Isotoping and Twisting Knots and Ribbons . .

By

lain Roderick Aitchison B.Sc. (Hon.) (University of Melbourne, Australia) 1979 M.Sc. (University of Melbourne, Australia) 1980 C.Phil. (University 0'£ California) 1983

DISSERTATION Submitted in partial satisfaction of the requirements for the degree of OOCTOR OF PHIlDSOPHY

in .. '

Mathematics in the .

GRADUATE DIVISION OF 1HE UNIVERSITY OF CALIFORNIA, BERKELEY

Approved:

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

1

Isotoping and Twisting Knots and Ribbons ABSTRACT

In higher dimensions the Seifert form determines up to isotopy the geometrically defined "simple" knots. Every classical knot is simple. However, there are numerous examples for which the Seifert form fails to provide sufficient distinguishing invariants. It was conjectured the Alexander polynomial would determine classical fibered knots to within finitely many possibilities. This paper addresses the queetd.om "Is there a geometrically defined class of classical knots for which the Seifert form acts as a complete invariant?" Geometric conditions known to place restrictions on the Seifert form are the properties of being fibered, slice and doubly slice. For fibered knots Thurston' s classification of diffeomorphisms of surfaces leads to further subclasses. For fibered ribbon knots, Casson and Gordon have shown such a knot bounds a ribbon in some homotopy ball whose complement fibers over the circle with fiber a handlebody. That this ball be standa.rd is .at present knowledge an additional requirement. Doubly slice

knots are known in higher dimensions to interact with codimension one embedding phenomena. Our main result is the construction of infinitely many distinct knots K in 83 which are prime, fibered, of genus 2, ribbon, bounding ribbons i

4 whose complements fiber over S1 with fiber a handlebody, have mono-

in B

dromy which is pseudo-Anosov, and thus have complements in S.3 admitting a complete hyperbolic structure, are symmetric slices of the O-spun figure 8 knot, are doubly slice, being constructed by ambient isotopy of

2

a closed surface in S3. and yet have identical Seifert forms. Thus the classical dimension is even further removed from the higher ones than previously expected. Along the way we construct fibered knots by ambient isotopy of submanifolds of spheres, give a short proof that every closed orlentable 3-manifold has a fibered knot and infinitely many hyperbolic fibered links and characterise geometrically the simple fibered doubly slice knots in higher odd dimensions. Finally we consider the twisting constuction of Stallings and relate this on the one hand to the Gluck construction on 2-knots and to the construction of different ribbons by isotopy on the other.

CONTENTS

o

Introduction

1

1

Definitions, Notation and Conventions

6

2

Fibered Knots

13

Diffeomorphisms of Surfaces

27

4

Slice Knots and Ribbons

43

5

Constructing Mapping Tori

46

6

Doubly Slice Knots

7

Further Aspects in Dimension 4

68

Bibliography

71

1

Isotoping and Twisting Knots arid Ribbons or Some Simple Samples of Simple Simple Knots

o

INTRODUCTION The fundamental problem of knot theory is to find sufficient comp-

utable means of distinguishing examples. In higher dimensions the Seifert form distinguishes all odd dimensional simple knots (31), and every classical knot is simple. However there are many examples in the classical dimension of knots which are not so distinguished. Simplicity is a geometric condition, and the natural question is whether there is a geometrically defined class of classical knots for which the abelian invariants arising from the Seifert form form a complete set. Geometric conditions manifest in the form the Seifert form may take, and again for higher dimensions, algebraic properties are realised geometrically.] The fundamental difference between the higher and lower dimensional cases is the availability of the techniques of the s-cobordism theorem and the Whitney trick. Failure of the Whitney trick in dimension 4 leads to much of the richness and pathology of the classical dimension. For example, in higher dimensions an algebraically slice simple knot is slice, whereas there are infinitely many examples of distinct classical knots with trivial abelian invariants, those with Alexander polynomial 1. The recent work of Freedman (16) and Donaldson (57) shows that on the

one hand these are all topologieally slice, but tttat there are many which cannot bound smooth slices, for reasons relating to possible smooth structures on 4-dimensional manifolds.

2

For f'ibered knots, the Alexander polynomial cannot be trivial except f'or the unknot. It has been long known that the only possible fibered knots of genus 1 are the trefoil and figure 8 knots. Based on the few examples known at the time, Burde and Neuwirth conjectured that there were only finitely many distinct examples of fibered knots with a given Alexander polynomial. Morton (36) showed this to be false at the same time as Stallings (49»), the latter introducing the notion of "twisting" for the purpose of generating more examples of fibered knots. These examples were distinguishable by their Alexander modules, from which the Alexander polynomial arises, and thus the obvious question was whether the Alexander module determined fibered knots to within finitely many possibilities. Quach (44) showed that there were abstractly infinitely many distinct examples with the same Alexander module structure, and Kanenobu (26) gave the first specific examples of this phenomena. Subse!i'f uently Morton (37) showed that there are infinitely many distinct fibered knots for any given possible Alexander module. Quach and Weber (45) rediscovered the examples of Stallings, and made the further observation that they are all ribbon knots. Recently, Bonahon has produced infinitely many distinct examples of fibered knots with the same Seifert form, with pseudo-Anosov monodromy, but which cannot 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 be ribbon is that the monodromy extend over a handlebody. There are no known examples of f'ibered ribbon knots not distinguished by their abelian invariants. The condition of being doubly slice places further restrictions on the Seifert form, and as before, these algebraic conditions are always

3

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

There are iminitely many distinct knots K in S3 suCh that n

a) Kn is prime ii) K is f'ibered of' genus 2 n

4 iii) K is ribbon, bounding a ribbon R in B n n 4 iV) B - R f'ibers over S1 with f'iber a handlebody n

v) K has pseudo-Anosov monodromy, and thus S3 - K admits a n n complete Riemannian metric of' constant curvature -1 vi) K is doubly slice n

vii) Kn is a SYmmetric slice of' the a-spun f'igure 8 knot viii) K is constructible by isotopy n

iX) the Seif'ert f'orm of' K is independent of' n n

A number of concepts arise in the statement of the theorem, and accordingly the paper is divided into sections • The proof' is constructive, and thus we reach the f'inal result only at the end after a number of' brief diversions along the way.

In the first section, we give some definitions, and describe briefly the f'undamental notion of twisting knots, arising from the calculus of' f'ramed links (30). This is 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 is to relate this concept to the work of Casson and Gordon (.58) and to put the results in the broader context of' four and higher dimensions.

4 Section 2 is devoted to fibrations, and we discuss in some detail the trefoil and figure 8 knots, crucial to our subsequent constructions. The notion of constructing diffeomorphisms of surfaces and handlebodies by isotopy in S3 is discussed in this context. We briefly mention the constructions of fibered knots due to Stallings (49) and apply this to give a new proof that every closed orientable 3-manifold has a fibered knot. We also prove Theorem 2.2

Every closed orientable 3-manifold has infinitely many

fibered links whose complements admit a hyperbolic structure. An interesting phenomena relating fibered links and hyperbolic structures in the 3-sphere is introduced. The monodromy of a fibered knot is a surface diffeomorphism, and so we discuss Thurston's work in section 3. The techniques of train tracks is described in the context of distinguishing diffeomorphisms. We prove Theorem 3.7

Infinitely many distinct pseudo-Anosov maps may be obtain-

ed by ambient isotopy of a genus 2 surface in S3. Section 4 briefly describes the relationship between the Seifert form and ribbon knots, . and the work of Casson and Gordon. In section 5 we use the construction of section 3 and apply it to prove Theorem 5.2

The knots K of Kanenobu are genus 2 fibered ribbon knots m,n

4

bounding ribbons in B whose complements fiber with fiber a handlebody. Theorem

5.4

The knots Sm of Stallings, Quach and Weber are genus 2

fibered ribbon knots, bounding ribbons as in Theo!em 5.2 We also introduce the examples for our main theorem, and the under lying technique for generating many more,

5 The techniques involved are handlebody decompositions OI 4-maniIolds and the calculus OI Iramed links. Doubly slice knots are discussed in section 6, and we prove Proposition 6.1

The monodromy Ior a simple Iibered doubly slice knot

in S2n+1 determines the SeiIert Iorm. We then give a new prooI OI a special case OI Zeeman's theorem Theorem 6.2 (56)

IIK is a Iibered knot, then K # -K is doubly slice.

The point OI this prOOI is to introduce a new idea into the study OI Iibered doubly slice knots, the notion OI "construction by isotopy": Theorem 6.4

There exist in£initely many prime Iibered doubly slice

knots in S3. Our construction gives rise to the generation OI zillions OI candidates Ior algebraically doubly slice Iibered classical knots which are ribbon, but probably not geometrically doubly slice. We conjecture Conjecture:

Every doubly slice Iibered knot is constructable by isotopy.

As evidence Ior this we prove Theorem 6.5

Every simple, Iibered doubly slice knot in S2n+1, IJf·~ 3,

may be constructed by isotopy. The construction gives a Iurther maniIestatfuon OI codimension 1 embedding phenomena in the study OI doubly slice knots. We also observe a phenomena we call synchronicity Theorem 6.7

In£initely many doubly slice knots arise synchronously

as Iinite dimensional slices OI in£inite dimensional doubly slice knots. ,11

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

6 1

DEFINITIONS, NOI'ATION

AND CONVENTIONS

We work in the smooth category unless otherwise stated. As general references, we use Gordon (22), Kervaire (29) and Rolfsen

et6).

n.· . n n+2 • An n-knot K as the amage of an embedded n-sphere S in S n+ n A Seifert manifold W 1 for K is any connected, orientable submanifold of Sn+2 with boundary Kn• Every knot has infinitely many Seifert manifolds. The Qomplement C( K) of K is Sn+2 - K, and the exterior X(K) is the complement of an open tubular neighbourhood of K.

In odd dimensions, the Seifert form of a knot K2n-1 is the integral bilinear form defined

o~ Hn (W2n; Z)/ Torsion X Hn (W2n; Z)/ Torsion by

2n the linking matrix V ::: (V..) with respect to the Seifert manifold W ~J

and choice of basis (a.) for H (W ~

2n;

n

Z)/Torsion. Here V.. ::: ~J

lk(a.,a~) ~ J

is the linking number in S2n+l determined by taking representative cycles a

i

2n, and a. for the classes (a.) and (a.), pushing a. off W J

~

J

J

using the orientation to choose a direction, to obtain ai:, and calculJ

ating the linking number. The matrix V is the Seifert matrix for K with respect to the Seifert manifold and basis chosen.

Examples:

In fig.i, we give the knots constructed by Stallings (49),

and rediscovered in Quach and Weber (45). These knots will be constructed as representatives of a much larger class later in this paper. We shall denote them by S • In fig.2, we give a similar class, K which m m,n were originally discovered in Kanenobe (26), and rediscovered by the author, and these too playa role in what follows.

7

fig. 1 The knots of Stallings, Quach and Weber, S • m

m full twists

a+

b+

c+

a

0

0

1

0

b

0

0

1

1

c

1

0

1

0

d

1

1

0

m-1

=

V

Seifert matrix

Under row and column operations, we have: t Alexander module presentation tv - V equivalent to

r

- t + 1 0

-mt t2 _ t +

J

8

b

a

+

b+

module

+

d+

a

0

0

-1

0

b

0

0

-1

-1

c

-1

0

n-1

-n

d

1

-1

-n

2

Alexander

c

[ t

_ :

+

1

m+n-1

t(n-m) ] t

fig. 2

2

- 3t + 1

Seifert matrix

9 For these examples, we see that there is a genus 2 punctured surface serving as Seifert manifold, and for each we have chosen the basis represented by the embedded circles a,b,c,d.

2n Since W is not unique, V is not well defined. However, the ThomPontryagin construction shows that every Seifert manifold is cobordant to any other. This leads to the notion of S-equivalence of Seifert matrices:

where a and b are respectively integer row and column vectors. An elementary reduction is the reverse operation. Two Seifert matrices V and V' are S-equivalent if each 'can be obtained from the other by a sequence of unimodular integral congruences, elementary enlargements and reductions. From V we obtain a presentation matrix

tv - v

t

for the Alexander

module over z[t,t- 1] for the knot K. This depends only on the S-equivalence class of V, and the presentation may often be simplified by row and column operations. For figures 1 and 2 we obtain the simpler matrices shown. From the Alexander module, the elementary ideals and i

th

Alexander polynomials may be defined, and are invariants of the knot type. These are so-called abelian invariants of the knot. In particular Definition: A(t)

= det

t) (tv - V is the Alexander polynomial.

10 For the examples of

1 and 2, only two Alexander polynomials arise.

In the first case, the knots are distinguished by their Alexander module but for the second class, when m = n we see that the knots have the same abelian invariants, and must be distinguished by other means. We shall see later that all these examples are fibered, the first class having been discovered as counterexamples "btl the conjecture of Burde and Neuwirth that there were at most finitely many distinct ftbered knots wi th the same Alexander polynomial. Other examples of

t~is

phenomena

have been constructed by Morton AJy n ~ vY, showing that the a

n

are strictly increasing with n, and we conclude finally that the

diffeomorphisms f

n

are mutually non-conjugate and non-isotopic.

43

4

SLICE KNDrS AND RIBBONS

When Quach and Weber rediscovered the knotsS

m

in fig.l, they made

the additional observation that they are all ribbon knot.s , Definition:

A knot Kn in Sn+2 is ribbon if it bounds an immersed

n+l-ball Bn+l all of whose self-intersections are transverse double n points, the connected components of which are embedded copies of B • The preimage of each such Bn is two copies , one of which is properly embedded in B n+l , the other lying entirely in the interior. Fig.l provides a configuration in which the immersed discs are apparant. We will shortly see that the knots Km,m are also ribbon, a fact the reader may easily discern

directly"

The interest in ribbon knots is that they are the boundaries of particularly nicely properly embedded balls in Bn +3 , To see thfus, merely push the self-intersections of the immersed B n+l into a collar of Sn+2 in Bn+3 , and use general position, Ribbon knots are the nicest examples of slice knots: Definition;

n

A knot K

is slice if the pair (Sn+2, Kn ) is the boundary

of some disc pair (B n+3 , Bn+l ) . In even dimensions, every knot is slice, though not all such are ribbon. In odd dimensions, an obstruction for a knot to be slice arises from the Seifert form: JDef'furr::kbion:

n

K is an algebraically slice knot if it has a Seifert

form S-equivalent to one of form

A necessary condition for a knot to be slice is that it is raically slice.

algeb~

44

In higher odd dimensions, Levine (31) has proved the remarkable Theorem: 4.1 A knot K2n-1 in S2n+1 is slice if and only if it is algeb2n-1 raically slice,. , for n~2 and K simple. Returning to the classiealdimension, it is a lorgstanding conjecture that a knot is .slice if and only if it is ribbon. Casson and Gordon have made two signifigant contributions to this problem. On the one hand (11), they have introduced invariants whose vanishing is necessary for a knot to be ribbon, although the possibility of the knot nonetheless being slice is not ruled out. These are finer obstructions than the algebraically slice condition, but we shall not have cause to calculate them here. The obstructions for our examples will vanish, as they are in fact all going to be ribbon. Their other contribution (58) has as starting point the observation that ribbon knots enjoy a nice property: there is a surjection of the fundamental group of the knot complement onto that of the ribbon complement in the ball. The latter complement having a handle decomposition with handles of index

& 2,

as may easily be seen by looking at the crit-

ical points of the embedded ribbon in the collar of Sn+2. n . Sn+2. h ti some h omo1 ogy A knot K In lS omo t ow rl.bbon on In

Definition:

Rn+1 such t hat t he na t ural incluSlon . · ball vn+3 if it bounds a s 1 lce

TI, (Sn+2

- K)

is a surjection. Casson and Gordon (58) then prove Theorem:4.2

A fibered knot in a homology 3-sphere is homotopy ribbon

if and only if the capped off monodromy extends over a handlebody.

Thus in particular, a necessary condition for a fibered knot to be ribbon is that its monodromy be null-cobordant (ie it must extend over some orientable .3-manifold). In his thesis, Bonahon ( 6) calculated the oriented cobordism group of diffeomorphisms of closed orientable surfaces and in (7 ), he applied this to produce infinitely many examples of fibered knots in S.3 which are not ribbon, but which nonetheless have the same algebraically .slice Seifert form. On the other hand, the knots K of fig.2 all have the same Alexn,n ander module, and thus the same abelian invariants. That these are distinct is proved in Kanenobe (26). We will now show how to construct the knots of figs. 1 and 2, and in so doing prove the stronger statement than that implied by Casson and 4 Gordon's work, namely tha.t the complement in B of the ribbons they bound f'Lber-eoven thEI·j3i±ele with fiber a genus two handlebody. Casson and Gordon's work shows that this property is satisfied in some homotopy 4-ball, but until the Poincare conjecture is settled in the smooth catB egory we may not conclude that our result is universally true. We should remark at this stage that the only class of fibered ribbon knots previously known to the author are the knots Sm' the 8

9

knot and

other 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 for constructing such knots there leads to the construction of Sand Km,n • m

46 .5

CONSTRUCTING MAPPING TORI We will show how the various knots Sand K arise, by reversing m m,n

the following observation: IfK is a fibered ribbon knot in S3, bounding 4 a ribbon R in B4 such that B - R fibers, over the circle with fiber a 4 handlebody, we can reconstruct B by first explicitly constructing the mapping torus M of the diffeomorphism f corresponding to the fibration, f and then adding a 2-handle to kill off the S1 factor, corresponding to glueing the ribbon back into the ball. If we were to use an arbitrary diffeomorphism of a handlebody, we of course do not in general expect to obtain the 4-ballo On the other hand, we will now show that the diffeomorphisms of the genus two handlebody constructed in section 2 all lead to B 4.: The following technique was first used by Akbulut and Kirby ( 2), generalised by Montesinos (3.5) and also used in Aitchison and Rubinstein. 2 2, For the diffeomorphism f: # s S1 X D --~> # s S1 X D we con-o.. 2 S1 X D by H • s 1. Since M arises by identifying the ends of H X I by f, we may f s isotope f so that it restricts to the identity in a neighbourhood N of

struct the mapping torus in stages. Denote #

s

some point p on the boundary. As O-spine forH

s

we take some other point

q in the interior of N. A 1-spine for H is then an embedded bouquet of s

circles with common point q, each circle intersecting Hs - N in an arc li running once around one of the handles as in fig/a. We will need to of the annuli A., as shown, under fo 1

Note: 1. lies on A.. 1

1

fig.41

47 This gives a handle decomposition for H with a O-handle and s 1s

= 1, ••• , s ,

handles b., 1 .i

Taking H X I with induced handle decomposition . .. . . £

we obtain M by identifying H X r-1~with H X {1} via f: f s s 2.

Identify N Xf-1} with N X

(11

by adding

a 1-handle b*. The ends of

the arcs 1. Xf-1} may be identified with their images under f by pairs of 1

arcs running over the 1-handle b*, and taking the union of these, the Ii Xf-1\ and their images in H X s ded circles a

i•

fq

we obtain a disjoint set of embed-

The mapping torus construction is completed by adding di,

s 2-handles with attaching circles the a., and with framings determined 1

by the A. and their images. The situation is depicted in fig.l42.. 1

.-

.. -.. .....

.....

",

. ' ....

...

..

...

"-

..... -.

·

,

\;

· .,. '10

~

~

: ••

·•

··•·

•,, , •,•

I

,• •• :• : ,•

· .· •. . ·• •• •• .. • ..... "-,....... .. ....... ....•· •

'

f

'

t

,.

~

H X I s

..... ..

'... . ... . ..

fig.42

I

48 We wish to apply this to the diffeomorphisms f n • Hence we need to "coordinatise" our description: 4 For O-handle, take B viewed as B3 X 1. The boundary 3-sphere thus naturally inherits a structure as the union of the two 3-balls B3 X[-1 ~ and

B3 X

f1~

,

take S3 as R3 U

and the S2 X I running between them.. It is convenient ttl:> 00

,

with B3 X f 1T the unit ball, and B3 Xf-1} the exte

erior of say the ball of radius 2. is a diffeomorphism of H2 ' Hg X I in th.is case has 2 1-handn les b and b thickened from a and a 2• These are attached to the unit 1 2, 1 ball, serving as O-handle for H with a Us ends at the "East ll and "West" 2, 1 As f

poles, and a

2

to the North and South. Thus in our model for H X I the 2

ends of b

and b are attached to th®3-balls 1 2 the "po'Lea" and poles, as in fig. 43:

in the neighbourhoods of

..d b

.• •

e. ·,tI_

.

.. '. 1If. ...... ..-.- ..._ .. ....Il.. .. -.. . ' .

....

..•.

q X

... ,:

..

b~ fig.43

.., -..

J

.Ob~

(1l

\... \ ..

/

Pi .

.. ... . . 'e

, ,

k

-, . ·.. .n.··

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

.. . .. ...-

.

"Ill .. ... .. ,...

..

'6

.f'

.. ..-

..

11'"

2

We have also drawn in the attaching balls for the 1-handle b*o For the moment, we consider only the case n

= O.

In this case, we have

a very convenient picture for the construction. The diffeomorphismmf O is exactly the figure 8 knot monodromy, thickened. Hence the attaching circles a

and a appear as in fig.44. The reader may wish to refer to 1 Z fig.16. Another consequence is that the framings are untwisted, being determined by annuli lying in the planes of the representation, for the identification by adding the Z-handles d

1

and d

Z

along a

1

anda Z'

fig.44

Inside Mf

Z

o

we see a neighbourhood D X S1 of the circle

p X I/~.

This appears as a neighbourhood of the arc z in the figure. We are going to attach another Z-handle d* along this circle, and show that the result 4• is B Firstly, we slide the Z-handles d

and d off b* using d*, and then Z 1 cancel these latter handles • The result is fig.45~ It will be convenient to keep track of the genus Z surface which appears as the boundary of the unit ball, surgered by tubes running over b

If we remove a disc Z' neighbourhood of the point p we obtain a punctured surface, with the knot 1

and b

50 K its boundary. When we finally arrive at the trivial handle decompos4 O 4, ition for B K will appear as .a knot. We will see that the image of W O under the cancelling of handles remains unchanged.

fig.45

In fig.46 , we have isotoped K over b so that the 2-handle d can O 1 1 be isotoped to reduce the number of times it runs over b

1

geometrically

to one.

fig.46

51 Now slide d

1

off b

1

using d

2,

and cancel b

1

with d

2

to obtain fig.

47- We must also slide KO as indicated. After further isotopy, it is clear thatb 2 is cancelled by d but in doing so, K must be isotoped O 1, 4, also. We finally obtain B in the boundary of which we see K and W, O as well as the Seifert surface for K on which Wlies. We have also labO elled two other unknotted circles

fig.47

fig.48

~

and Q which lie on the surface. 2

52

fig.49

:fig·50 fig.51

53 In fig.51 we isotopecthe knot K to a more recognisable form - we O obtain figure 8 # -figure 8. Furthermore, the circles Q and Q satisfy 1 2 the conditions of Stallings, and so twisting on each of them we obtain the knots K of Kanenobe , m,n That K is a fibered knot may be deduced from the construction. O Let f be the restriction of f to ~H2' Thus aMf is the mapping torus Mf , 1 0 a genus 2 surface bundle over S • Adding the 2-handle ~* has the effect of surgering M along the circle swept out by p, and thus S3 is obtain-

f

2 1 1 2 ed by removing D X S and replacing by S X D • Hence K O is isotopic to

S1 X

t oh

proving that the complement of KOfibers.

4

A similar conclusion holds whenever we obtain B by adding a 2-hand.Le

to H X S1, for any diffeomorphism g: H ---J)o H fixing a neighbours g s s

hood of a point p on

~Hs'

along the circle swept out by p. The boundary

K of a disc neighbourhood of p on

~H

s

will always be a fibered ribbon

knot in S3, bounding a ribbon R. Suppose now that Cis an embedded circle on aH ,bounding a disc s D in H , and that after handle cancellation C appears unknotted in S3= s

d(Mg

U d*) as constructed above.

Proposition:5.1

Any knot K' obtained from K by twisting along C isa

4

genus 2 fibered ribbon knot, bounding a ribbon R V whOse complement in B fibers over 8 Proof;

1

with fiber H • s Let g* be any diffeomorphism of H obtained from g by follows

ing with a cut along D, twisting a number of times and reglueing. We

4

claim this is the monodromy of B - R v, 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" this s g disc in the representation of the mapping torus construction).

Twisting along C changes the mondromy of K by a number of Dehn twists along C on the fiber surface. Considered as 3H , this can be achieved s

by cutting along D and twisting the handlebody. Hence in the 4-dimensional perspective, a neighbourhood of the disc D in B4 is a 2-handle attached to the complementc the twisting is achieved by removing this 2-handle and replacing it with a new 2-handle with the same attaching circle but with different framing. We are "blowing down" the ribbon D•• Hence R' is exactly R, twisted along C. This is because the pair

K, C bounds a ribbon link arising naturally from the construction of the mapping torus - in our initial handlebody picture for Mg U d* both Rand D appear as "ribbon" discs, and under isotopy and handle sliding, only ribbon intersections are introduced. Twisting along C thus twists R to v

a ribbon R • Applying this to the cases above, w.e have Theorem: 5.2 The knots K of Kanenobe are genus 2 fibered ribbon knots m,n 4 such that B - R fibers over the circle with which bound ribbons R m,n m,n fiber the

~enus

2 handlebody.

Proof;

It is clear that the circles Q1and Q2 satisfy the conditions

of the discussion above.

Now observe we may do the same -t-lith W. In this case, denoting the knot obtained by twisting n times by K , we see that the monodromy for n

Kn is exactly f n , and hence the Kn are all distinct for nz O. Moreover I'

Wand KO form a boundary link,and so all of the1