice, i t admits a unimodular form (from the

Consequently, we may conjugate:

fibration)ible .

[: ~ I Jlnimodularity : L m p l ~ e s that both A and B are invert ,[I ~ 1 f o AJ.(1 t Fo AB "1.OElE a.OEtJ lr 0

to find an equivalent form. Up to sign,

~ the monodromy is given by

{B':At B ~ A ' 1 ~

. .Hence i f we are given the monodromy of such a knot, we know i t i sand to this we may associate the

Given another possibili ty for M, such as

similar to one of formSeifert form r: :] 0_.t].!it

rp 0 , ] ,Lo P t

we mU'st show


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60the Seifert forms are equivalent; Since the two matrix representationsfor the monodromy are s ~ m i l a r , we can find X,Y, Z and W such tha.t

Remark: I f we assume that the knot is only fibered, slice and simple,we may take a Seifert form of forminvertible . As above,

for C arbitrary, and A, B

Ft-FE:-1J-1

E

corresponding monodromy is. [ _ F E ~ l . I. ]fo I] =[ .Et

E 1 0 LEt Ft 0

The

Hence i f D i s a solution ofthe Seifert formsdromy.

and give r ise to the same mono-

Examples: I f we take E= [f i]find that the map

as arises with the figure 8 knot, we

i s an isomorphism of the space of2X2 matrices. On the other hand, for

In th is case, direct calculation shows that the Alexander

the t refoi l , the matrix E = .[1 -11 in (*) has the non-trivial solution10j

modules arising from the Seifert matricesare isomorphic. [: :J and [ : :J


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In the classical dimension, this analysis shows that the only poss-ible S-equivalence classes for fibered genus 2 doubly slice knots arerealised by the knots S a n d K We will give a proof that they are ino 0fact doubly slice;Theorem (Zeeman); 6.2I f K i s a fibered knot, thenK#-K is doubly slice.

61

Proof; Let F be the f iber Seifert manifold, and take a small bal ln+2neighbourhood B of a point p on the knot, invariant under the f ibra t -

ion. We obtain a bal l pair (Bn+2 , Lf") by removing th is invariant ball ,where A. n = K - nhdj p}, This ball pai r f ib er s over S1 with f iber F = F,as in fig .54:

K

fig. 54n+2Denote the monodromy by f . Now take B X I , in which we see a

slice disc l in X I , whose boundary in a(B n+2 X I) = Sn+2 i s K # -K.The complement of th is slice R in Bn+2 X I fibers over S1 with f iberF X I and monodromy f X ide Thus K # -K is a fibered slice knot.

~ ] ~ 2 1The fibration of B - R may be embedded in B X Sn+2 n+2 - 'n+2

Take B C S , thickenF inside B , to obtain F Y, and push F Yaround the f ibrat ion as we go around the S1 factor of Bn+2 X S1.

w2 1 ~ 2 ~ Now glue together B X Sand S X D to obtain S We do.. n+1 2 Qtn+1 2 n+1 2th is in two stages, f i r s t spll t t lng S X D as P+ XDUD X D ,n+1 1: n+2 1 -n+1 2and adding D+ X D as a 2-handle attached to B X S along D+ X D ,

-n+1 '" .,. n+2 n+] I n+1 2 n+]where D+ = FYI I \JB 0 This gives B ,and adding D_ X D gives S I t is clear that OB n+2 i s unknotted in Sn+], and intersects OB n+]


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in 0:5;+1. But th is i s just ?}B n+3=o(F v Xf Xi d S1 U D;+1 X D2) =

Thus K # -K i s doubly slice.

n+2S viewed as62

From th is we also observe that not only i s K # -K doubly slice, butthat it i s t r ivial ly the double of the bal l pair (Bn+2 , ..6.n). We quote aresul t of Levine (32):Theorem: 6.3 For n= 1, 2 there exists a doubly slice knot which i s notthe double of a bal l pair.

In the classical dimension, a double of a ball pair i s merely aconnect sum, and so 946 i s an appropriate example. We proveTheorem: 6.4 There exist infinitely many prime, fibered doubly sliceknots in S3. In particular, K i s doubly slice for a ll n.nProof; The proof of the special case of Zeemanus theorem given aboveinvolves in the classical dimension an isotopy of a thickened Seifertsurface - a handlebody - in S3. This argument works equally well whenwe allow twisting of the handlebody in addition to that induced by push-ing around the f ibrat ion, In other words, i f f : H --=lloH i s a diffeomors sphism obtained by isotopy in S3, we see Hs Xf S1 embedded in S3 X S1,and surgering the la t te r to obtain 84 has the effect of adding a 2-hand-

Definition:,

When a doubly slice knot arises via th is isotopy construct-ion, we say i t has been constructed by isotopY.

The knots Kn are obtained from Ko by twisting along W. On the otherhand, we will now show there are many alternative circles along which wemay twist without effecting the Seifert form.


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63Algebraically doubly sl ice prime fibered ribbon knots in S3 of genus 2.:Examples: We will show how to construct what appear to be zill ionsof distinct knots sa ti sfy ing the above t i t l e , The twisting technique hasin the past rel ied on ad hoc methods for finding unknotted circle embed-ded on a fiber surface. The method of the preceding section gives a meansof seeing infinitely many unknotted, non-isotopic embedded circles onthe fiber:

In the mapping torus construction, we saw 5 discs on the unit spherecorresponding tcxthe attaching discs for the two i-handles, and a neigh-bourhood of the fixed point p. We obtain a thrice punctured disc D] byremoving these disc and an arc joining p to the "West" pole, as in fig,55:

fig.55

The circle Wi s embedded on D3, and i s unknotted in S3 because unl ike K we never need to slide Wovet any handles in the reduction ofothe decomposition by handles to the t r iv ia l one for B4 . The same is thustrue for any embedded circle on D3, such as Qi and Q2 , These la t te r cir-cles represent non-trivial homology classes in H ~ t a H 2 ; Z), which is whytwisting about them transforms K to knots with different Alexander mod-oules. On the other hand, i f we take any null-homologous embedded circleon D

3, such as W, we obtain together with K

oa boundary l ink. Twisting


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w i l l thus leave th e S e i f e r t form i n v a r i a n t . Many examples of such knotsmay be obtained by viewing D3 as a 4-punctured sphere and taking th eimages o f Wunder th e a ctio n of the b r a i d group on these punctures. Wei l lust rate in th e f ol lo wi ng f ig .5 6

We see i n f a c t t h a t by ar r anging any s e q u e n d ~ of such embedded c i r -

64

c l e s on concentr ic spheres, we ca n t w i s t K about a l l of them any numberoof times, simultaneously. I t i s c le a r t h a t u n t i l techniques are improved,case by case d i s t i n c t i o n of these knots w i l l be a d i f f i c u l t t a s k . Notet h a t we may proceed i n exactly the same way f o r th e knot S o


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We offer the following conjecture65

Conjecture; Every doubly slice fibered knot is constructible byisotopy.

This would imply that very few of the examples of algebraicallydoubly sl ice knots just given are in fact doubly sl ice. A ful l under-standing r equir es the classif ication of diffeomorphisms of a surface orhandlebody obtainable by isotopy in S3. In this respect we are againseeing a manifestation of codimension 1 embedding phenomena in theunderstanding of doubly slice knots.

As support for our conjecture, we proveTheorem: 6.5 Every simple, fibered doubly sl ice odd dimensional knotarises by isotopy.Proof. We general ise the construction of K n

Consider the standard decompositionu # Dr+1 X Sn-r-1k '

the two submanifolds being giliued together along their common boundary#k Sr X Sn-r-1. Choose as .reference point X o E #k Sr X Sn-r-1 = F ~ ~ rand a small bal l neighbourhood Bn of x which intersects each ofo 0#k Sr X Dn-r and #k Dr+1 X Sn-r-1 in an n-ball.

Now le t ht : Sn Sn , t E [0,1J , be an ambient isotopy from the,identity to h1, which preserves the decomposition (*) , and such thatht preserves B: for a l l t , with h1 restr icting to the identity. Hencewe may construct the mapping torus of h1 by isotopy inside Sn X S1.

Since B: is preserved by h1, we may remove B X S1 and glue inn-1 2 n-1 1 n n rS X D along the common boundary S X S Now aB . intersects F.Ie'o \


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66in an equatorial (n-2)-sphere Kn-2, which i s the boundary of the

hemispheres D ~ - 1 = '0B i h # k i Sr X Dn-rand Dn-1 = QBn rn # ' nr+1 X Sn-r-1- 0 k

Thus Sn- 1 X D2 -- Dn-1 X D2 U D n ~ 1 X D2 d . b f add'+ _ ' an we Vlew as e ore lngSn-1 X D2 as adding a 2-handle to '

n-1 1along D+ X S1 n-1 1S along D_ X S nd to

(# Sr X Dn-r _ Bn ) X S1k 0 h1(# Dr+l X Sn-r-1 _ Bn ) Xk .0 h1Thus Kn-2 i s a knot in the common boundary !:" of these two man-

A =n-r-1In particular, i f det(A -I) =+1r -

i folds. Note that we have a choice o:f glueing maps here.Now o n a (#k Sr X Dn-r) , Poincare duality allows us t.o choose bases

ai forHr( F ~ , r ; z) = Zk and b j :for Hn_r_1( F ~ , r ; Z) = zk, suchthat under the intersection pairing (a . , b.) = b.. , and furthermorel J l Jsuch that the a. are represented by the spheres (Sr X *). in the i t h_l l

( n - r -1 ) thsummand, and the b . represented by the spheres * X S . in the j .J JLet A be the induced automorphism of H ( Fkn,r; Z) . Hencex x

(a., b .) = ~ . . (A (a.), A 1(b.)) = S..a J a,J r l . n-r - J l JA-tr +then det(A - I) =-1n-r-1

The following isa t r iv ia l consequence o:f higher dimensional handlebody theory;Proposition: 6. 6 I f r 2, n-r-1 '2 , then !: i s a codimensi.on 1sphere Sn smoothly embedded in Sn+1, and Kn-2 i s a doubly slice :fiberedknot in ~ i f and only i f det(A - I)r += ~

From th is we may easily concludeTheorem; Every simple doubly slice fibered knot can be constructedby i s o t o p y ~ in odd dimensions, as well as many even dimensional,knots.Proo:f: As we have seen, the monodromy of a fibered doubly slice knot


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determines the Seifert form. This in turn, for simple odd dimensionalknots, determines the knot isotopy class. Hence we need only show thata l l possible monodromies may be realised by isotopy.a) Consider an unknotted handlebody Hk in S3. By ambiently sl iding handles, any desired automorphism of H1(Hk; z) = Zk may be realised byisotopy.b) Now consider the inclusion # Sr X Dn-r Sn


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7 FURTHER ASPECTS IN DIMENSION 468

We would l ike to point out how the construction of the knots K t iesnin with the Gluck construction (18) . Gluck has shown t h a t , given a knotK in S4 with neighbourhood S2 X D2, we can pbtain at most two homotopy4-spheres by removing S2 X D2 and reglueing by spinning the S2 factor121times as we go around the S factor of the boundary S X S He proves

2 2th is by showing that spinning twice extends over S X D , in which casethe possible 4-manifolds obtainable depend on m mod 2.

I t i s a major problem whether every homotopy 4-sphere arising inth is way is standard. Some of' the s trongest candidates for counter-exam-pIes to the Poincare conjecture are known to be of this form, most nota-bly the double covers of' Cappell and Shaneson vs (9 ) exotic projectivespaces. We ref'er for more details to Akbulut and Kirby (2 ) and Aitchisonand Rubinstein (1 ) . On the other hand, Gordon has shown that f'or twistspun knots, even though the Gluck construction gives S4 again, the knot

2 2obtaaned as the core of the reglued S X D i s not isotopic to the orig-inal knot (21)

Our purpose here i s to point out an integral manifestation of the'Gluck construction, ie where the phenomena does not reduce mod 2.Theorem: 7.1 The knots K are a l l symmetric slices of the O-spun fign

I

ure 8 k n o t j c ~ . .and arise by perf'orming the Gluck construction on a symm-etr ic unknot in the complement of K.Proof': I f we "double" the dif'f'eomorphism f n on H2 we obtain a dif'f'eo-morphism of' #2 S2 X S1. As bef'ore we may construct the mapping torus,and perform surgery along the circle swept out by the fixed point p toobtain S4. For f' we obtain the O-spin of the figure 8 knot, arising aso


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the image in 84 of' the boundary Z-sphere of' the ball neighbourhood of' pin #Z sZ X S1. We can see this alternatively as the double of' the ballpair (B4 , R ) .o

To consturctwe need only make

a handlebody description f'or the surgered mapping torusminor additions to the picture obtained f'or (B4, R ) .n

In doubling HZ we introduce two additional Z-handles, which in the pict ure f'or the mapping torus thicken to give 4-dimensional Z-handles attach-ed to unknotted meridional circles f'or d1 and .dZ To ident if 'y these twohandles with the ir images under f' we must add 3-handles, but as theoattaching maps f'or these do not change the outcome, we need not draw themexplicit ly, as in ( Z). Now notice that the circle Wmay be unlinked f'romthe res t of' the diagram, using the two new Z-handles, and so any amountof' twisting around W cannot change the 4-manif'01d. In other words, thedouble of' the bal l pair (B4, R ) i s always (s4, K).n

What we are dUng here to obtain K iSiDblowing down the disc boundned by Win B4, and when we double, we are perf'orming the Gluck construc-t ion on the double of' W,which i s asymmetric unknotted Z-sphere in thecomplement of' K.

Finally we can summarise a l l of' the properties of' theK innTheorem 7.Z There exist inf' initely many distinct prime knots K in s3nwhich are i ) fibered of' genus Z

i i ) ribbon, bounding'ribbonsR in B4, such thatni i i ) B4 - R f'ibers ov

lain Roderick Aitchison- Isotoping and Twisting Knots and Ribbons

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Transcript of lain Roderick Aitchison- Isotoping and Twisting Knots and Ribbons