Annals of Mathematics
On the Total Curvature of Knots
Author(s): J. W. Milnor
Reviewed work(s):
Source: The Annals of Mathematics, Second Series, Vol. 52, No. 2 (Sep., 1950), pp. 248-257
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1969467 .
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OF MATHEMATICS
ANNwAzs
Vol. 52, No. 2, September,1950
ON THE TOTAL CURVATURE OF KNOTS
BY J. W. MILNOR
(Received October5, 1949)
f
The totalcurvature
Introduction
S"(s) I ds of a closed curve C of class C", a quantity
which measures the total turningof the tangent vector, was studied by W.
Fenchel,who proved,in 1929,that, in threedimensionalspace,
I]"(s)
I ds _
2'n,equalityholdingonlyforplane convexcurves.K. Borsuk,in 1947,extended
this resultto n dimensionalspace, and, in the same paper, conjecturedthat the
total curvatureof a knot in threedimensionalspace must exceed 47r.A proof
ofthis conjectureis presentedbelow.'
suggestedby R. H.
use willbe made ofa definition,
In provingthisproposition,
Fox, of total curvaturewhich is applicable to any closed curve. This general
definitionis validated by showingthat the generalizedtotal curvatureK(C) is
equal to
I
\"()
I ds for any closed curve C of class C". Furthermore,the
theoremofFencheland Borsukis trueforany closed curve,ifthe new definition
of total curvatureis used.
( ), the
Closelyrelatedto the conceptof total curvatureis a new invarient/Z
ina
is
either
This
positive
curves.
(
of
closed
of the isotopytype
crookedness
In
a
is
or
is
polygon.
as
by
the
not
represented
(E
type
tegeror oc, according
alternative
an
to
is
it
provide
of
crookedness
possible
the
of
concept
terms
formulationof the generalizedtotal curvatureas a Lebesgue integralover an
(n - 1) dimensionalsphere.The crookedness,4(Cs)of a type (S of simpleclosed
curvesis connectedwith the total curvaturesof its representativecurves C by
the fundamentalrelation2,ru( S) = g.l.b. K(C). Generallyspeaking this lower
bound is not attained.
In the courseof the paper several interestingincidentalresultsare obtained:
ifthe total curvatureof a simpleclosed curveis finite,thenthereis an inscribed
polygonequivalentto it by isotopy,and also if the curveis knottedtheremust
be a plane whichintersectsit in at least six points.
I am indebtedto R. H. Fox forsubstantialassistance in the preparationof
thispaper.
1. The Total Curvatureof a Closed Polygon
By a closedpolygonP in Euclidean n-space H', n > 1, will be meant a finite
, ctm-i, am = ao , of which is required only that
sequence of points a, ,
.
f
1 Since the completionof this paper, therehas appeared an independantproof,by I.
FAry,that the relation
"(s) I ds 2 47rholds forall knots [6].
248
ON THE
TOTAL
CURVATURE
249
OF KNOTS
$i5 ai+, , and the linesegmentsaiai+1 fori = 0,1, ... , m - 1. For convenience
let ai, wherei is any integer,signifya(X), where (i) is the least positiveresidue
mod m. The terms point and vector will be used synonymously,with every
to a commonorigin,so that ai+1 - ai means the vectorequal in
vectorreferred
lengthand parallel to the line segmentaiai+1 . Denote by ac the angle between
0 < as ir. By thetotalcurvature
thevectorsa+?1- as and a, - ai1 satisfying
sum
P is meanttheangle
K(P) ofa closedpolygon
E., ai.
1.1 LEMMA.2 The adjunctionof a new vertexto a closedpolygoncannotdecrease
its totalcurvature.The curvaturemay remainconstantif eitherthenew vertexa3
and thetwoadjacentvertices
aj-1 and aj+l are collinearor ap-2 , 1Ia X ,aj+Xl , and
increase.
it mustdefinitely
aj+2 are coplanar.Otherwise
Let P' be the closed polygonwithverticesa1 , a2 , - , aC-1 , aj+, * , amC.
aX
,
,
? X. *X
, a
Let P be the closed polygonwithverticesaL, a2 , .* X1
,jl
=1, 2,
i
a'
for
by
Denote
.
vertex
the
P'
adjoining
obtained from by
aj
-2(X.
FIG.
1
j - 1,j + 1, *, m the respectiveexteriorangles of P', and by ai for i =
, m the respectiveexterioranglesof P. Denote by
1, 2, *
j - 1,j, j + 1,
, the angle betweena -cai1 and aj+l - j , and by 3+ the angle between
aj+l- aj- and aj+l - aj. (See Fig. 1.)
> aj-a , wherethe
4By the triangleinequalityforsphericaltrianglesai-, +
equality can hold only if the threeangles lie in a plane, that is, only if aj.2,
j , cja, and aj+l are coplanar.Similarlyaj+l + A+ > aia+, wherethe equality
can hold only if aj-1, aj, a+ja , and aj42 are coplanar. From the trianglewith
verticesaj1 , ai , and aj+l we have /- + /+ = a3 . Therefore
K(P) -
K(P')
=
>
(aj-1 -O-
ai-1)
+
+ex
aj +
(aj+l
-
ai4l)
= O.
Hence K(P) > K(P') and the equality can hold only if either aj2 ,
aj+l , and aj+2 are coplanar or aj-, aj , and aj+l are collinear.
2
aj-1,
This proofis essentiallythe same as a proofgivenby Borsuk [1. pp. 254-256].
a ,
250
J. Wv.MILNOR
1.2 COROLLARY. If thevertices
,j-1 , aj , and aj+1 of a closedpolygonP
ai-2 ,
are notcoplanar,and thevertexaj is replacedby a vertexa' whichlies on theline
segment
ajaj+1 , thenK(P) is decreased.
2. The Total Curvatureof a Curve
By a closedcurveC in Euclideann-spaceH' will be meant a continuousvector
, Xn(t))of period 1 which is not constant in any tfunctionX(t) = (&l(t),
interval.In particularany polygoncan be describedin this manner;it will be
convenientto regarda polygonas a closed curve,ignoringthe distinctionbetween
parameterizations.A closed curve X(t) is simple if X(t1)= X(t2) only
different
when (t1 - t2)/lis an integer.
'
rn/
FIG. 2
, am is said to be inscribedin a closed
A closed polygonP withverticesal ,
values
tisuch that ti < ti+1 , ti+m-ti + 1,
curve X(t)ifthereis a set ofparameter
=
i.
of
values
and as X(ti)forall integral
2.1 LEMMA. For any closed polygon P, K(P) = l.u.b. {K(P')} where P' ranges
overall polygonsinscribedin P.
If P is a polygonhavingtwo or moreverticescoincident,it can be represented
as the limitof a sequence of polygonswith all verticesdistinct;hence it only
remainsto provethe lemmaforP withall verticesdistinct.
If PO is a representativeinscribedpolygonwhose verticesinclude all but m
of the verticesof P, we may adjoin the remainingverticesof P one by one to
By 1.1, K(Po) <
P.
P, producing a sequence of polygons PO' , P'
...
l.u.b. {iK(P')}.
=
Therefore
=
<
but
<?
K(P)
K(P).
K(P')
K(Pf);
*
K(Pf)
TOTAL
THE
ON
OF
CURVATURE
251
KNOTS
by K(C) = l.u.b. {K(P) } where
For each closedcurveC definethetotalcurvature
P rangesover all polygonsinscribedin C. If C is itselfa polygon,the preceding
lemma shows that this definitionis consistentwith the definitionof Section 1.
by arclengths,
2.2 rTHEOREM. If C is a closedcurveof class C"' pararneterized
then K(C)
I & (s) I ds.
=
If act -(S
=
,,
'),
=
X(s') are the vertices of a polygon Pm inscribed in
C, suchthatlim, axmaxi { (sim - s')}
=
boK(Pm)
0, it willfirstbeshownthatlimm
(s) ds (Compare Fig. 2).
Define s2
and
,'(Sm l)
=
2 (si
t'(S2).
+ S'+1) foreveryi and m. Denote by 62 the angle between
The vector X'(s) describesa curve L of length
X" (s) I ds
f
on theunitsphereS'-'. The vectors '(s2) formthe verticesof a sphericalpolym
gon of length O2= 62 which is inscribedin L. Thereforelimnb m 62 =
I
c
X'(s) Ids.
continuous,foreach E > 0 thereis a 6 > 0 suchthat
Since X"(s) is uniformly
< e forall u - v < 6. From the identity
"(v)
m
+
(S i+
(s2)
(s2
sT),,(,m)
[X"(u)- X"(0)]du dv
i
S
i+
m~~~~~~~S
~
l "(u)-
-
f f
-
Li
fn
?
E""(s2) &(u)]du d
-
we have
si+i-si m
|(s)m
-
)
(-
<
X(9)
+-
s) 'E
4 whenever maxi { (sm+, - s7) }
<
6.
is the angle between (sm+) - (sm)and '(S2) then sin m2< (s2+, - s')
If 2m
s2) lies withina
E/4,since the end point of the vector (T(s524+i)- X(s2if))(sm2+
sphereof radius (Smi+, - S'i) E/4 about the end point of the unitvector X'(V2).
small E wehave '2 < 2 sin '2 < (s2i
sit) E/2. The angle
Hence forsufficiently
between X(si+i) - ~(sT,) and X(sT) - x(s2m-)is aT , while that between X'(9T) and
so
m+
,mi- and 62 < aim + sp? +'pi,
ai <= O +
l) is ri . Therefore
X'(S,
62 < lE, where
Z
am -ZEr=,
Em.=1
that I 0 - 62 < ('in+, - sm,-) E/2, and
1 is the length of C. Therefore
-
-
lim K(Pm)
m-4o00
lim
In orderto showthat
m
m
E
m-00 i=1
=
I
remainsto show that K(P)
ai
= lim E
mn-c00
= I
Jc
|(s)
ds.
C (s) ds = l.u.b. {K(P) } forP inscribedin C, it only
<
I
"
'(s)
tds.
252
J. W. MILNOR
Given any polygonPk inscribedin C, we may forma sequence of polygons
Pmform = k, k + 1, ... by adjoiningverticesto Pk so that
limma.,maxi {(si+1 - s,)} = 0.
By 1.1
K(Pk)
thereforeK(P7-)
?
K(Pk+1)
?
fs
<
--*
but limid..
K(Pm)
(s)
=
ds, and
"(s) ds.
3. The Crookednessof a Closed Curve
For each closedcurveC and each unitvectorb, defineA(C, b) to be the number
of maxima of the functionb- (t) (i.e. the numberof parametervalues tofor
which b* (to) ? b- (t) for t withinsome neighborhoodof to) in a fundamental
period. For each closed curve C definebt(C) = minb{Wt(C,b)}. We may call
of C.
,u(C) the crookedness
For everyvectorai+1 - ai inthe space H' definebi = (ai+1 - ai)/ I ai+1 - ai
Accordingto the conventionintroducedearlier,bi also denotes a point on the
unit sphere S"1, the sphericalimage of ai+1 - ai. Given a polygonP with
verticesal , a2, ... , am, a spherical polygon Q is formed on Sn'- by joining
each bi-, to bi by a great circle arc of length ai . This spherical polygon Q
is called a sphericalimage of P, and is unique unless for some j the vector
bj = - j+l . Note that it may happen that bj = bj+l .
3.1 THEOREM.3 For any closedcurveC in Hn, n >
2, the Lebesgueintegral
u(C, b)dS, whereb ranges over the unit sphere, exists and is equal to
is themeasureof S" 1.
where3in_ = (27r"12)/F(n/2)
(Mn_1K(C))/21r,
curve is a polygonP. For every
which
the
in
case
the
We will firstconsider
of S"'_ whichhas a pole at b.
the
S,-2
denote
sphere
great
point b of S"', let
if
if
Sn-2
and
b
only
crosses
An edge bj-lbj ofQ
(aj+l - aj) and b (aj - aj1) have
of b (t). Therefore,if
or
minimum
is
maximum
a
that
b
opposite sign, so
aj
to b) the number
P
is
of
perpendiuclar
no
edge
of
(i.e.
vertex
Q,
Sb-2 containsno
b
for which Sb2
The
of
set
points
is
b).
2,u(P,
with
of
Q
of intersections
Sb2
of
the
finite
collection
great spheres
of
is
union
the
of
Q
vertex
containssome
of
Sn-1
to
Ui Sbn,2the
with
respect
the
complement
n2;
of
in
each
component
Sb
|
f
function2M(P, b) is constant.The integral
2p(P, b)dS, where dS is the
sn-1
elementof surfaceon Sn-1,is thereforedefined.The set of points b forwhich
Sb-2 meets a given segmentbi-,bi of length0 ? ai _ r is a "double lune"
boundedby the greatspheresS'-27 and Ssn2. Thus the contributionof bi-,bi to
2,4(P, b) is 1 if b is an interiorpoint of this lune and 0 if b is an exteriorpoint.
whereMn,1 is the measure of the entire
The measureof this lune is (afiMIn,1)/r
sphere. Consequently
f
,-1
8
2y(P, W)dS =
__E
m
it 1
~~~~7r
ai
7r
This theoremis relatedto Crofton'sformula.Cf. [3. p. 811.
K(P
ON THE
TOTAL
CURVATURE
253
OF KNOTS
If C is an arbitraryclosed curve g(t), let Pm be a set of inscribedpolygons
= g(t') such that each Pm containsall
a',
,
gm(t)with verticesa' =
the vertices of Pm-1 and satisfyinglimnm.AK(Pm) = K(C) and limm.xmaxi
(tm,-tm) } = 0. The values of b forwhich b (t) or any b*m(t) has an interval
of constancyforma set of measure zero, and thereforehave no effecton the
integral.Such values willbe ignoredforthe remainderofthe proof.
We firstshow that ,t(C, b) = limm-oo A(Pm b). It is certainly true that
j(C, b) > i(Pm , b) > i(Pm-i X b). If ,t(C, b) < o, it is possibleto selecta neighborhoodof each of the A(C, b) maximaof b*g(t) and of each of its minimasufficientlysmall so that a polygonwith a vertex in each of these neighborhoods
musthave at least A(C, b) maxima;whichis certainlytrueofPmform sufficiently
large.If A(C, b) = Oo,the set of values of t forwhichb (t) is a maximummust
containa denumerablesubset {t2i}such that eitherto< t2< . .. < liming t2i <
to+ 1orto> t2> .*. > limli- t2i > to - 1. In eithercase we may selecta series
ofintermediatevalues t2i+1such that each b (t2i) > b (t2i+1) and b (t2i) >
b-*(t2i-1). Given any 2j < oc we may select neighborhoodsof the g(ti), for
i < 2j, so small that any polygonwithat least one vertexin each neighborhood
large.Therefore
has at leastj - 1 maxima;whichis trueof Pmform sufficiently
A(Pm, b) increases without finitebound as m -* o. Each of the integrals
-
,
f
Pu(Pm,b) dS exists; and the nondecreasingsequence of positive functions
Ai(Pm X b) approaches,4(C,b). Therefore4theintegral
equals
3.2
Since
1ime-.t(Pm,
K(C)
COROLLARY.
C
Kt_1 K(C)
2ir
b) dS
=
limm_:+
(Mn-1)
fI
(C, b) dS
K(Pm) =(Mn1)
exists and
K(C)-
? 2irA(C).
=
f-1 1(Ca)
dS?
fn- lu(C)ds =
Mn11i(C)
By a convexcurvewill be meant a closed plane curve describedby g(t) such
that any line contains g(t) eitherfor not more than two values of t withina
fundamentalperiodor forall values of t withinsome interval.
conditionthata closedpolygonP in
3.3 LEMMA. The necessaryand sufficient
H2 be convexis thatforeveryb eitherA(P, b) = 1 or,A(P,b) =
It is clear that this conditionis necessary.Suppose that P is a closed plane
polygonsuch that eitherli(P, b) = 1 or 4(P, b) = oo foreach b in the plane.
For any b such that A(P, b) = 1, any line perpendicularto b will intersectP in
at most two points.If b is a vectorforwhichlu(P, b) = Oo, and if H' is a line
perpendicularto b whichintersectsP a finitenumber,say r, of times,then it is
withP in the
always possibleto rotateH1 about one of its pointsof intersection
properdirectionso that r is not decreased. Hence thereis a b and a H' perpendicular to b such that Az(P,b) < cc and the number of intersectionsof H'
r ? 2.
withP is f ? r. But by the firstcase, f < 2, and therefore
o
4
[4. Theorem12.6p. 281.
J. W. MILNOR
254
3.4 THEOREM. For any closedcurveC, K(C) ? 2r. The equalityholdsifand only
if C is convex.
Since ,g(C) ? 1 foreverycurve,or since everycurvehas an inscribedpolygon
P forwhichK(P) = 2r, we have K(C) ? 2r. Since any curvewhichis not convex
has an inscribedpolygonwhichis not convex,and sincea polygoninscribedin a
convexcurvemustbe convex,it onlyremainsto provethe secondportionof the
theoremfor polygons.
It is provedin plane geometrythat the sum of the exteriorangles of a convex
polygonis 2r. If therewere a non-planarpolygonP forwhich K(P) = 2r, we
could selectfourconsecutivenon-coplanarvertices(neglectingverticesforwhich
ai = 0). By 1.2 therewould be a new polygonP' such that K(P') < K(P) = 27,
which is impossible.If there were a non-convexplane polygon P for which
K(P) = 2w, then 3.3 states that there would be a directionb for which 1 <
o but thereis a neighborhoodof any such 6 withinwhich1u(P,b) is
so,
,u(P, b)
constant.This meansthat K(P)
(P,b) dS
=
>
f
dS = 2X.
4. The Curvatureand Crookednessof IsotopyTypes of Curves
In H' the closed curve describedby X(t)of period1,and the closed curve describedby ;(t) of period 1 are said to be equivalentby isotopyif there is some
isotopyof Hn onto itself,whichtransforms
X(ul) into ~(ul) forall u.
By a curvetype S in Hn is meant an equivalence class of closed curvesunder
isotopy.A curve type is simpleif the representativeclosed curves are simple.
if (E is that
A simplecurve type S and its membersare said to be unknotted
type whichcontainsall circles.If a simple curve type containsno circlesthen
the type and its membersare said to be knotted.
A curvetype C and its membersare said to be tame'if G containsa polygon.
Otherwisethey are said to be wild.' It is well known that in H' every simple
closed curve is unknotted.In H nforn > 3, every simple tame curve is unknotted.
For each curvetype A, defineK(G) = g.l.b. K(C) and A(G-)= min ,(C), where
C rangesover all membersof (S.
<K r, thereis an iso4.1 LEMMA. For each c and p in Jfn-1such thatI cn-1
c into P and leaves
ontoitselfwhichtransforms
tu 1, of H
topy,f (T),I 0
2)-sphereof radius r and centerc, such
fixedall pointsof H'-1 outsidethe (n
thatfco(T) is a continuousfunctionof u, I, c, and p.
For example:
=
-
ui
X
7
]
(P
-
c)
for I
-c
forj
- c! _ r.
4.2 THEOREM. For any simple closed curve C, such that 1t(C) <
toC byisotopy.
polygonP inscribedin C and equivalent
5 This definitionwas given by Fox and Artin [51.
I
r.
xc, there is a
ON THE
TOTAL
CURVATURE
OF KNOTS
255
If b is a unit vectorforwhich AL(C,b) < xo,thereare a finiteset of values
< t2A(Cb) < t&+ 1,forwhichb* (t) has a maximumor minimum.
< t2 < .
About each point X(t-)constructa cylinderZ-1(0) with generatorsparallel to
b whichintersectsC in exactlytwo pointsX(Q) and i(tt) such that both lie on a
base of the cylinderand such that ((t+) is the centerof this base. It will firstbe
shown that there is an isotopy of the closed n-cell bounded by Zi'-(0) onto
itselfwhich leaves Z '(0) fixed and transformsthe curve segment X(t) for
C < t < t+ onto the polygonalline X(t), X(ti),X(tt).
Each hyperplaneH'-1 perpendicularto b whichintersectsZ,-1(O) intersects
it in a sphereS,-2. Performthe isotopyofeach H'-' onto itselfwhichtransforms
the curve segmentX(t)forti _ t <tt into the axis of the cylinder,and which
leaves all pointsoutside of the n-cellbounded by the cylinderfixed,as defined
by 4.1. Select a continuoussequence of coaxial cylindersZ"'(v), 0 ? v < xc,
such that any Zl'G(3) is containedwithinall Z'-1(v) withv < v, such that each
cylinderintersectsC onlyin the centerof one base and in one otherpointofthat
base, and such that Z-'(v) tends to the point x(ti) as v -* oo. Rotate each
into
Z -l(v) about its axis so that each point &(t)forC ? t < ti is transformed
the plane determinedby g(t7) and the axis ofthe cylinders.Since we have transformedg(t) fortU ? t < tt onto a plane curve withinZn-1(0), it is certainly
possible to transformit onto the polygonalline (t), g(ti), g(tj), still within
Z-1 (0), producingan equivalentcurveC describedby f(t). This curve is divided
into 4,g(C,b) = 4g(C, b) distinctsegmentsby the points i(t+) and g(t7). If
g > 0 is the g.l.b. of the distancesbetweendistinctand nonconsecutivecurve
Zn-1
we may
segments,thenforeach point of C whichis not withinany of the Zi
a
in
lies
hyperplane
constructthe sphereSn 2 whichhas its centerat the point,
perpendicularto b, and has radius g/3. Since no two of these spherescan interconstructeda tube around each curvesegmentoutsidethe
sect,we have in effect,
It is now possibleto inscribea polygcylinders,withno two tubes intersecting.
onal line,lyingcompletelywithinthe tube, in each segmentof C. Performthe
each ofthese
isotopyofeach H'-1 perpendicularto b ontoitselfwhichtransforms
leaves fixed
which
and
line
curve segmentsonto the correspondingpolygonal
and
C, into
therefore
thus
have
transformed
C,
of
the
Sn2. CWte
all pointsoutside
that
reference
for
future
(Note
isotopy.
by
an inscribedpolygonP, equivalent
t1
(C, b) = A (P, b).)
4.3 COROLLARY. The necessary and sufficientcondition that a simple curve type
(S be tameis that4(S) < oo
4.4 COROLLARY.The total curvatureof a tame knot cannot equal the curvatureof
its type.
Assumethat C is a tame knotoftype (EwithK(C) = K(Q,). Let P be a polygon
of type S inscribedin C. Then K(P) < K(C). Since P cannotlie in any plane, we
may selectfourconsecutivenon-coplanarvertices(ifwe ignoreverticesforwhich
ai= 0). By 1.2 we may selecta new polygonP, stilla memberof A, and having
K(C) = K(G); which is impossible.
4.5 COROLLARY.The crookednessof any knotis greaterthan or equal to 2.
K(P) < K(P) <
If C is a curvewith,u(C) = 1, then,in the proofof 4.2 we can select the two
256
J. W. MILNOR
cylinderswith a commonbase. The firstisotopywill then transformC into a
plane quadrilateral,whichis certainlyunknotted.
4.6 COROLLARY. The totalcurvature
ofany knotis greaterthan4%r.
4.7. THEOREM. If ( is a simplecurvetype,thenK( A) = 2iry(C).
It has already been shownthat K(C) > 2iryu(C)
forany C e G and therefore
that K(G) > 2ry(G). If A(G) = co, this provesthe proposition.If A(G) < co,
we may select a curve C of G and a directionb such that ,u(C, b) = Au(G). By
4.2 thereis a polygonP whichis a memberof G such that ,A(P,b) = ,A(C,b).
For conveniencewe will select a new coordinatesystemso that b is parallel to
the xl axis. We may then definethe isotopy FU(x, X2, X3,I...,
Xn) =
,
P into an
(X1 , UX2 , UX3, ... ux*,)
for0 < u < 1. This evidentlytransforms
\
tt\V
;F:AR:AEPZ
7': AE$C
FIG. 3
equivalent polygon Pu, such that ,'(P , b) = ,u(P, b). If ai , 1 < i < m, is the
set ofverticesof P. , we may divideit intofoursubsets:
(a) verticesa'Usuch that b*a,_I < b*ai < b*as+X
(b) vertices ai' such that b*ait 1 > b *aS' > b *a
(c) verticesai' such that braid < b a~' > b-a4X
(d) verticesas such that b*a!' > b*at < b*a'+.
(If an equalitywereto hold,we wouldhave Au(P., b) = co.) Evidentlythenumber
of verticesin (c) equals the numberin (d) equals Au( ). Howeverformembersof
(c) and (d), lim~o (a') = ir; whereasfor(a) and (b), limb~o(a') = 0. Therefore
limb-o K(Pu) = 2rp( C).
As anotherinterestingconsequence of Theorem 4.2, we have the following.
4.8 THEOREM. Givena knotC in H3 forwhichji(C) < oo,thereis a plane whose
withC consistsof at leastsix components.
intersection
Since every such C has an inscribedpolygonwhich is knotted,and since a
plane intersectsa curve at least as many times as it intersectsan inscribed
polygon,it onlyremainsto provethe theoremforknottedpolygons.If thereis a
polygonwhichdoes not satisfythetheorem,theremustbe one havinga minimum
numberof sides. If P is such a polygon,we have: 4r <
K(P)
= 2
Aj(P, b)dS.
Thereforethere must be some unit vector b such that 2 < ji(P, b) and also
ON THE
TOTAL
CURVATURE
OF KNOTS
257
b a3 # ba*k foreverypair of distinctverticesaj and ak of P. If we selecta plane
perpendicularto b and move it parallel to itselfin the directionof b until it
intersectsP, it must firstintersectP in a minimum(i.e. an aj such that b aj <
b -a.,). Afterthis it will intersectP in two points,until it intersectsanother
minimum,afterwhichit will have fourintersections.If it next reachesanother
minimum,the theoremis proved.If it next reachesa maximum,therewill then
be only two intersections.
Jointhese two pointsby a line segment,so that two
new polygonsare formedby this segmentand the sides of P. (See Fig. 3.) At
least one of these new polygons,P1 , must be knotted.Since /u(P) > 3, each of
thenewpolygonsP, and P2 musthave at least fivesides.Since P2 has fiveor more
sides,P1 musthave fewersides than P; and thereforetheremust be some plane
intersectingP, in six or more components.It is clear that this plane must
intersectthe originalpolygonP itselfin six or more components.
BIBLIOGRAPHY
[1] BORSUK, K., Sur la courburetotaledes courbesfermtes.Annales de la Soc. Polonaise 20
(1947),251-265.
Math. Ann.
Raumkurven.
[2] FENCHEL, W., Uber Krummungund Windunggeschlossener
101 (1929),238-252.
[3] BLASCHKE, W., Vorlesungen uber Integralgeometrie, zweites heft.HamburgerMathe-
matische Einselschriften, 22 Heft, 1937.
[4] SAKS, S., Theory of the Integral (2ndrevised edition). Warsaw 1937.
space. Ann.
[5] Fox, R. H. and ARTIN, E., Some WildCells and Spheresin three-dimensional
of Math. Vol. 49 (1948), 979-990.
Bulletinde
[6] FXRY, M. I., Sur la CourbureTotaled'une CourbeGaucheFaisant un Nweud.
la Soc. Math. de France LXXVII (1949), 128-138.