Fujitsugu Hosokawa and Takaaki Yanagawa- Is Every Slice Knot a Ribbon Knot?

8/3/2019 Fujitsugu Hosokawa and Takaaki Yanagawa- Is Every Slice Knot a Ribbon Knot?

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Hosokawa, F. and Yanagawa, T.Osaka J. Math .2 (1965), 373-384IS EVERY SLICE KNOT A RIBBON KNOT?

Dedicated to Professor H. Terasaka on his 60-th birthdayF UJ IT S UGU HOSOKAWA and TAKAAKI YANAGAWA

(Received September 4, 1965)

Let S2 be a locally flat 2-sphere in a4-dimensional euclidean spaceR 4, then the knot obtained by slicing S2 with a hyperplane in R 4 iscalled a slice knot [ 1 ] .A singular disk that is a continuous image of the unit 2-cell in a3-dimensional euclidean space R 3 will be called a ribbon, if and only ifeach of its singularities is of the following t y p e :

and a knot that is the boundary 0 of a ribbon is called a ribbon knot.R. H. Foxpresented a problem "Is every slice knot a ribbon knot ?"in his paper [ 2 ] . The purpose of this paper is to give an affirmativeanswer to the problem.In this paper we will consider everything from thesemilinear pointof view.1. In this paper, we shall use themotion picture method of des-cribing surfaces in a 4-space R 4.Let K be a slice knot in a 3-space R 3, and H 4[t l, /2] be a subspace

# 3 x[^, / 2 ] in a 4-space R 4 =R3x( co, oo), where ( ,) means open in-terval and [ , ] closed.1) The boundary of a singular disk (a ribbon) means the image of the boundary ofthe inverse image.


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374 F. HOSOKAWA and T. YANAGAWA

Then , wecan easily seethat there exists alocally flat, non- singular2- cell e2 in H4( l, 0] with boundary / c. Inorder todecribe e2, weconsider the intersections of e2with thehyperplanes R * =R*x[t^\ anddescribe the changing ofthe configuration as/ increases from 1 to 0.We can modify e2 isotopically inH\ 1, 0] so that these intersections,except at a finite number of levels, consist of collections oforiented,simple closed polygons which vary continuously with / . In each ex-ceptional level of themodified e2 there are a finite number ofcriticalpoints called elementary critical points, but these can be classified inthree types, that is, astincreases through thecritical value 0, theconfiguration changes asfollows(elliptic critical point of type I att= t0:) asmall unknotted simple closedpolygon appears as infigure1

nothing a P i n t

t < t t==t t o < tFig. 1

(elliptic critical point oftype II att=t0:) asmall unknotted simple closedpolygon shrinks to a point and disappears as infigure2a P i n t nothing

t


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S L I C E K N O T 375

say that the 2-cell e2

has a (1, l)- canonical form. Therefore, we havethe followingLemma 1. / / a knot K in /?? = ? 3 x [0] is a slice knots then there is

a locally flat, non- singular 2- cell e2 with boundary K in H4[_ 1, 1] whichhas a ( 1, \)- canonical form.

Now, we may assume that the p hyperbolic critical points of e2 arefound one by one at the values tiy i = l , 2, , A where I


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3 7 6 F. HOSOKAWA and T. YANAGAWA

Repeating these processes, we have the deformations g{

from, - , / t+ 1 - ] (Q


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Fig. 5c

Fig. 5d

where k, cl, , cr are inverse images of K , aly - , ar.A singular perforated disk in R 3 will be called a perforated ribbonif the singularities are all of the above four types.F r o m the above considerations we have

Lemma 2. / / a knot K is a slice knot in R 3, there exists in R 3 / ca collection of r mutually disjoint, non- singular 2-cells 19


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Let D, D0, D

ly, D

rbe the inverse images of ,

0

w-, > , and

let ky cly czy , r be the inverse images of K , aly a2, , r , that is dD=k,dDt=c{ ( = l , 2 , - , r ) .Now, let us consider the intersections of


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S L I C E KN OT 379

line of or it arrives at the point B and forms a closed double lineof or the case I occurs.

We remark that if a 6- line meets an / - line, then the junction- pointmust be a branch point.

T h e remaining singular lines of arise from th e intersection ofG O 0 with f d i O' O). Such a singular line must be a closeddouble curve.

We will call a double line, whose endpoints are on the boundary,a double line of ribbon type.

F r o m the above considerations, the singularities of are of thefollowing types3)

i) double lines of ribbon type.ii) closed double curves.

iii) triple points which are crossing points of double lines.iv) branch points.3.

(I ) Branch pointsSince the interiors of 0 and have no branch points, a branchp o i n t appears only in case I of section 2 th at is, all branch poin ts are

o n f ( i = l , 2, , r}. As a branch point is the result of local winding ofcr2 around ai9 we may modify , so that there is only one double linethrough the branch point. Then the double lines through the pointsare of the following two types, and figures 7a, 7b show their inverseimages.

Fig. 7We can eliminate such a bran ch point by cu tt ing along a double

line through the branch point, as shown in figure 8 [ 4 ] .2) By a slight modification we can put into general position,


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Fig. 8[ I I ] Triple pointsA triple point is a crossing of double lines that is, if A is a triplepoint then there are just three double lines through A. Let these

double lines be EG, HI, JK in a neighborhood of A, and the inverseimage of A be A, A", A", then the inverse image of in a neigh-borhood of A is as in figure 9.

Fig. 9Now, we will consider triple points on double lines of ribbon type.

By the considerations about singularities of in section 2, one of thet h r e e inverse images of every triple point does not belong to / - l ine .By cutting along the image of an arc starting from a boundary pointan d disjoint with / - lines, we may suppose that E'G' is a subarc of a in-line, say E'G' again, and H' is of a 6- line or of a closed double curve,and G does not contain any inverse image of triple points as in figure10- left upside.

We take such a point G1on EA that the double line GG containsno triple points except A. Now we cut off along GGl as in figur 10,then the triple point A vanishes.T h e cut has no singularities of new types and the knot type ofth e boundary does not change. We will denote the cut and its boun-dary by and K again.


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SL ICE K N O T 381

E '

Fig. 10Repea t in g these processes, we can remove all the triple points on

th e double lines of ribbon type.N o w , let the co l lec t ion {A{B{, A'2B'2, -, 'mB'm} be the - lines on D ;t h a t is, the i n v e r se im a g es of the double l ines of r i bb o n t y p e w h o s ee n d p o i n t s are on k. Let d{ be a n a r r o w b a n d in D w h i c h c o n t a i n sA'tB't in it s i n t e r i o r and d o e s not c o n t a i n the i n v e r s e i m a g e s of the o t h e rs in g u la r i t i e s of ( / = ! , , ra). By c u t t i n g D a lo n g the b o u n d a r i e s ofd19 dzy , dm, the cel l is s e p a r a t e d i n t o 2m- fl disks d19 d2y , dm anddm+i, dm+2, , d2m+l as in figure 11.L e t l f 2, -, 2 m + 1 be the i m a g e s of dl9 d29 , rf2;/ /+1, t h e n l f - , wa re m u tu a l ly d i sjo in t , n o n - s in g u la r d i sk s, and m + 1 , , 2 m + 1 are s in g u la rd i sks . However m + 1 L ) m + 2 U L J 2 m + 1 has no s in g u la r i t i e s in a neigh -b o r h o o d of 3 W + 1 U 3 W + 2 U U 9 2 m + 1 . T h e r e f o r e , by a s i m p l e e x t e n s i o nof D e h n ' s l e m m a ( p r o o f in the A p p e n d i x ) we can r e p l a c e m + 1 , m + 2 , ,


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A'm

Fig. 11 2w + by mutu llay disjoint, non- singular disks # m + 1 , $M + 2,differ from W + 1 U U 2 w + 1 only on a c o m p a c t subset of (?

, 92 w + 1

that

Now, by identifying the disks , , , m , tfm + 1, - , $ 2 W+ 1 along thecuts as before, we have a new singular disk with boundary / c. Let usagain denote this new disk by .

We will now consider the singularities of the new disk .Since

x, ,

w, are mutually disjoint non- singular disks, and

$ w + 1 , , $2m+ i are mutually disjoint non- singular disks, all the singulari-ties occur only as intersections of f and &j ( i = l , , m, j = m + l9,2m + 1).After a slight modification, only double lines, triple points and

branch point occur as singularities.When a double line crosses a cut, the inverse images of the cross-

ing point must be identical, and such a singular point must be a branchpoint. F urth erm ore every branch point arises in this way.

If th ere is a double line which is neither closed nor of ribbon type,the inverse images of the double line are two arcs A'B" and A"B' andboth A and Bf are on 9 where A! is on the boundary of } and Bf ison the boundary of e (as in figure 12). But t does not intersect

Fig. 12


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\$j, so this is a contradiction.If there is a triple point, it must be contained in three different

disks of the collection 1? ~ , m , #m+19 # 2m + . But th is is impossible.Since we can eliminate the branch points as in the beginning ofthis section, we can modify3) so that the singularities are all of thefollowing type

1) double lines of ribbon type2) closed doublecurves.As there exist no triple points, all the closed double curves are

simple and mutually disjoint.Therefore we can easily remove these simple closed double curves

by the well- known cut- and- exchange method.Thus we can obtain a ribbon modified from with the boundary

K . Therefore we haveTheorem. Every slice knot is a ribbon knot.H . Terasaka proved t h a t the Alexander polynomial of a ribbon knot

is of the form tmf(f) f(t^\ [7]. Therefore it follows from the aboveT h e o r e m t h a t the Alexander polynomial of a slice knot is of the forml) , [8].

AppendixLemma. / / D19 D2, , Dr is a set of normal, canonical Dehn- diskssuch that dD{ \Dj= (i^FJ>iy j = l,2,- ,r) in a ^- manifold M, then thereexists a set of mutually disjoint, non- singular disks $ j, $2> , dr such that&i is identical with D t in a sufficiently small neighborhood of dD{ ( = l ,2, - , r\Proof. By making use of Dehn's lemma repeatedly, we have a set

of non- singular disks D(, Z>2, -,D'r such that D


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( = 2 , - , r ) .Repeating this process r1 times produces a set of disks #19z?2, t?r having the required property.

Bibliograpy[ 1 ] R. H . Fox: A quick trip through knot theory, To p. of 3- manifolds and

related topics, edited by M. K. Fort Jr., Prentice Hall, 1962.[ 2 ] R. H . F ox : Some problems in Knot theory, ibid.[ 3 ] J. Milnor: Morse theory, Ann. of Math. Studies N o. 51, 1963.[ 4 ] C. D . Papakyriakopoulos: OnDehrts lemma and asphericity of knots,Ann .

of M a t h . 66 (1957), 1-26.[ 5 ] A. Shapiro an d J. H . C. Whitehead: A proof and extension of Dehrfs

lemma, Bull. Amer. M a t h . Soc . 64 (1958), 174-178.[ 6 ] T . H o m m a : OnDehn's lemma for S3, Yokohama M a t h . J. 5 (1957), 223- 244.[ 7 ] H . Terasaka: On null- equivalent knots, Osaka M a t h . J. 11 (1959), 95- 113.[ 8 ] R. H . F ox an d J. Milnor: Singularities of 2- spheres in 4- space and equi-

valence of knots, unpublished. Cf. Bull. Amer . M a t h . Soc. 63 (1957), 406.

Fujitsugu Hosokawa and Takaaki Yanagawa- Is Every Slice Knot a Ribbon Knot?

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