ROSEMAN, DENNIS

Roseman, Dennis

Osaka University Knowledge Archive

Woven knots are spun knots

Osaka City University Repository

Roseman, D.
Osaka J. Math.
11 (1974), 307-312
WOVEN KNOTS ARE SPUN KNOTS
DENNIS 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 Π^i?3-^) is isomorphic to Il^JR^-K). In [1], Artin constructs
one such example called a spun knot; in [3], Yajima also gives an example which
we will refer to as a woven knot. The object of this paper is to show that these
knots are, in fact, the same; that is, given k, the corresponding spun knot and
the woven knot constructed from the mirror image of k are ambiently isotopic.
By a knotted w-sphere in Rn+2y we will mean an ambient isotopy class of
embeddings of Sn into Rn+2. Sometimes, in order to avoid proliferation of
notations 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 higher
dimensional knots.
We will use PL spheres in our constructions. We will use the following
notion of general position: if γ is a PL rc-sphere in Rn+2, we will say y is in general
position if for each vertex, υ> and β-simplex σ of γ, with v not a vertex of σ, y is
not contained in the &-plane of Rn+2 determined by σ.
(b)
Figure 1
308 D. ROSEMAN
Suppose y is an w-sphere in Rn+2; let Rn+
+2
={(xly •••, xn+2)^Rn+2 with
x^0}y let dRl+2=:{(x19 .-, x»+2)ξΞRn+2 with ^ = 0 } . Also, define h: RT^R1
by h(xly •••, # Λ + 2 )=# M + 2 ; we may think of h as a height function. Without loss
of generality, we may assume that γ is the union of two n-disks a and β such
that aΠβ is an (n— l)-sphere, and (1) γ(5 n)ei?^+ 2, such that Aoγ>0 (i.e., γ
lies above the half-(n+l)-plane in Rn++2 given by xn+2>0; (2) γ(Sn) Π 3i?++2=/5;
(3) Up: R%+2^Rl+1 is given by p(x19 —, * r t + 1, Λ?ff+2)=(Λ?1, —, * w + 1 ) , then we will
require that p\β is an embedding (all that we will ever use is thatp\dβ=p\da
is an embedding) (μ) γ is in general position. If γ is a circle in i?3, a is an arc
as in figure 1 (a).
To describe the spun knot, we will write points of Rn+k+2^Rk+1χ Rn+1 in
the form (zρy xk+2y ~yxn+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 of
all points of the form (zpy xk+2y •••, xn+k+2). Then the maps hp defined by
hp(xly ••-,xn+2)=(x1 p, x2- ,xn+2) are embeddings of i?++2 into Rn+k+2y and
U hp(RT2)=Rn+k+2.
P
We will need the following notations for subsets of the (τz+&)-sphere. We
will consider Sn+k to be the unit sphere Rn+k+1^Rk+1χRn and denote points by
(zp, Xfg+2> •••> χn+k+i) where p is a unit vector in the first k-\-l coordinates, # ^ 0 ;
we will consider Dn to be the unit disk in Rn+1. Let λp be the w-disk in Sn+k
which is the image of the map \p(xly •••, xn)=(\/l—Σ x\ p, xly •••, xn); λp is
the intersection of Sn+k with the set of all points of the form (zp, xk+2, •••, xn+k+i)-
For each point a^Dn, a=(alf •••, an), define a map μa: Sk-+Sn+k by μa(x19 •••,
^Λ, ••*, V
a
x*+i> ai> *••> ««) where η
a
=Vl—Σ βf. Thus μ
α
 is the
intersection of *SW+* with the set of points (x19 •••, ^.+ 1, a n •••, an); also we may
see that μ
a
 is a ^-sphere of radius η
a
 if α e l n t Dn, μ
a
 is a point if a^dDn. If
we are spinning an arc, then Sn+k is a 2-sphere, and λp is a longitudinal are, μ
α
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 each
p, S^oXp^Apoα. 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 Rn+2 and at is an
isotopy with a
o
=a, a
x
=a' and for all t, O ^ ί ^ l , α, n#++ 2=a#(3Dw), then there
is an isotopy, Kt, between the sphere obtained ^-spinning a and that obtained
by ^-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 last
coordinate, x
n+Jt+2. Let Π be this projection; U(zρ, xk+2y ---yxn+k+ly xn+k+2)=
(zρy xk+2y •••, xn+k+i). Let/>: Rn++2->Rn++1 be as before; let a*=p(a). For each
p , w e m a y def ine e m b e d d i n g s h p : l P i
+ 1
- > j R f l + * + 1 b y hp'(xly •••, Λ ? I I + 1 ) = ( Λ 1 P , X2, •••,
x
n + ί ) . S i n c e Πohμ=hμΌp9 n ( S * ) = n ( U A P (α)) U Π A P ( α ) = U Ap' ία*) . W e m a y
P P P
state this as follows: The projection of the /j-spinning of a is the same as the k-
WOVEN KNOTS ARE SPUN KNOTS 309
spinning of the projection of a (for the spinning of the arc of figure 1, see figure
2; figure 2(b) shows Ώ(Sl) with U V(«*) removed where 0<ψ<Π/2). We
may also describe Π(S^) as follows; if i G α with b=a(a) with # e ί ) Λ , let
^4^= (j Ap(ό), 4^& will be a ^-sphere if αe lntZΓ, a point if a^dD", let
P
At=n{Ab)= U V(έ), then Π(S*)= U Af. If Λf r is the set of points of multi-
plicity r of a under p, that is, M
r
={x^a* such that )^~1(ΛJ) Π a consists of
exactly r points}, and if M/ is the set of points of multiplicity r of Sk under
Π, M
r
'={x^ΐl(S£) such that Π" 1 ^) Π 5* consists of exactly r points}, then
MJ is obtained by A-spinning M
ry i.e., M/={ U h/(x) where x^Mr}. In the
P
case of spinning a 1-sphere, each double point of the projection will correspond
to a circle of double points of the spun knot. Furthermore, suppose that byb'^a
with p(b)=p(b') and h(b)<h{b'), then for all p, the #
rt+ife+2-coordinate of hp(b)
will be less than the x
rt+A?+2-coordinate of hp(b') (since these will be equal to h(b)
and h(b')> respectively), denote this by Ab<Ab'.
(b)
Figure 2
We next describe another embedding of Sn+k into Rn+k+2y the woven knot.
As before, we begin with a. Recall that h(b) > 0 for all b e a let M be a number
such that M>h(b) for all i e α . By our general position, we may find an 6
such that if υ is a vertex of a, σ a ^-simplex of a with ϋ $ σ , then £ is less than
the distance between v and the β-plane of Rn+2 determined by σ. Now suppose
that a is given by a ( a ) = ( ^ ( a ) , —, Λ?
Λ+2(α)), let Λ:1 /(^)=^1(α)(l+(£xΛ + 2(α))/M,
and for ί, 0<ΐ< 1, (Λ?i)
ί
(α)=Λ?1(α)(l+(ίfΛ?li+2(α))/Λ/). Next define α/(α)=(Λ?1/(α),
Λ?2(«)> — >^+2(«)), a*(«)=((^i)#(e), Λ?2(α), —,Λ?ll+2(α)), then «,(«) is an isotopy in
Rl+2 from α to a' fixed on 3α. If a<=Dn, a=(a19 •••, αw), let Ha be the ( * + l ) -
hyperplane of Rn+k+1=Rk+1χRn of the form (xly •••, ΛA + 1, ^ , •••, an)y then
A t Λ = 5
W +
 * Π H
a
. Let Λ
Λ
: H
a
->Rn+ k+2 be the map which takes H
a
 to a hyperplane
of i?w +*+ 2 by a map which takes μ
Λ
 to a circle of radius #/(#) defined as follows:
310 D . ROSEMAN
let v
a
=x1'(a)lva if ^ β φ0, va=0 if va=0 (i.e., if a^dDn), then define ka(x19 •••,
XM+I> ai> —> <*n)=(?aXij — , ^ Λ + i > Λ?2(α), a? 8(«), *••> ^ + i W , * i ( a ) ) . N o t e t h a t t h e
last coordinate is given by x^a).
Now we define an embedding Wk
a
: Sn+k->Rn+k+2 by requiring that Wk
Λ
oμ
a
=
k
a
oμ
a
, or Wk(μ
a
)=k
a
(μ
a
). The isotopy class of W% will be called the &-woven
knot 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 α*,
bends a on the overpasses away from dR% more than on the underpasses. Thus
(α7)* looks like figure 3(a). If a(a)=(x1(a)9 x2(a)y #3(<z)), with a^D1, α*(α)=
(x
x
(a), x2(ά)) Let P 3 be the hyperplane in R4 with last coordinate zero. Let Ra
be the set of points of the form (0, y> x^a), x2(ά)) with \y\ <#/(#), see figure
3(b). Then R
a
 is a ribbon in P 3 and if Π7, Π7: R4->P\ is defined by
U'(xlf x2, X3, ^0=(0, 2^> ^ 3> #4) then Π/(Wl)=Ra. In fact, we may see that W\
is the symmetric ribbon knot of R
a
, see Yajima [4]. Furthermore, it is clear
from the discussion in Yajima [4], page 137, that W\ is the same as the 2-sρhere
similar to the knot 7, defined in Yajima [3]. From the discussion which is to
follow, we will see that W]> will be a spun knot; thus the knots defined in Yajima
[3] are all spun knots.
For convenience we will describe Yajima's construction [3] and illustrate it
with the trefoil knot. Given a knot γ and the corresponding knotted arc, a, we
construct a self-intersecting tube around the projection, α*, of α, narrowing the
tube along the arc at the underpasses and closing off the tube at the end points
of α* (see figure 3). This describes the projection of a knotted 2-sphere; to
Figure 3 Figure 4
WOVEN KNOTS ARE SPUN 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* is
as in figure 4a, then the double point set consists of two circles c
λ
 and c2 and we
will define our embedded sphere so that the smaller tube passes under the large
one at c
λ
 and the smaller tube passes over the large tube at c2; the projection of
these 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 knot
and its generalizations.)
(a) (b)
Figure 5
We now wish to examine the projection H(Wl'). For each b^a\ with
b=a\a)y we define Bb=Wk(μa); then Bb is a ^-sphere of radius #/(#) if
αeΞlnt Dn, a point if atΞdD". If Ab'= U hp(b), (Ab')*=n(Ab')9 Bf=Π(Bb)
P
then we see that for all b, (Ab')*=Bf, since each set consists of a ^-sphere of
radius #/(#) in the hyperplane (xly •••, xk+1> x2(a)y •••, xn+1{a)) with center
(0, " ,0, x2(a), -~,xn+1(a)). Thus U(S^)=U(W^)\ however, this does not
imply that Sfr is ambiently isotopic to Wfr, we need to check the height relations
in the x
n+k+2 coordinate. We note that for any Bby the xn+k+2 coordinate of
points of Bb are the same, namely x^a). Now suppose that Bf=B%' and thus
(Abγ=(Aί')*=Bf9 then (α/)*(*)=(α/)(*0» a n d t h u s * / ( β ) = * / ( 4 w h e r e
a(a')—b'. Now suppose that A(έ)<A(ό/), then as we have seen, Ab <A&\ how-
ever, Bb>Bb' since the xn+k+2 coordiate of points in Bb and Bh' is given by xx(a)
and x^a'), respectively, and from the definition of #/ we see that if #/(α)=
χi(a') w ^ t n h(b)<.h{b'), then Λ?1(α)>Λ?1(α/). We may summarize this by saying
that although Ii{S^)=Ii{W^)y the height relations of Sa are the opposite of
312 D. ROSEMAN
those of
Let —a! be the mirror image of a! obtained by reflection in the last coordi-
nate of i?++2; (—ar){a)=(xί(a), x2(a), •••, — xn+2(a)+M) (we need to add the M
to the last coordinate in order that —-a! satisfy condition (1) in the definition of
a). For mirror images of circles in i?3, see Crowell-Fox, Chapter 1, Section 4
[2]. Now the height relations of SL
Λ
' are the reverse of those of S#/, and
n(Sfr)=n(Sij). Thus SI*' is ambiently isotopic to W^\ in fact, by an
ambient isotopy which translates Bb in the Λ?M+Λ+2 coordinate until it coincides
with —Ah'= U h{-a\ά)).
P
THE UNIVERSITY OF IOWA
References
[1] E. Artin: Zur Isotopie zweidimensionaler Flάchen in RAt Hamburg 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. Math. 1 (1964), 133-152.


WOVEN KNOTS ARE SPUN KNOTS

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WOVEN KNOTS ARE SPUN KNOTS

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