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Roseman, D.Osaka J. Math.11 (1974), 307- 312

WOVEN KNOTS ARE SPUN KNOTSD E N N I S ROSEMAN

(Received November 10, 1971)(Revised December 1, 1973)

Given a knotted 1-sphere, k> in R3 it is possible to find a knotted 2- sphere,Ky in i?4 such that ^i?3- ^) is isomorphic to Il^JR^- K). In [1], Artin constructson e such example called a spun knot; in [3], Yajima also gives an example whichwe will refer to as a woven knot. The object of this paper is to show that theseknots are, in fact, the same; that is, given k, the corresponding spun knot andth e woven knot constructed from the mirror image of k are ambiently isotopic.

By a knotted w-sphere in R n+ 2y we will mean an ambient isotopy class ofembeddings of Sn into R n+ 2. Sometimes, in order to avoid proliferation ofnotat ions we will use the same letter to denote a map and the image of that map.We will also generalize this construction to other types of spinnings of higherdimensional knots.

We will use PL spheres in our constructions. We will use the followingnotion of general position : if is a PL rc- sphere in R n+ 2, we will say y is in generalposition if for each vertex, > and - simplex of , with v not a vertex of , y isn ot contained in the &-plane of R n+ 2 determined by .

(b)

Figure 1

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308 D. ROSEMAN

S u p p o s e y is an w - sp h e r e in R n + 2 ; le t R n++2={ ( x ly , x n+ 2 ) ^ R n + 2 w i t hx ^ 0 } y let dRl+2=:{(x19 . - , x+2) Rn+2 with ^ = 0 }. Also, define h: RT^R1by h(x ly , # + 2 ) = # M + 2 ; we may think of h as a height function. Without lossof generality, we may assume that is the union of two n- disks a and sucht h a t a is an (n l)- sphere, and (1) (5 n)ei?^ + 2, such that Ao >0 (i.e., lies above the half- (n+ l)- plane in R n++2 given by xn+2>0; (2) (S n) 3i?+ + 2 = / 5 ;(3) Up: R%+2^Rl+1 is given by p(x19 , *r t + 1, ?ff+2)=( ?1, , * w + 1 ) , then we willrequire that p\ is an embedding (all that we will ever use is that p\d =p\dais an embedding) () is in general position. If is a circle in i?3, a is an arcas in figure 1 (a).

T o describe the spun knot, we will write points of Rn+k+2^Rk+1 Rn+1 int h e form (z y x k+2y ~ yx n+k+2) where p is a unit vector in the first ( &+ 1 ) -coordinates and #^>0. For each p, let Hp denote the half- (w+ 2)-hyperplane ofall points of the form (zpy x k+2y , x n+k+2). Then the maps hp defined byh p( x ly - , x n+ 2)= ( x 1 p, x2- ,x n+ 2) are e m b e d d i n g s of i? + + 2 in to R n+k+2y andU h p(RT 2)= R n+k+2.PW e will n e e d the following n o t a t i o n s for s u b s e t s of the ( z + & ) - sp h e r e. We

wil l consi d e r Sn+k to be th e u n i t s p h e r e R n+k+1^ R k+ 1 R n and d e n o t e p o i n t s by( z p , Xfg+2> > n+k+i) w h e r e p is a u n i t v e c t o r in the first k - \ - l c o o r d i n a t e s , # ^ 0 ;w e will consi d e r D

nto be the u n i t disk in R

n+1. Let p be the w- disk in S

n+k

w h i c h is the i m a g e of the map \ p ( x ly , x n)= ( \ / l x\ p, xly , xn); p ist h e inte rsec t ion of Sn+k w i t h the set of all p o i n t s of the fo rm (zp, x k+ 2, , x n+k+i)-F o r e a c h p o i n t a^Dn, a= ( a lf , an), define a map a: S k - + S n+ k by a( x 19 ,^ , * Vax*+i> ai> *> ) where a=Vl f. Thus is theintersection of *S W+ * with the set of points (x19 , ^.+ 1, a n , an); also we maysee that a is a ^- sphere of radius a if e l n t Dn, a is a point if a^dDn. Ifwe are spinning an arc, then Sn+k is a 2-sphere, and p is a longitudinal are, is a meridian circle, or a pole, see figure l(b).

We will now define an embedding *: sn*k- >Rn+k+2 by requiring for eachp , S^oXp^Apo . The isotopy class of S# will be called the knot obtained by^- spinning a. We remark that if a and a! are two w-disks in R n+ 2 and at is anisotopy with ao=a, ax=a' and for all t, O ^ ^ l, , n # + + 2 = a # ( 3 D w ) , then thereis an isotopy, Kt, between the sphere obtained ^- spinning a and that obtainedby ^- spinning a!\ the isotopy is defined so that for all t, hp(at)=Kt(Xp).

We will want to examine the projection of *S by projection along the lastc o o r d i n a t e , x n+Jt+2. Let be t h i s p r o j e c t i o n ; U(z , x k+2y - - - yx n+k+ly x n+k+2)=( z y x k+2y , x n+k+i). L e t / > : R n++2 - >R n+ +1 be as b e f o r e ; let a*=p(a) . F or eac hp , we m a y d e f i n e e m b e d d i n g s h p : l P i + 1 - > j R f l + * + 1 b y h p '( x ly , ? I I + 1 ) = ( 1 P , X2, ,x

n + ) . S i n c e o h = h p 9 n ( S * ) = n ( U A P ( ) ) U A P ( ) = U A p ' * ) . W e m a yP P Pstate this as follows: The projection of the / j- spinning of a is the same as the k-

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WOVEN KNO TS ARE S P U N KNOTS 309

spinning of the projection of a (for the spinning of the arc of figure 1, see figure2; figure 2(b) shows (Sl) with U V(*) removed where 0< < / 2 ). Wemay also describe (S^) as follows; if i G with b=a(a) with # e ) , let^4^= (j Ap( ), ^4& will be a ^- sphere if e ln t Z , a point if a^dD", letPAt=n{Ab)= U V( ) , then (S*)= U Af. If f r is the set of points of multi-plicity r of a under p, that is, Mr={x^a* such that ^)~1( J) a consists ofexactly r points}, and if M/ is the set of points of multiplicity r of Sk under , Mr'={x^l(S) such that "1^) 5* consists of exactly r points}, thenMJ is obtained by A- spinning Mry i.e., M/ ={ U h/(x) where x^Mr}. In thePcase of spinning a 1- sphere, each double poin t of the projection will correspondt o a circle of double point s of the spun knot. F ur thermo re, suppose that byb'^awith p(b)=p(b') and h(b) 0 for all be a let M be a number

such that M>h(b) for all i e . By our general position, we may find an 6such that if is a vertex of a, a ^- simplex of a with $ , then is less thant h e distance between v and the - plane of R n+ 2 determined by . N ow supposet h a t a is given by a ( a ) = ( ^ ( a ) , , ? +2( )), let : 1 / ( ^ ) = ^ 1 ( ) ( l+ ( x + 2 ( ) ) / M ,and for , 0 > ^ + 2 ( )) , a * ( ) = ( ( ^ i) # ( e ) , ?2( ), , ?ll+2( )), th en ,() is an isotopy inR l+ 2 from to a' fixed on 3 . If aRn+ k+2 be th e map which takes Ha to a hyperplaneof i?w+ *+ 2 by a map which takes to a circle of radius # / (# ) defined as follows:

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310 D . ROSEMAN

let va=x1'(a)lva if ^ 0, va=0 if va=0 (i.e., if a^dDn), then define ka(x19 ,X M + I > ai> > ?2( ), a ?8(), * > ^ + i W , * i ( a ) ) . N o t e t h a t t h elast coordinate is given by x^a).

Now we define an embedding W ka: Sn+k- >Rn+k+2 by requiring that W k oa=kaoa, or W k(a)=k a(a). The isotopy class of W % will be called the &-wovenknot corresponding to 7.We will now discuss the special case of 1-weaving a 1-sphere, illustrating

with the particular example of the trefoil knot of figure l(a). In this case, a!can be described as a slight distortion of a which, above the doublepoints of *,bends a on the overpasses away from dR% more than on the underpasses. Thus(

7) * looks like figure 3(a). If a(a)=(x1(a)9 x2(a)y# 3( x^a), x 2( )) with \ y\ < # / ( # ) , see figure3(b) . Then Ra is a ribbon in P 3 and if 7, 7: R4- >P\ is defined byU'(xlf x2, X3, ^ 0 = ( 0 , 2> 3> #4) then /(Wl)=Ra. In fact, we may see that W\is the symmetric ribbon knot of Ra, see Yajima [4]. Furthermore, it is clearfrom the discussion in Yajima [4], page 137, that W\ is the same as the 2-s here

similar to the knot 7, defined in Yajima [3]. From the discussion which is tofollow, we will see that W]> will be a spun knot; thus the knots defined in Yajima[3] are all spun knots.

F or convenience we will describe Yajima's construction [3] and illustrate itwith the trefoil knot. Given a knot and the corresponding knotted arc, a, weconstruct a self- intersecting tube around the projection, *, of , narrowing thetube along the arc at the underpasses and closing off the tube at the end pointsof * (see figure 3). This describes the projection of a knotted 2- sphere; to

Figure 3 Figure 4

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WOVEN KNOTS ARE S P U N KNOTS 311

determine the height relations at the double points we use the following rule:choose a direction for a indicated by arrows, if the crossing at a point of a* isas in figure 4a, then the double point set consists of two circles c and c2 and wewill define our embedded sphere so that the smaller tube passes under the largeon e at c and the smaller tube passes over the large tube at c2; the projection ofthese tubes will look like figure 4b. (This over- under alternation at each cross-ing point accounts for our choice of the term "weaving" to describe this knotan d its generalizations.)

(a) (b)Figure 5

We now wish to examine the projection H(Wl'). For each b^a\ withb=a\a)y we define Bb=W k(a); then Bb is a ^- sphere of radius # / (#) if e lnt D n, a point if at dD". If Ab'= U hp(b), (Ab')*=n(Ab')9 Bf= (Bb)

P

t h e n we see that for all b, (Ab')*=Bf, since each set consists of a ^- sphere ofradius #/ (#) in the hyperplane (xly , xk+1> x 2( a) y , xn+1{a)) with center(0, " ,0, x 2(a) , -~,xn+1(a)). Thus U(S^)=U(W^)\ however, this does notimply that Sfr is ambiently isotopic to Wfr, we need to check the height relationsin the x n+k+2 coordinate. We note that for any Bby the x n+k+2 coordinate ofpoints of Bb are the same, namely x^a). Now suppose that Bf=B%' and thus(Ab =(A')*=Bf9 then ( /)*(*)= ( / )(*0 a n d t h u s * / ( ) = * / ( 4 w h e r ea(a')b'. Now suppose that A( )

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312 D. ROSEMAN

those ofLet a! be the mirror image of a! obtained by reflection in the last coordi-

nate of i? + + 2 ; (ar){a)=(x(a), x 2(a ) , , xn+2(a)+M) (we need to add the Mto the last coordinate in order that a! satisfy condition (1) in the definition ofa). For mirror images of circles in i?3, see Crowell-Fox, Chapter 1, Section 4[2]. Now the height relations of SL ' are the reverse of those of S#/ , andn(Sfr)=n(Sij). Thus SI*' is ambiently isotopic to W^\ in fact, by anambient isotopy which translates Bb in the ?M+ + 2 coordinate until it coincideswith Ah'= U h{- a\ )).P

TH E UNIVERSI TY OF IOWA

References[ 1] E. Artin: Zur Isotopie zweidimensionaler Fl chen in RAt H amburg Abh. 4 (1925),

174- 177.[ 2] R.H. Crowell and R.H. Fox: Introduction to Knot Theory, Ginn and Co. 1963.[ 3] T. Yajima: On the fundamental groups of knotted 2- manifolds in the 4- space, J.

Math. Osaka City Univ. 13 (1962), 63- 71.[ 4] T. Yajima: On simply knotted spheres in RA, Osaka J. Ma th. 1 (1964), 133- 152.