arxiv:1101.2305v1 [math.dg] 12 jan 2011

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011 TOTAL CURVATURE OF GRAPHS AFTER MILNOR AND EULER

ROBERT GULLIVER AND SUMIO YAMADA

Abstract. We define a new notion of total curvature, callednet total curvature,for finitegraphs embedded inRn, and investigate its properties. Two guiding principles are givenby Milnor’s way of measuring the local crookedness of a Jordan curve via a Crofton-typeformula, and by considering the double cover of a given graphas an Eulerian circuit. Thestrength of combining these ideas in defining the curvature functional is (1) it allows us tointerpret the singular/non-eulidean behavior at the vertices of the graph as a superpositionof vertices of a 1-dimensional manifold, and thus (2) one cancompute the total curva-ture for a wide range of graphs by contrasting local and global properties of the graphutilizing the integral geometric representation of the curvature. A collection of results onupper/lower bounds of the total curvature on isotopy/homeomorphism classes of embed-dings is presented, which in turn demonstrates the effectiveness of net total curvature as anew functional

measuring complexity of spatial graphs in differential-geometric terms.

1. INTRODUCTION: CURVATURE OF A GRAPH

The celebrated Fary-Milnor theorem states that a curve inRn of total curvature

at most 4π is unknotted.As a key step in his 1950 proof, John Milnor showed that for a smooth Jordan

curveΓ in R3, the total curvature equals half the integral overe ∈ S2 of the numberµ(e) of local maxima of the linear “height” function〈e, ·〉 alongΓ [M]. This equalitycan be regarded as a Crofton-type representation formula oftotal curvature wherethe order of integrations over the curve and the unit tangentsphere (the space ofdirections) are reversed. The Fary-Milnor theorem follows, since total curvatureless than 4π implies there is a unit vectore0 ∈ S2 so that〈e0, ·〉 has a unique localmaximum, and therefore that this linear function is increasing on an interval ofΓand decreasing on the complement. Without changing the pointwise value of this“height” function,Γ can be topologically untwisted to a standard embedding ofS1

into R3. The Fenchel theorem, that any curve inR3 has total curvature at least 2π,also follows from Milnor’s key step, since for alle ∈ S2, the linear function〈e, ·〉assumes its maximum somewhere alongΓ, implying µ(e) ≥ 1. Milnor’s proof isindependent of the proof of Istvan Fary, published earlier, which takes a differentapproach [Fa].

We would like to extend the methods of Milnor’s seminal paper, replacing thesimple closed curve by a finitegraph Γ in R3. Γ consists of a finite number ofpoints, thevertices, and a finite number of simple arcs, theedges, each of whichhas as its endpoints one or two of the vertices. We shall assume Γ is connected.

Date: December 31, 2010.Supported in part by JSPS Grant-in-aid for Scientific Research No.17740030.Thanks to the Korea Institute for Advanced Study for invitations.

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http://arxiv.org/abs/1101.2305v1

2 ROBERT GULLIVER AND SUMIO YAMADA

Thedegreeof a vertexq is the numberd(q) of edges which haveq as an endpoint.(Another word for degree is “valence”.) We remark that it is technically not neededthat the dimensionn of the ambient space equals three. All the arguments can begeneralized to higher dimensions, although in higher dimensions (n ≥ 4) there areno nontrivial knots. Moreover, any two homeomorphic graphsare isotopic.

The key idea in generalizing total curvature for knots to total curvature forgraphs is to consider the Euler circuits of the given graph, namely, parameteri-zations byS1, of the doublecover of the graph. We note that given a graph ofeven degree, there can be several Euler circuits, or ways to “trace it without lift-ing the pen.” A topological vertex of a graph of degreed is a singularity, in thatthe graph is not locally Euclidean. However by considering an Euler circuit of thedouble of the graph, the vertex becomes locally the intersection point of d paths.We will show (Corollary 2) that at the vertex, each path through it has a (signed)measure-valued curvature, and the absolute value of the sumof those measures iswell-defined, independent of the choice of the Euler circuitof the double cover.We define (Definition 1) thenet total curvature(NTC) of a piecewiseC2 graph tobe the sum of the total curvature of the smooth arcs and the contributions from thevertices as described.

This notion of net total curvature is substantially different from the total curva-ture, denoted TC, as defined by Taniyama [T]. (Taniyama writes τ for TC.) Seesection 2 below.

This is consistent with known results for the vertices of degree d = 2; withvertices of degree three or more, this definition helps facilitate a new Crofton-type representation formula (Theorem 1) for total curvature of graphs, where thetotal curvature is represented as an integral over the unit sphere. Recall that thevertex is now seen asd distinct points on an Euler circuit. The way we pick upthe contribution of the total curvature at the vertices identifies thed distinct points,and thus the 2d unit tangent spheres on a circuit. As Crofton’s formula in effectreverses the order of integrations — one over the circuit, the other over the space oftangent directions — the sum of thed exterior angles at the vertex is incorporatedin the integral over the unit sphere. On the other hand the integrand of the integralover the unit sphere counts the number of net local maxima of the height functionalong an axis, where net local maximum means the number of local maxima minusthe number of local minima at thesed points of the Euler circuit. This establishesa correspondence between the differential geometric quantity (net total curvature)and the differential topological quantity (average number of maxima) of the graph,as stated in Theorem 1 below.

In section 2, we compare several definitions for total curvature of graphs whichhave appeared in the recent literature. In section 3, we introduce the main tool(Lemma 1) which in a sense reduces the computation of NTC to counting intersec-tions with planes.

Milnor’s treatment [M] of total curvature also contained animportant topolog-ical extension. Namely, in order to define total curvature, the knot needs only tobecontinuous. This makes the total curvature a geometric quantity definedon any

TOTAL CURVATURE OF GRAPHS AFTER MILNOR AND EULER 3

homeomorphic image ofS1. In this article, we first define net total curvature (Def-inition 1) on piecewiseC2 graphs, and then extend the definition to continuousgraphs (Definition 3.) In analogy to Milnor, we approximate agiven continuousgraph by a sequence of polygonal graphs. In showing the monotonicity of the totalcurvature (Proposition 2) under the refining process of approximating graphs weuse our representation formula (Theorem 1) applied to the polygonal graphs.

Consequently the Crofton-type representation formula is also extended (Theo-rem 2) to cover continuous graphs. Additionally, we are ableto show that continu-ous graphs with finite total curvature (NTC or TC) are tame. Wesay that a graphis tamewhen it is isotopic to an embedded polyhedral graph.

In sections 5 through 8, we characterize NTC with respect to the geometry andthe topology of the graph. Proposition 5 shows the subadditivity of NTC underthe union of graphs which meet in a finite set. In section 6, theconcept of bridgenumber is extended from knots to graphs, in terms of which theminimum of NTCcan be explicitly computed, provided the graph has at most one vertex of degree> 3. In section 7, Theorem 6 gives a lower bound for NTC in terms of the widthof an isotopy class. The infimum of NTC is computed for specificgraph types: thetwo-vertex graphsθm, the “ladder”Lm, the “wheel”Wm, the complete graphKm onmvertices and the complete bipartite graphKm,n.

Finally we prove a result (Theorem 7) which gives a Fenchel type lower bound(≥ 3π) for total curvature of a theta graph (an image of the graph consisting of acircle with an arc connecting a pair of antipodal points), and a Fary-Milnor typeupper bound (< 4π) to imply the theta graph is isotopic to the standard embedding.A similar result was given by Taniyama [T], referring to TC. In contrast, for graphsof the type ofKm (m ≥ 4), the infimum of NTC in the isotopy class of a polygononm vertices is also the infimum for a sequence of distinct isotopy classes.

Many of the results in our earlier preprint [GY2] have been incorporated intothe present paper.

We thank Yuya Koda for his comments regarding Proposition 7,and JaigyoungChoe and Rob Kusner for their comments about Theorem 7, especially about thesharp case NTC(Γ) = 3π of the lower bound estimate.

2. DEFINITIONS OF TOTAL CURVATURE

The first difficulty, in extending the results of Milnor’s classic paper, is to un-derstand the contribution to total curvature at a vertex of degreed(q) ≥ 3. We firstconsider the well-known case:

Definition of Total Curvature for Knots

For a smooth closed curveΓ, the total curvature is

C(Γ) =∫

Γ

|~k|ds,

where s denotes arc length alongΓ and~k is the curvature vector. Ifx(s) ∈ R3

denotes the position of the point measured at arc lengths along the curve, then


~k = d2xds2 . For a piecewise smooth curve, that is, a graph with verticesq1, . . . , qN

having always degreed(qi ) = 2, the total curvature is readily generalized to

(1) C(Γ) =N∑

i=1

c(qi) +∫

Γreg

|~k|ds,

where the integral is taken over the separateC2 edges ofΓ without their endpoints;and where c(qi ) ∈ [0, π] is the exterior angle formed by the two edges ofΓ whichmeet atqi . That is, cos(c(qi )) = 〈T1,−T2〉, whereT1 =

dxds(q

+i ) andT2 = −

dxds(q

−i )

are the unit tangent vectors atqi pointing into the two edges which meet atqi . Theexterior angle c(qi ) is the correct contribution to total curvature, since any sequenceof smooth curves converging toΓ in C0, with C1 convergence on compact subsetsof each open edge, includes a small arc nearqi along which the tangent vectorchanges from neardx

ds(q−i ) to neardx

ds(q+i ). The greatest lower bound of the contri-

bution to total curvature of this disappearing arc along thesmooth approximatingcurves equals c(qi ).

Note thatC(Γ) is well defined for animmersedknotΓ.

Definitions of Total Curvature for Graphs

When we turn our attention to agraphΓ, we find the above definition for curves(degreed(q) = 2) does not generalize in any obvious way to higher degree (see[G]). The ambiguity of the general formula (1) is resolved ifwe specify the re-placement for c(0) whenΓ is the cone over a finite set{T1, . . . ,Td} in the unitsphereS2.

The earliest notion of total curvature of a graph appears in the context of thefirst variation of length of a graph, which we callvariational total curvature ,and is called themean curvatureof the graph in [AA]: we shall write VTC. Thecontribution to VTC at a vertexq of degree 2, with unit tangent vectorsT1 andT2,is vtc(q) = |T1 + T2| = 2 sin(c(q)/2). At a non-straight vertexq of degree 2, vtc(q)is less than the exterior angle c(q). For a vertex of degreed, the contribution isvtc(q) = |T1 + · · · + Td|.

A rather natural definition of total curvature of graphs was given by Taniyama in[T]. We have called thismaximal total curvature TC(Γ) in [G]. The contributionto total curvature at a vertexq of degreed is

tc(q) :=∑

1≤i< j≤d

arccos〈Ti ,−T j〉.

In the cased(q) = 2, the sum above has only one term, the exterior angle c(q) atq. Since the length of the Gauss image of a curve inS2 is the total curvature of thecurve, tc(q) may be interpreted as adding to the Gauss image inRP2 of the edges,a complete great-circle graph onT1(q), . . . ,Td(q), for each vertexq of degreed.Note that the edge between two vertices does not measure the distance inRP2 butits supplement.


In our earlier paper [GY1] on the density of an area-minimizing two-dimensionalrectifiable setΣ spanningΓ, we found that it was very useful to apply the Gauss-Bonnet formula to the cone overΓ with a point p of Σ as vertex. The relevantnotion of total curvature in that context iscone total curvature CTC(Γ), definedusing ctc(q) as the replacement for c(q) in equation (1):

(2) ctc(q) := supe∈S2

d∑

i=1

(π

2− arccos〈Ti , e〉

) .

Note that in the cased(q) = 2, the supremum above is assumed at vectorse lyingin the smaller angle between the tangent vectorsT1 and T2 to Γ, so that ctc(q)is then the exterior angle c(q) at q. The main result of [GY1] is that 2π timesthe area density ofΣ at any of its points is at most equal to CTC(Γ). The sameresult had been proven by Eckholm, White and Wienholtz for the case of a simpleclosed curve [EWW]. TakingΣ to be the branched immersion of the disk givenby Douglas [D1] and Rado [R], it follows that ifC(Γ) ≤ 4π, thenΣ is embedded,and thereforeΓ is unknotted. Thus [EWW] provided an independent proof of theFary-Milnor theorem. However, CTC(Γ) may be small for graphs which are farfrom the simplest isotopy types of a graphΓ.

In this paper, we introduce the notion ofnet total curvature NTC(Γ), which isthe appropriate definition for generalizing —to graphs— Milnor’s approach toisotopy and total curvature ofcurves. For each unit tangent vectorTi at q, 1 ≤ i ≤d = d(q), let χi : S2 → {−1,+1} be equal to−1 on the hemisphere with center atTi, and+1 on the opposite hemisphere (modulo sets of zero Lebesgue measure).We then define

(3) ntc(q) :=14

∫

S2

d∑

i=1

χi(e)

+

dAS2(e).

We note that the function∑d

i=1 χi(e) is odd, hence the quantity above can be writtenas

ntc(q) :=18

∫

S2

∣∣∣∣∣∣∣

d∑

i=1

χi(e)

∣∣∣∣∣∣∣dAS2(e).

as well. In the cased(q) = 2, the integrand of (3) is positive (and equals 2) onlyon the set of unit vectorse which have negative inner products with bothT1 andT2, ignoring e in sets of measure zero. This set is bounded by semi-great circlesorthogonal toT1 and toT2, and has spherical area equal to twice the exterior angle.So in this case, ntc(q) is the exterior angle. Thus, in the special case whereΓis a piecewise smooth curve, the following quantity NTC(Γ) coincides with totalcurvature, as well as with TC(Γ) and CTC(Γ):


Definition 1. We define thenet total curvatureof a piecewise C2 graph Γ withvertices{q1, . . . , qN} as

(4) NTC(Γ) :=N∑

i=1

ntc(qi ) +∫

Γreg

|~k|ds.

For the sake of simplicity, elsewhere in this paper, we consider the ambient spaceto beR3. However the definition of the net total curvature can be generalized for agraph inRn by defining the vertex contribution in terms of an average over Sn−1:

ntc(q) := π(?

Sn−1

d∑

i=1

χi(e)

+

dASn−1(e)),

which is consistent with the definition (3) of ntc whenn = 3.Recall that Milnor [M] defines the total curvature of a continuous simple closed

curveC as the supremum of the total curvature of all polygons inscribed inC. Byanalogy, we define net total curvature of acontinuousgraphΓ to be the supre-mum of the net total curvature of all polygonal graphsP suitably inscribed inΓ asfollows.

Definition 2. For a given continuous graphΓ, we say a polygonal graph P⊂ R3

is Γ-approximating, provided that its topological vertices (those of degree, 2) areexactly the topological vertices ofΓ, and having the same degrees; and that thearcs of P between two topological vertices correspond one-to-one to the edges ofΓ between those two vertices.

Note that ifP is aΓ-approximating polygonal graph, thenP is homeomorphicto Γ. According to the statement of Proposition 2, whose proof will be given inthe next section, ifP and P are Γ-approximating polygonal graphs, andP is arefinement ofP, then NTC(P) ≥ NTC(P). Here P is said to be a refinement ofP provided the set of vertices ofP is a subset of the vertices ofP. AssumingProposition 2 for the moment, we can generalize the definition of the total curvatureto non-smooth graphs.

Definition 3. Define thenet total curvatureof a continuous graphΓ by

NTC(Γ) := supP

NTC(P)

where the supremum is taken over allΓ-approximating polygonal graphs P.

For a polygonal graphP, applying Definition 1,

NTC(P) :=N∑

i=1

ntc(qi ),

whereq1, . . . , qN are the vertices ofP.Definition 3 is consistent with Definition 1 in the case of a piecewiseC2 graph

Γ. Namely, as Milnor showed, the total curvatureC(Γ0) of a smooth curveΓ0 is thesupremum of the total curvature of inscribed polygons ([M],p. 251), which givesthe required supremum for each edge. At a vertexq of the piecewise-C2 graphΓ, as


a sequencePk of Γ-approximating polygons become arbitrarily fine, a vertexq ofPk (and ofΓ) has unit tangent vectors converging inS2 to the unit tangent vectorstoΓ atq. It follows that for 1≤ i ≤ d(q), χPk

i → χΓi in measure onS2, and thereforentcPk(q)→ ntcΓ(q).

3. CROFTON-TYPE REPRESENTATION FORMULA FOR TOTALCURVATURE

We would like to explain how the net total curvature NTC(Γ) of a graph is relatedto more familiar notions of total curvature. Recall that a graphΓ has an Euler circuitif and only if its vertices all have even degree, by a theorem of Euler. An Eulercircuit is a closed, connected path which traverses each edge ofΓ exactly once. Ofcourse, we do not have the hypothesis of even degree. We can attain that hypothesisby passing to thedoubleΓ of Γ: Γ is the graph with the same vertices asΓ, but withtwo copies of each edge ofΓ. Then at each vertexq, the degree as a vertex ofΓis d(q) = 2d(q), which is even. By Euler’s theorem, there is an Euler circuit Γ′

of Γ, which may be thought of as a closed path which traverses eachedge ofΓexactlytwice. Now at each of the points{q1, . . . , qd} alongΓ′ which are mapped toq ∈ Γ, we may consider the exterior angle c(qi ). The sum of these exterior angles,however, depends on the choice of the Euler circuitΓ′. For example, ifΓ is theunion of thex-axis and they-axis in Euclidean spaceR3, then one might chooseΓ′

to have four right angles, or to have four straight angles, orsomething in between,with completely different values of total curvature. In order to form a version oftotal curvature at a vertexq which only depends on the original graphΓ and not onthe choice of Euler circuitΓ′, it is necessary to consider some of the exterior anglesas partially balancing others. In the example just considered, whereΓ is the unionof two orthogonal lines, two opposing right angles will be considered to balanceeach other completely, so that ntc(q) = 0, regardless of the choice of Euler circuitof the double.

It will become apparent that the connected character of an Euler circuit of Γ isnot required for what follows. Instead, we shall refer to aparameterizationΓ′ of thedoubleΓ, which is a mapping from a 1-dimensional manifold without boundary,not necessarily connected; the mapping is assumed to cover each edge ofΓ once.

The nature of ntc(q) is clearer when it is localized onS2, analogously to [M].In the cased(q) = 2, Milnor observed that the exterior angle at the vertexq equalshalf the area of thosee∈ S2 such that the linear function〈e, ·〉, restricted toΓ, has alocal maximum atq. In our context, we may describe ntc(q) as one-half the integralover the sphere of the number ofnet local maxima, which is half the difference oflocal maxima and local minima. Along the parameterizationΓ′ of the double ofΓ, the linear function〈e, ·〉 may have a local maximum at some of the verticesq1, . . . , qd over q, and may have a local minimum at others. In our construction,each local minimum balances against one local maximum. If there are more localminima than local maxima, the number nlm(e, q), the net number of local maxima,will be negative; however, our definition uses only the positive part [nlm(e, q)]+.


We need to show that ∫

S2[nlm(e, q)]+ dAS2(e)

is independent of the choice of parameterization, and in fact is equal to 2 ntc(q);this will follow from another way of computing nlm(e, q) (see Corollary 2 below).

Definition 4. Let a parameterizationΓ′ of the double ofΓ be given. Then a vertexq of Γ corresponds to a number of vertices q1, . . . , qd of Γ′, where d is the degreed(q) of q as a vertex ofΓ. Choose e∈ S2. If q ∈ Γ is a local extremum of〈e, ·〉, thenwe consider q as a vertex of degree d(q) = 2. Let lmax(e, q) be the number of localmaxima of〈e, ·〉 along Γ′ at the points q1, . . . , qd over q, and similarly letlmin(e, q)be the number of local minima. We define the number ofnet local maximaof 〈e, ·〉at q to be

nlm(e, q) =12

[lmax(e, q) − lmin(e, q)]

.

Remark 1. The definition ofnlm(e, q) appears to depend not only onΓ but on achoice of the parameterizationΓ′ of the double ofΓ: lmax(e, q) and lmin(e, q) maydepend on the choice ofΓ′. However, we shall see in Corollary 1 below that thenumber ofnet local maximanlm(e, q) is in fact independent ofΓ′.

Remark 2. We have included the factor12 in the definition ofnlm(e, q) in orderto agree with the difference of the numbers of local maxima and minima along aparameterization ofΓ itself, if d(q) is even.

We shallassumefor the rest of this section that a unit vectorehas been chosen,and that the linear “height” function〈e, ·〉 has only a finite number of critical pointsalongΓ; this excludese belonging to a subset ofS2 of measure zero. We shallalso assume that the graphΓ is subdivided to include among the vertices all criticalpoints of the linear function〈e, ·〉, with degreed(q) = 2 if q is an interior point ofone of the topological edges ofΓ.

Definition 5. Choose a unit vector e. At a point q∈ Γ of degree d= d(q), let theup-degreed+ = d+(e, q) be the number of edges ofΓwith endpoint q on which〈e, ·〉is greater (“higher”) than〈e, q〉, the “height” of q. Similarly, let thedown-degreed−(e, q) be the number of edges along which〈e, ·〉 is less than its value at q. Notethat d(q) = d+(e, q) + d−(e, q), for almost all e in S2.

Lemma 1. (Combinatorial Lemma)For all q ∈ Γ and for a.a. e∈ S2, nlm(e, q) =12[d−(e, q) − d+(e, q)].

Proof.Let a parameterizationΓ′ of the double ofΓ be chosen, with respect to whichlmax(e, q) and lmin(e, q) are defined. Recall the assumption above, thatΓ has beensubdivided so that along each edge, the linear function〈e, ·〉 is strictly monotone.

Consider a vertexq of Γ, of degreed = d(q). ThenΓ′ has 2d edges with anendpoint among the pointsq1, . . . , qd which are mapped toq ∈ Γ. On 2d+, resp.2d− of these edges,〈e, ·〉 is greater resp. less than〈e, q〉. But for each 1≤ i ≤ d,the parameterizationΓ′ has exactly two edges which meet atqi . Depending on the


up/down character of the two edges ofΓ′ which meet atqi , 1 ≤ i ≤ d, we cancount:(+) If 〈e, ·〉 is greater than〈e, q〉 on both edges, thenqi is a local minimum point;there are lmin(e, q) of these amongq1, . . . , qd.(-) If 〈e, ·〉 is less than〈e, q〉 on both edges, thenqi is a local maximum point; thereare lmax(e, q) of these.(0) In all remaining cases, the linear function〈e, ·〉 is greater than〈e, q〉 alongone edge and less along the other, in which caseqi is not counted in computinglmax(e, q) nor lmax(e, q); there ared(q) − lmax(e, q) − lmin(e, q) of these.

Now count the individual edges ofΓ′:(+) There are lmin(e, q) pairs of edges, each of which is part of a local minimum,both of which are counted among the 2d+(e, q) edges ofΓ′ with 〈e, ·〉 greater than〈e, q〉.(-) There are lmax(e, q) pairs of edges, each of which is part of a local maximum;these are counted among the number 2d−(e, q) of edges ofΓ′ with 〈e, ·〉 less than〈e, q〉. Finally,(0) there ared(q) − lmax(e, q) − lmin(e, q) edges ofΓ′ which are not part of a localmaximum or minimum, with〈e, ·〉 greater than〈e, q〉; and an equal number of edgeswith 〈e, ·〉 less than〈e, q〉.

Thus, the total number of these edges ofΓ′ with 〈e, ·〉 greater than〈e, q〉 is

2d+ = 2 lmin+ (d − lmax− lmin) = d + lmin − lmax.

Similarly,

2d− = 2 lmax+ (d − lmax− lmin) = d + lmax− lmin.

Subtracting gives the conclusion:

nlm(e, q) :=lmax(e, q) − lmin(e, q)

2=

d−(e, q) − d+(e, q)2

.

Corollary 1. The number of net local maximanlm(e, q) is independent of thechoice of parameterizationΓ′ of the double ofΓ.

Proof. Given a directione ∈ S2, the up-degree and down-degreed±(e, q) at avertexq ∈ Γ are defined independently of the choice ofΓ′.

Corollary 2. For any q∈ Γ, we haventc(q) = 12

∫S2

[nlm(e, q)

]+dAS2.

Proof. Considere ∈ S2. In the definition (3) of ntc(q), χi(e) = ±1 whenever±〈e,Ti〉 < 0. But the number of 1≤ i ≤ d with ±〈e,Ti〉 < 0 equalsd∓(e, q), so that

d∑

i=1

χi(e) = d−(e, q) − d+(e, q) = 2 nlm(e, q)

by Lemma 1, for almost alle ∈ S2.

Definition 6. For a graphΓ in R3 and e∈ S2, define themultiplicity at e as

µ(e) = µΓ(e) =∑{nlm+(e, q) : q a vertex ofΓ or a critical point of〈e, ·〉}.


Note thatµ(e) is a half-integer. Note also that in the case whenΓ is a knot, orequivalently, whend(q) ≡ 2, µ(e) is exactly the integerµ(Γ, e), the number of localmaxima of〈e, ·〉 alongΓ as defined in [M], p. 252.

Corollary 3. For almost all e∈ S2 and for any parameterizationΓ′ of the doubleof Γ, µΓ(e) ≤ 1

2µΓ′(e).

Proof.We haveµΓ(e) = 12

∑q[lmaxΓ′(e, q) − lminΓ′(e, q)],≤ 1

2

∑q lmaxΓ′(e, q) =

12µΓ′ .

If, in place of the positive part, we sum nlm(e, q) itself overq located above aplane orthogonal toe, we find a useful quantity:

Corollary 4. For almost all s0 ∈ R and almost all e∈ S2,

2∑{nlm(e, q) : 〈e, q〉 > s0} = #(e, s0),

the cardinality of the fiber{p ∈ Γ : 〈e, p〉 = s0}.

Proof. If s0 > maxp∈Γ〈e, p〉, then #(e, s0) = 0. Now proceed downward, usingLemma 1 by induction.

Note that the fiber cardinality of Corollary 4 is also the value obtained for knots,where the more general nlm may be replaced by the number of local maxima [M].

Remark 3. In analogy with Corollary 4, we expect that an appropriate general-ization ofNTC to curved polyhedral complexes of dimension≥ 2 will in the futureallow computation of the homology of level sets and sub-level sets of a (general-ized) Morse function in terms of a generalization ofnlm(e, q).

Corollary 5. The multiplicity of a graph in direction e∈ S2 may also be computedasµ(e) = 1

2

∑q∈Γ |nlm(e, q)|.

Proof. It follows from Corollary 4 withs0 < minΓ〈e, ·〉 that∑

q∈Γ nlm(e, q) = 0,which is the difference of positive and negative parts. The sum of these partsis∑

q∈Γ |nlm(e, q)| = 2µ(e).

It was shown in Theorem 3.1 of [M] that, in the case of knots,C(Γ) = 12

∫S2 µ(e) dAS2,

where Milnor refers to Crofton’s formula. We may now extend this result tographs:

Theorem 1. For a (piecewise C2) graphΓmapped intoR3, the net total curvaturehas the following representation:

NTC(Γ) =12

∫

S2µ(e) dAS2(e).

Proof. We have NTC(Γ) =∑N

j=1 ntc(q j ) +∫Γreg|~k|ds, where q1, . . . , qN are

the vertices of Γ, including local extrema as vertices of degreed(q j ) = 2, and

where ntc(q) := 14

∫S2

[∑di=1 χi(e)

]+dAS2(e) by the definition (3) of ntc(q). Apply-

ing Milnor’s result to eachC2 edge, we haveC(Γreg) = 12

∫S2 µΓreg(e) dAS2. But


µΓ(e) = µΓreg(e) +∑N

j=1 nlm+(e, qj), and the theorem follows.

Corollary 6. If f : Γ→ R3 is piecewise C2 but is not an embedding, then the net to-tal curvatureNTC(Γ) is well defined, using the right-hand side of the conclusion ofTheorem 1. Moreover,NTC(Γ) has the same value when points of self-intersectionof Γ are redefined as vertices.

Fore ∈ S2, we shall use the notationpe : R3→ eR for the orthogonal projection〈e, ·〉. We shall sometimes identifyR with the one-dimensional subspaceeR of R3.

Corollary 7. For any homeomorphism type{Γ} of graphs, the infimumNTC({Γ})of net total curvature among mappings f: Γ → Rn is assumed by a mappingf0 : Γ→ R.

For any isotopy class[Γ] of embeddings f: Γ → R3, the infimumNTC([Γ]) ofnet total curvature is assumed by a mapping f0 : Γ→ R in the closure of the givenisotopy class.

Conversely, if f0 : Γ → R is in the closure of a given isotopy class[Γ] ofembeddings intoR3, then for allδ > 0 there is an embedding f: Γ → R3 in thatisotopy class withNTC( f ) ≤ NTC( f0) + δ.

Proof. Let f : Γ → R3 be any piecewise smooth mapping. By Corollary 6 andCorollary 4, the net total curvature of the projectionpe ◦ f : Γ → R of f onto theline in the direction of almost anye ∈ S2 is given by 2πµ(e) = π(µ(e) + µ(−e)). Itfollows from Theorem 1 that NTC(Γ) is the average of 2πµ(e) overe in S2. But thehalf-integer-valued functionµ(e) is lower semi-continuous almost everywhere, asmay be seen using Definition 4. Lete0 ∈ S2 be a point whereµ attains its essentialinfimum. Then NTC(Γ) ≥ πµ(e0) = NTC(pe0 ◦ f ). But (pe0 ◦ f )e0 is the limitasε → 0 of the mapfε whose projection in the directione0 is the same as thatof f and is multiplied byε in all orthogonal directions. Sincefε is isotopic to f ,(pe0 ◦ f )e0 is in the closure of the isotopy class off .

Conversely, givenf0 : Γ→ R in the closure of a given isotopy class, letf be anembedding in that isotopy class uniformly close tof0 e0; fε as constructed aboveconverges uniformly tof0 asε→ 0, and NTC(fε)→ NTC( f0).

Definition 7. We call a mapping f: Γ → Rn flat (or NTC-flat) if NTC( f ) =NTC({Γ}), the minimum value for the topological type ofΓ, among all ambientdimensions n.

In particular, Corollary 7 above shows that for anyΓ, there is a flat mappingf : Γ→ R.

Proposition 1. Consider a piecewise C2 mapping f1 : Γ→ R. There is a mappingf0 : Γ → R which is monotonic along the topological edges ofΓ, has valuesat topological vertices ofΓ arbitrarily close to those of f1, and hasNTC( f0) ≤NTC( f1).

Proof. Any piecewiseC2 mapping f1 : Γ → R may be approximated uniformlyby mappings with a finite set of local extreme points, using the compactness of


Γ. Thus, we may assume without loss of generality thatf1 has only finitely manylocal extreme points. Note that for a mappingf : Γ→ R = Re, NTC(f ) = 2πµ(e):hence, we only need to compareµ f0(e) with µ f1(e).

If f1 is not monotonic on a topological edgeE, then it has a local extremum at apointz in the interior ofE. For concreteness, we shall assumez is a local maximumpoint; the case of a local minimum is similar. Writev,w for the endpoints ofE.Let v1 be the closest local minimum point toz on the interval ofE from z to v(or v1 = v if there is no local minimum point between), and letw1 be the closestlocal minimum point toz on the interval fromz to w (or w1 = w). Let E1 ⊂ Edenote the interval betweenv1 andw1. ThenE1 is an interval of a topological edgeof Γ, having end pointsv1 andw1 and containing an interior pointz, such thatf1is monotone increasing on the interval fromv1 to z, and monotone decreasing onthe interval fromz to w1. By switchingv1 andw1 if needed, we may assume thatf1(v1) < f1(w1) < f1(z).

Let f0 be equal tof1 except on the interior of the intervalE1, and mapE1 mono-tonically to the interval ofR betweenf1(v1) and f1(w1). Then for f1(w1) < s <f1(z), the cardinality #(e, s) f0 = #(e, s) f1 − 2. Fors in all other intervals ofR, thiscardinality is unchanged. Therefore, nlmf1(w1) = nlm f0(w1) − 1, by Lemma 1.This implies that nlm+f1(w1) ≥ nlm+f0(w1) − 1. Meanwhile, nlmf1(z) = 1, a termwhich does not appear in the formula forµ f0 (see Definition 6).Thusµ f0 ≤ µ f1, andNTC( f0) ≤ NTC( f1).

Proceeding inductively, we remove each local extremum in the interior of anyedge ofΓ, without increasing NTC.

4. REPRESENTATION FORMULA FOR NOWHERE-SMOOTH GRAPHS

Recall, while defining the total curvature for continuous graphs in section 2above, we needed the monotonicity of NTC(P) under refinement ofpolygonalgraphsP. We are now ready to prove this.

Proposition 2. Let P andP be polygonal graphs inR3, having the same topo-logical vertices, and homeomorphic to each other. Suppose that every vertex ofP is also a vertex ofP: P is a refinementof P. Then for almost all e∈ S2, themultiplicity µP(e) ≥ µP(e). As a consequence,NTC(P) ≥ NTC(P).

Proof. We may assume, as an induction step, thatP is obtained fromP byreplacing the edge having endpointsq0, q2 with two edges, one having endpointsq0, q1 and the other having endpointsq1, q2. Choosee ∈ S2. We consider variouscases:

If the new vertexq1 satisfies〈e, q0〉 < 〈e, q1〉 < 〈e, q2〉, then nlmP(e, qi ) =nlmP(e, qi ) for i = 0, 2 and nlmP(e, q1) = 0, henceµP(e) = µP(e).

If 〈e, q0〉 < 〈e, q2〉 < 〈e, q1〉, then nlmP(e, q0) = nlmP(e, q0) and nlmP(e, q1) = 1.The vertexq2 requires more careful counting: the up- and down-degreed±

P(e, q2) =

d±P(e, q2) ± 1, so that by Lemma 1, nlmP(e, q2) = nlmP(e, q2) − 1. Meanwhile, foreach of the polygonal graphs,µ(e) is the sum overq of nlm+(e, q), so the change


from µP(e) to µP(e) depends on the value of nlmP(e, q2):(a) if nlmP(e, q2) ≤ 0, then nlm+

P(e, q2) = nlm+P(e, q2) = 0;

(b) if nlmP(e, q2) = 12 , then nlm+

P(e, q2) = nlm+P(e, q2) − 1

2;(c) if nlmP(e, q2) ≥ 1, then nlm+

P(e, q2) = nlm+P(e, q2) − 1.

Since the new vertexq1 does not appear inP, recalling that nlmP(e, q1) = 1, wehaveµP(e)− µP(e) = +1,+1

2 or 0 in the respective cases (a), (b) or (c). In any case,µP(e) ≥ µP(e).

The reverse inequality〈e, q1〉 < 〈e, q2〉 < 〈e, q0〉 may be reduced to the casejust above by replacinge ∈ S2 with −e, sinceµP(−e) = −µP(e) for any polhedralgraphP. Then, depending whether nlmP(e, q2) is ≤ −1, = −1

2 or ≥ 0, we find thatµP(e) − µP(e) = nlm+

P(e, q2) − nlm+P(e, q2) = 0, 1

2, or 1. In any case,µP(e) ≥ µP(e).These arguments are unchanged ifq0 is switched withq2. This covers all cases

except those in which equality occurs between〈e, qi〉 and 〈e, q j〉 (i , j). Theset of such unit vectorse form a set of measure zero inS2. The conclusionNTC(P) ≥ NTC(P) now follows from Theorem 1.

We remark here that this step of proving the monotonicity forthe nowhere-smooth case differs from Milnor’s argument for the knot total curvature, where itwas shown by two applications of the triangle inequality forspherical triangles.

Milnor extended his results for piecewise smooth knots to continuous knots in[M]; we shall carry out an analogous extension to continuousgraphs.

Definition 8. We say a point q∈ Γ is critical relative to e ∈ S2 when q is atopological vertex ofΓ or when〈e, ·〉 is not monotone in any open interval ofΓcontaining q.

Note that at some points of a differentiable curve,〈e, ·〉may have derivative zerobut still not be considered a critical point relative toe by our definition. This isappropriate to theC0 category. For a continuous graphΓ, when NTC(Γ) is finite,we shall show that the number of critical points is finite for almost alle in S2 (seeLemma 4 below).

Lemma 2. Let Γ be a continuous, finite graph inR3, and choose a sequencePk

of Γ-approximating polygonal graphs withNTC(Γ) = limk→∞ NTC(Pk). Then foreach e∈ S2, there is a refinement Pk of Pk such thatlimk→∞ µPk(e) exists in[0,∞].

Proof.First, for eachk in sequence, we refinePk to include all vertices ofPk−1.Then for alle ∈ S2, µPk

(e) ≥ µPk−1(e), by Proposition 2. Second, we refinePk

so that the arc ofΓ corresponding to each edge ofPk has diameter≤ 1/k. Third,given a particulare ∈ S2, for each edgeEk of Pk, we add 0, 1 or 2 points fromΓas vertices ofPk so that maxEk

〈e, ·〉 = maxE〈e, ·〉 whereE is the closed arc ofΓ

corresponding toEk; and similarly so that minEk〈e, ·〉 = minE〈e, ·〉. Write Pk for

the result of this three-step refinement. Note that all vertices ofPk−1 appear amongthe vertices ofPk. Then by Proposition 2,

NTC(Pk) ≤ NTC(Pk) ≤ NTC(Γ),


so we still have NTC(Γ) = limk→∞NTC(Pk).Now compare the values ofµPk(e) =

∑q∈Pk

nlmPk+(e, q) with the same sum for

Pk−1. SincePk is a refinement ofPk−1, we haveµPk(e) ≥ µPk−1(e) by Proposition 2.Therefore the valuesµPk(e) are non-decreasing ink, which implies they are ei-

ther convergent or properly divergent; in the latter case wewrite limk→∞ µPk(e) =∞.

Definition 9. For a continuous graphΓ, define themultiplicity at e ∈ S2 asµΓ(e) := limk→∞ µPk(e) ∈ [0,∞], where Pk is a sequence ofΓ-approximatingpolygonal graphs, refined with respect to e, as given in Lemma2.

Remark 4. Note that any twoΓ-approximating polygonal graphs have a commonrefinement. Hence, from the proof of Lemma 2, any two choices of sequences{Pk}

of Γ-approximating polygonal graphs lead to the same valueµΓ(e).

Lemma 3. Let Γ be a continuous, finite graph inR3. ThenµΓ : S2 → [0,∞]takes its values in the half-integers, or+∞. Now assumeNTC(Γ) < ∞. ThenµΓ isintegrable, hence finite almost everywhere on S2, and

(5) NTC(Γ) =12

∫

S2µΓ(e) dAS2(e).

For almost all e∈ S2, a sequence Pk of Γ-approximating polygonal graphs, con-verging uniformly toΓ, may be chosen (depending on e) so that each local extremepoint q of〈e, ·〉 alongΓ occurs as a vertex of Pk for sufficiently large k.

Proof. Given e ∈ S2, let {Pk} be the sequence ofΓ-approximating polygonalgraphs from Lemma 2. IfµΓ(e) is finite, thenµPk(e) = µΓ(e) for k sufficiently large,a half-integer.

Suppose NTC(Γ) < ∞. Then the half-integer-valued functionsµPk are non-negative, integrable onS2 with bounded integrals since NTC(Pk) < NTC(Γ) < ∞,and monotone increasing ink. Thus for almost alle ∈ S2, µPk(e) = µΓ(e) for ksufficiently large.

Since the functionsµPk are non-negative and pointwise non-decreasing almosteverywhere onS2, it now follows from the Monotone Convergence Theorem that

∫

S2µΓ(e) dAS2(e) = lim

k→∞

∫

S2µPk(e) dAS2(e) = 2NTC(Γ).

Finally, the polygonal graphsPk have maximum edge length→ 0. For almost alle∈ S2, 〈e, ·〉 is not constant along any open arc ofΓ, andµΓ(e) is finite. Given suchane, chooseℓ = ℓ(e) sufficiently large thatµPk(e) = µΓ(e) andµPk(−e) = µΓ(−e) forall k ≥ ℓ. Then fork ≥ ℓ, along any edgeEk of Pk with corresponding arcE of Γ,the maximum and minimum values of〈e, ·〉 alongE occur at the endpoints, whichare also the endpoints ofEk. Otherwise, asPk is further refined, new interior localmaximum resp. local minimium points ofE would contribute a new, positive valueto µPk(e) resp. toµPk(−e) ask increases. Since the diameter of the correspondingarcE of Γ tends to zero ask→ ∞, any local maximum or local minimum of〈e, ·〉must become an endpoint of some edge ofPk for k sufficiently large, and fork ≥ ℓ


in particular.

Our next lemma focuses on the regularity of a graphΓ, originally only assumedcontinuous, provided it has finite net total curvature, or another notion of totalcurvature of a graph which includes the total curvature of the edges.

Lemma 4. Let Γ be a continuous, finite graph inR3, with NTC(Γ) < ∞. ThenΓhas continuous one-sided unit tangent vectors T1(p) and T2(p) at each point p, nota topological vertex. If p is a vertex of degree d, then each ofthe d edges whichmeet at p have well-defined unit tangent vectors at p: T1(p), . . . ,Td(p). For almostall e ∈ S2,

(6) µΓ(e) =∑

q

{nlm(e, q)}+,

where the sum is over thefinite number of topological vertices ofΓ and criticalpoints q of〈e, ·〉 alongΓ. Further, for each q,nlm(e, q) = 1

2[d−(e, q)−d+(e, q)]. Allof these critical points which are not topological verticesare local extrema of〈e, ·〉alongΓ.

Proof. We have seen in the proof of Lemma 3 that for almost alle ∈ S2, thelinear function〈e, ·〉 is not constant along any open arc ofΓ, and by Lemma 2there is a sequence{Pk} of Γ-approximating polygonal graphs withµΓ(e) = µPk(e)for k sufficiently large. We have further shown that each local maximumpointof 〈e, ·〉 is a vertex ofPk, possibly of degree two, fork large enough. Recall thatµPk(e) =

∑q nlm+Pk

(e, q). Thus, each local maximum pointq for 〈e, ·〉 along Γprovides a non-negative term nlm+Pk

(e, q) in the sum forµPk(e). Fix such an integerk.

Consider a pointq ∈ Γ which is not a topological vertex ofΓ but is a criticalpoint of 〈e, ·〉. We shall show, by an argument similar to one used by van Rooijin[vR], that q must be a local extreme point. As a first step, we show that〈e, ·〉 ismonotone on a sufficiently small interval on either side ofq. Choose an ordering ofthe closed edgeE of Γ containingq, and consider the intervalE+ of points≥ q withrespect to this ordering. Suppose that〈e, ·〉 is not monotone on any subinterval ofE+ with q as endpoint. Then in any interval (q, r1) there are pointsp2 > q2 > r2 sothat the numbers〈e, p2〉, 〈e, q2〉, 〈e, r2〉 are not monotone. It follows by an inductionargument that there exist decreasing sequencespn → q, qn → q, andrn → q ofpoints ofE+ such that for eachn, rn−1 > pn > qn > rn > q, but the value〈e, qn〉 liesoutside of the closed interval between〈e, pn〉 and〈e, rn〉. As a consequence, there isa local extremumsn ∈ (rn, pn). Sincern−1 > pn, thesn are all distinct, 1≤ n < ∞.But by Lemma 3, all local extreme points, specificallysn, of 〈e, ·〉 alongΓ occuramong thefinite number of vertices ofPk, a contradiction. This shows that〈e, ·〉 ismonotone on an interval to the right ofq. An analogous argument shows that〈e, ·〉is monotone on an interval to the left ofq.

Recall that for acritical point q relative toe, 〈e, ·〉 is not monotone on anyneighborhood ofq. Since〈e, ·〉 is monotone on an interval on either side, the sense


of monotonicity must be opposite on the two sides ofq. Therefore every criticalpoint q alongΓ for 〈e, ·〉, which is not a topological vertex, is a local extremum.

We have chosenk large enough thatµΓ(e) = µPk(e). Then for any edgeEk ofPk, the function〈e, ·〉 is monotone along the corresponding arcE of Γ, as well asalongEk. Also, E andEk have common end points. It follows that for eacht ∈ R,the cardinality #(e, t) of the fiber{q ∈ Γ : 〈e, q〉 = t} is the same forPk as forΓ. We may see from Lemma 1 applied toPk that for each vertex or critical pointq, nlmPk(e, q) = 1

2[d−Pk(e, q) − d+Pk

(e, q)]; but nlm(e, q) andd±(e, q) have thesamevalues forΓ as forPk. The formulaµΓ(e) =

∑q{nlmΓ(e, q)}+ now follows from the

corresponding formula forPk, for almost alle ∈ S2.Consider an open intervalE of Γ with endpointq. We have just shown that for

a.a.e ∈ S2, 〈e, ·〉 is monotone on a subinterval with endpointq. Choose a sequencepℓ from E, pℓ → q, and writeTℓ := pℓ−q

|pℓ−q| ∈ S2. Then limℓ→∞ Tℓ exists. Otherwise,

sinceS2 is compact, there are subsequences{Tmn} and {Tkn} with Tmn → T′ andTkn → T′′ , T′. But for an open set ofe ∈ S2, 〈e,T′〉 < 0 < 〈e,T′′〉. For suche,〈e, qmn〉 < 〈e, q〉 < 〈e, qkn〉 for n >> 1. That is, asp → q, p ∈ E, 〈e, p〉 assumesvalues above and below〈e, q〉 infinitely often, contradicting monotonicity on aninterval starting atq for a.a.e∈ S2.

This shows thatΓ has one-sided tangent vectorsT1(q), . . . ,Td(q) at each pointq ∈ Γ of degreed = d(q) (d = 2 if q is not a topological vertex). Further, ask→ ∞,TPk

i (q) → TΓi (q), 1 ≤ i ≤ d(q), since edges ofPk have diameter≤ 1k .

The remaining conclusions follow readily.

Corollary 8. LetΓ be a continuous, finite graph inR3, withNTC(Γ) < ∞. Then foreach point q ofΓ, the contribution at q to net total curvature is given by equation(3), where for e∈ S2, χi(e) = the sign of〈−Ti(q), e〉, 1 ≤ i ≤ d(q). (Here, if q is nota topological vertex, we understand d= 2.)

Proof. According to Lemma 4, for 1≤ i ≤ d(q), Ti(q) is defined and tangent toan edgeEi of Γ, which is continuously differentiable at its end pointq. If Pn is asequence ofΓ-approximating polygonal graphs with maximum edge length tendingto 0, then the corresponding unit tangent vectorsTPn

i (q) → TΓi (q) asn→ ∞. ForeachPn, we have

ntcPn(q) =14

∫

S2

d∑

i=1

χiPn(e)

+

dAS2(e),

andχiPn → χi

Γ in measure onS2. Hence, the integrals forPn converge to those forΓ, which is equation (3).

We are ready to state the formula for net total curvature, by localization onS2,a generalization of Theorem 1:


Theorem 2. For a continuous graphΓ, the net total curvatureNTC(Γ) ∈ (0,∞]has the following representation:

NTC(Γ) =14

∫

S2µ(e) dAS2(e),

where, for almost all e∈ S2, the multiplicityµ(e) is a positive half-integer or+∞,given as the finite sum(6).

Proof. If NTC(Γ) is finite, then the theorem follows from Lemma 3 and Lemma4.

Suppose NTC(Γ) = sup NTC(Pk) is infinite, wherePk is a refined sequence ofpolygonal graphs as in Lemma 2. ThenµΓ(e) is the non-decreasing limit ofµPk(e)for all e ∈ S2. ThusµΓ(e) ≥ µPk(e) for all e andk, andµΓ(e) = µPk(e) for k ≥ ℓ(e).This implies thatµΓ(e) is a positive half-integer or∞. Since NTC(Γ) = ∞, theintegral

NTC(Pk) =12

∫

S2µPk(e) dAS2(e)

is arbitrarily large ask→ ∞, but for eachk is less than or equal to

12

∫

S2µΓ(e) dAS2(e).

Therefore this latter integral equals∞, and thus equals NTC(Γ).

We turn our attention next to the tameness of graphs of finite total curvature.

Proposition 3. Let n be a positive integer, and write Z for the set of n-th roots ofunity inC = R2. Given a continuous one-parameter family St, 0 ≤ t < 1, of setsof n points inR2, there exists a continuous one-parameter familyΦt : R2 → R2 ofhomeomorphisms with compact support such thatΦt(St) = Z, 0 ≤ t < 1.

Proof. It is well known that there is an isotopyΦ0 : R2→ R2 such thatΦ0(S0) =Z andΦ0 = id outside of a compact set. This completes the caset0 = 0 of thefollowing continuous induction argument.

Suppose that [0, t0] ⊂ [0, 1) is a subinterval such that there exists a continuousone-parameter familyΦt : R2 → R2 of homeomorphisms with compact support,with Φt(St) = Z for all 0 ≤ t ≤ t0. We shall extend this property to an interval[0, t0 + δ]. Write Bε(Z) for the union of ballsBε(ζi) centered at then roots of unityζ1, . . . ζn. Forε < sin π

n , these balls are disjoint. We may choose 0< δ < 1− t0 suchthatΦt0(St) ⊂ Bε(Z) for all t0 ≤ t ≤ t0+δ.Write the points ofSt asxi(t), 1 ≤ i ≤ n,whereΦt0(xi(t)) ∈ Bε(ζi). For eacht ∈ [t0, t0 + δ], each of the ballsBε(ζi) may bemapped onto itself by a homeomorphismψt, varying continuously witht, such thatψt0 is the identity,ψt is the identity near the boundary ofBε(ζi) for all t ∈ [t0, t0+δ],andψt(Φt0(xi(t))) = ζi for all sucht. For example, we may constructψt so that foreachy ∈ Bε(ζi), y− ψt(y) is parallel toΦt0(xi(t)) − ζi . We now defineΦt = ψt ◦Φt0for eacht ∈ [t0, t0 + δ].

As a consequence, we see that there is no maximal interval [0, t0] ⊂ [0, 1) suchthat there is a continuous one-parameter familyΦt : R2→ R2 of homeomorphisms


with compact support withΦt(St) = Z, for all 0≤ t ≤ t0. Thus, this property holdsfor the entire interval 0≤ t < 1.

In the following theorem, the total curvature of a graph may be understood interms of any definition which includes the total curvature ofedges and which iscontinuous as a function of the unit tangent vectors at each vertex. This includesnet total curvature, TC of [T] and CTC of [GY1].

Theorem 3. SupposeΓ ⊂ R3 is a continuous graph with finite total curvature.Then for anyε > 0, Γ is isotopic to aΓ-approximating polygonal graph P withedges of length at mostε, whose total curvature is less than or equal to that ofΓ.

Proof.SinceΓ has finite total curvature, by Lemma 4, at each topological vertexof degreed the edges have well-defined unit tangent vectorsT1, . . . ,Td, which areeach the limit of the unit tangent vectors to the corresponding edges. If at eachvertex the unit tangent vectorsT1, . . . ,Td are distinct, then any sufficiently fineΓ-approximating polygonal graph will be isotopic toΓ; this easier case is proven.

We consider thereforen edgesE1, . . . ,En which end at a vertexq with commonunit tangent vectorsT1 = · · · = Tn. Choose orthogonal coordinates (x, y, z) for R3

so that this common tangent vectorT1 = · · · = Tn = (0, 0,−1) andq = (0, 0, 1).For someε > 0, in the slab 1− ε ≤ z≤ 1, the edgesE1, . . . ,En project one-to-oneonto thez-axis. After rescaling aboutq by a factor≥ 1

ε, E1, . . . ,En form a braidB

of n strands in the slab 0≤ z< 1 of R3, plus the pointq = (0, 0, 1). Each strandEi

hasq as an endpoint, and the coordinatez is strictly monotone alongEi, 1≤ i ≤ n.Write St = B∩ {z= t}. ThenSt is a set ofn distinct points in the plane{z= t} foreach 0≤ t < 1. According to Proposition 3, there are homeomorphismsΦt of theplane{z= t} for each 0≤ t < 1, isotopic to the identity in that plane, continuous asa function oft, such thatΦt(St) = Z× {t}, whereZ is the set ofnth roots of unity inthe (x, y)-plane, andΦt is the identity outside of a compact set of the plane{z= t}.

We may suppose thatSt lies in the open disk of radiusa(1 − t) of the plane{z = t}, for some (arbitrarily small) constanta > 0. We modifyΦt, first replacingits values with (1− t)Φt inside the disk of radiusa(1 − t). We then modifyΦt

outside the disk of radiusa(1 − t), such thatΦt is the identity outside the disk ofradius 2a(1− t).

Having thus modified the homeomorphismsΦt of the planes{z = t}, we maynow define an isotopyΦ of R3 by mapping each plane{z = t} to itself by thehomeomorphismΦ−1

0 ◦ Φt, 0 ≤ t < 1; and extend to the remaining planes{z = t},t ≥ 1 andt < 0, by the identity. Then the closure of the image of the braidB is theunion of line segments fromq = (0, 0, 1) to then points ofS0 in the plane{z= 0}.Since eachΦt is isotopic to the identity in the plane{z = t}, Φ is isotopic to theidentity ofR3.

This procedure may be carried out in disjoint sets ofR3 surrounding each unitvector which occurs as tangent vector to more than one edge ata vertex ofΓ.Outside these sets, we inscribe a polygonal arc in each edge of Γ to obtain aΓ-approximating polygonal graphP. By Definition 3,P has total curvature less thanor equal to the total curvature ofΓ.


Artin and Fox [AF] introduced the notion oftameand wild knots inR3; theextension to graphs is the following

Definition 10. We say that a graph inR3 is tameif it is isotopic to a polyhedralgraph; otherwise, it iswild.

Milnor proved in [M] that knots of finite total curvature are tame. More gener-ally, we have

Corollary 9. A continuous graphΓ ⊂ R3 of finite total curvature is tame.

Proof.This is an immediate consequence of Theorem 3, since theΓ-approximatingpolygonal graphP is isotopic toΓ.

Observation 1. Tameness does not imply finite total curvature.

For a well-known example, considerΓ ⊂ R2 to be the continuous curve{(x, h(x)) :x ∈ [−1, 1]} where the function

h(x) = −xπ

sinπ

x,

h(0) = 0, has a sequence of zeroes±1n → 0 asn→ ∞. Then the total curvature of

Γ between (0, 1n) and (0, 1

n+1) converges toπ asn→ ∞. ThusC(Γ) = ∞.On the other hand,h(x) is continuous on [−1, 1], from which it readily follows

thatΓ is tame.

5. ON VERTICES OF SMALL DEGREE

We are now in a position to illustrate some properties of net total curvatureNTC(Γ) in a few relatively simple cases, and to make some observations regardingNTC({Γ}), the minimum net total curvature for the homeomorphism type of a graphΓ ⊂ Rn (see Definition 7 above).

5.1. Minimum curvature for given degree.

Proposition 4. If a vertex q hasodd degree, thenntc(q)≥ π/2. If d(q) = 3, thenequality holds if and only if the three tangent vectors T1,T2,T3 at q are coplanarbut do not lie in any open half-plane. If q haseven degree2m, then the mini-mum value ofntc(q) is 0. Moreover, the equalityntc(q) = 0 only occurs whenT1(q), . . . ,T2m(q) form m opposite pairs.

Proof.Letqhave odd degreed(q) = 2m+1. Then from Lemma 1, for anye ∈ S2,we see that nlm(e, q) is a half-integer±1

2, . . . ,±2m+1

2 . In particular,|nlm(e, q)| ≥ 12.

Corollary 2 and the proof of Corollary 5 show that

ntc(q) =14

∫

S2

∣∣∣∣nlm(e, q)∣∣∣∣ dAS2.

Therefore ntc(q) ≥ π/2.If the degreed(q) = 3, then|nlm(e, q)| = 1

2 if and only if bothd+(q) andd−(q)are nonzero, that is,q is not a local extremum for〈e, ·〉. If ntc(q) = π/2, then thismust be true for almost every directione ∈ S2. Thus, the three tangent vectorsmust be coplanar, and may not lie in an open half-plane.


If d(q) = 2m is even and equality ntc(q)= 0 holds, then the formula above forntc(q) in terms of |nlm(e, q)| would require nlm(e, q) ≡ 0, and henced+(e, q) =d−(e, q) = m for almost alle ∈ S2: whenevere rotates so that the plane orthogonalto epassesTi , another tangent vectorT j must cross the plane in the opposite direc-tion, for a.a.e, which impliesT j = −Ti .

Observation 2. If a vertex q of odd degree d(q) = 2p+ 1, has the minimum valuentc(q) = π/2, and a hyperplane P⊂ Rn contains an even number of the tangentvectors at q, and no others, then these tangent vectors form opposite pairs.

The proof is seen by fixing any (n−2)-dimensional subspaceL of P and rotatingP by a small positive or negative angleδ to a hyperplanePδ containingL. SincePδ must havek of the vectorsT1, . . . ,T2p+1 on one side andk+ 1 on the other side,for some 0≤ k ≤ p, by comparingδ > 0 with δ < 0 it follows that exactly halfof the tangent vectors inP lie nonstrictly on each side ofL. The proof may becontinued as in the last paragraph of the proof of Proposition 4. In particular, anytwo independent tangent vectorsTi andT j share the 2-plane they span with a third,the three vectors not lying in any open half-plane: in fact, the third vector needs tolie in any hyperplane containingTi andT j .

For example, a flatK5,1 in R3 must have five straight segments, two being oppo-site; and the remaining three being coplanar but not in any open half-plane. Thisincludes the case of four coplanar line segments, since the four must be in oppo-site pairs, and either opposing pair may be considered as coplanar with the fifthsegment.

5.2. Non-monotonicity of NTC for subgraphs.

Observation 3. If Γ0 is a subgraph of a graphΓ, thenNTC(Γ0) might not be≤ NTC(Γ).

For a simple polyhedral example, we may consider the “butterfly” graph Γ inthe plane with six vertices:q±0 = (0,±1), q±1 = (1,±3), andq±2 = (−1,±3). Γ hasseven edges: three vertical edgesL0, L1 andL2 are the line segmentsLi joining q−ito q+i . Four additional edges are the line segments fromq±0 to q±1 and fromq±0 toq±2 , which form the smaller angle 2α at q±0 , where tanα = 1/2, so thatα < π/4.

The subgraphΓ0 will be Γ minus the interior ofL0. Then NTC(Γ0) = C(Γ0) =6π − 8α. However, NTC(Γ) = 4(π − α) + 2(π/2) = 5π − 4α, which is< NTC(Γ0).

The monotonicity property, which is shown in Observation 3 to fail for NTC(Γ),is a virtue of Taniyama’s total curvature TC(Γ).

5.3. Net total curvature , cone total curvature , Taniyama’s total curva-ture. It is not difficult to construct three unit vectorsT1,T2,T3 in R3 such thatthe values of ntc(q), ctc(q) and tc(q), with these vectors as thed(q) = 3 tan-gent vectors to a graph at a vertexq, have different values. For example, wemay takeT1,T2 andT3 to be three unit vectors in a plane, making equal angles


2π/3. According to Proposition 4, we have the contribution to net total curvaturentc(q) = π/2. But the contribution to cone total curvature is ctc(q) = 0. Namely,ctc(q) := supe∈S2

∑3i=1

(π2 − arccos〈Ti , e〉

). In this supremum, we may choosee to

be normal to the plane ofT1,T2 andT3, and ctc(q) = 0 follows. Meanwhile, tc(q)is the sum of the exterior angles formed by the three pairs of vectors, each equal toπ/3, so that tc(q) = π.

A similar computation for degreed and coplanar vectors making equal angles

gives ctc(q) = 0, and tc(q) = π2

[(d−1)2

2

](brackets denoting integer part), while

ntc(q) = π/2 for d odd, ntc(q) = 0 for d even. This example indicates that tc(q)may be significantly larger than ntc(q). In fact, we have

Observation 4. If a vertex q of a graphΓ has degree d= d(q) ≥ 2, thentc(q) ≥(d − 1)ntc(q).

This follows from the definition (3) of ntc(q). Let T1, . . . ,Td be the unit tangentvectors atq. The exterior angle betweenTi andT j is

arccos〈−Ti ,T j〉 =14

∫

S2(χi + χ j)

+ dAS2.

The contribution tc(q) atq to total curvature TC(Γ) equals the sum of these integralsover all 1≤ i < j ≤ d. The sum of the integrands is

∑

1≤i< j≤d

(χi + χ j)+ ≥

[ ∑

1≤i< j≤d

(χi + χ j)

]+= (d − 1)

[ d∑

i=1

χi

]+.

Integrating overS2 and dividing by 4, we have tc(q) ≥ (d − 1)ntc(q).

5.4. Conditional additivity of net total curvature under taking union. Obser-vation 3 shows the failure of monotonicity of NTC for subgraphs due to the cancel-lation phenomena at each vertex. The following subadditivity statement specifiesthe necessary and sufficient condition for the additivity of net total curvature undertaking union of graphs.

Proposition 5. Given two graphsΓ1 andΓ2 ⊂ Rn with Γ1 ∩ Γ2 = {p1, . . . , pN}, the

net total curvature ofΓ = Γ1 ∪ Γ2 obeys the sub-additivity law

NTC(Γ) = NTC(Γ1) + NTC(Γ2) +

+12

N∑

j=1

∫

S2[nlm+Γ (e, p j) − nlm+Γ1

(e, p j) − nlm+Γ2(e, p j)] dAS2(7)

≤ NTC(Γ1) + NTC(Γ2).

In particular, additivity holds if and only if

nlmΓ1(e, p j) nlmΓ2(e, p j ) ≥ 0

for all points pj of Γ1 ∩ Γ2 and almost all e∈ S2.


Proof. The edges ofΓ and vertices other thanp1, . . . , pN are edges and ver-tices ofΓ1 or of Γ2, so we only need to consider the contribution at the verticesp1, . . . , pN to µ(e) for e ∈ S2 (see Definition 6). The sub-additivity follows fromthe general inequality (a + b)+ ≤ a+ + b+ for any real numbersa andb. Namely,let a := nlmΓ1(e, p j) andb := nlmΓ2(e, p j ), so that nlmΓ(e, p j) = a + b, as fol-lows from Lemma 1. Now integrate both sides of the inequalityoverS2, sum overj = 1, . . . ,N and apply Theorem 1.

As for the equality case, suppose thatab ≥ 0. We then note that eithera > 0& b > 0, or a < 0 & b < 0, or a = 0, or b = 0. In all four cases, we havea+ + b+ = (a+ b)+. Applied witha = nlmΓ1(e, p j) andb = nlmΓ2(e, p j ), assumingthat nlmΓ1(e, p j)nlmΓ2(e, p j ) ≥ 0 holds for all j = 1, . . . ,N and almost alle ∈ S2,this implies that NTC(Γ1 ∪ Γ2) = NTC(Γ1) + NTC(Γ2).

To show that the equality NTC(Γ1 ∪ Γ2) = NTC(Γ1) + NTC(Γ2) implies theinequality nlmΓ1(e, p j)nlmΓ2(e, p j) ≥ 0 for all j = 1, . . . ,N and for almost alle∈ S2, we suppose, to the contrary, that there is a setU of positive measure inS2,such that for some vertexp j in Γ1 ∩ Γ2, whenevere is in U, the inequalityab < 0is satisfied, wherea = nlmΓ1(e, p j ) andb = nlmΓ2(e, p j). Then fore in U, a andb are of opposite signs. LetU1 be the part ofU wherea < 0 < b holds: we mayassumeU1 has positive measure, otherwise exchangeΓ1 with Γ2. OnU1, we have

(a+ b)+ < b+ = a+ + b+.

Recall thata+ b = nlmΓ(e, p j). Hence the inequality between half-integers

nlm+Γ (e, p j) < nlm+Γ1(e, p j ) + nlm+Γ2

(e, p j)

is valid on the set of positive measureU1, which in turn implies that NTC(Γ1∪Γ2) <NTC(Γ1) + NTC(Γ2), contradicting the assumption of equality.

5.5. One-point union of graphs.

Proposition 6. If the graphΓ is the one-point union of graphsΓ1 andΓ2, where thepoints p1 chosen inΓ1 and p2 chosen inΓ2 are not topological vertices, then theminimumNTC among all mappings is subadditive, and the minimumNTC minus2π is superadditive:

NTC({Γ1}) + NTC({Γ2}) − 2π ≤ NTC({Γ}) ≤ NTC({Γ1}) + NTC({Γ2}).

Further, if the points p1 ∈ Γ1 and p2 ∈ Γ2 may appear as extreme points on map-pings of minimumNTC, then the minimum net total curvature among all mappings,minus2π, is additive:

NTC({Γ}) = NTC({Γ1}) + NTC({Γ2}) − 2π.

Proof.Write p ∈ Γ for the identified pointsp1 = p2 = p.Choose flat mappingsf1 : Γ1 → R and f2 : Γ2 → R, adding constants so that

the chosen pointsp1 ∈ Γ1 and p2 ∈ Γ2 have f1(p1) = f2(p2) = 0. Further, byProposition 1, we may assume thatf1 and f2 are strictly monotone on the edgesof Γ1 resp. Γ2 containingp1 resp. p2. Let f : Γ → R be defined asf1 on Γ1

and as f2 on Γ2. Then at the common point ofΓ1 andΓ2, f (p) = 0, and f iscontinuous. But sincef1 and f2 are monotone on the edges containingp1 and


p2, nlmΓ1(p1) = 0 = nlmΓ2(p2), so we have NTC({Γ}) ≤ NTC( f ) = NTC( f1) +NTC( f2) = NTC({Γ1}) + NTC({Γ2}) by Proposition 5.

Next, for allg : Γ→ R, we shall show that NTC(g) ≥ NTC({Γ1})+NTC({Γ2})−2π. Giveng, write g1 resp.g2 for the restriction ofg to Γ1 resp.Γ2. Thenµg(e) =µg1(e) − nlm+g1

(p1) + µg2(e) − nlm+g2(p2) + nlm+g (p). Now for any real numbersa

andb, the difference (a + b)+ − (a+ + b+) is equal to±a, ±b or 0, depending onthe various signs. Leta = nlmg1(p1) andb = nlmg2(p2). Then sincep1 andp2 arenot topological vertices ofΓ1 resp.Γ2, a, b ∈ {−1, 0,+1} anda + b = nlmg(p) byLemma 1. In any case, we have

nlm+g(p) − nlm+g1(p1) − nlm+g2

(p2) ≥ −1.

Thus,µg(e) ≥ µg1(e) + µg2(e) − 1, and multiplying by 2π, NTC(g) ≥ NTC(g1) +NTC(g2) − 2π ≥ NTC({Γ1}) + NTC({Γ2}) − 2π.

Finally, assumep1 and p2 are extreme points for flat mappingsf1 : Γ1 → R

resp. f2 : Γ2 → R. We may assume thatf1(p1) = 0 = min f1(Γ1) and f2(p2) =0 = max f2(Γ2). Then nlmf2(p2) = 1 and nlmf1(p1) = −1, and hence using Lemma1, nlmf (p) = 0. Soµ f (e) = µ f1(e) − nlm+f1(p1) + µ f2(e) − nlm+ f2(p2) + nlm+f (p) =µ f1(e)+µ f2(e)−1.Multiplying by 2π, we have NTC({Γ}) ≤ NTC( f ) = NTC({Γ1})+NTC({Γ2}) − 2π.

6. NET TOTAL CURVATURE FOR DEGREE 3

6.1. Simple description of net total curvature.

Proposition 7. For any graphΓ and any parameterizationΓ′ of its double,NTC(Γ) ≤12C(Γ′). If Γ is a trivalent graph, that is, having vertices of degree at most three,thenNTC(Γ) = 1

2C(Γ′) for any parameterizationΓ′ which does not immediatelyrepeat any edge ofΓ.

Proof.The first conclusion follows from Corollary 3.Now consider a trivalent graphΓ. Observe thatΓ′ would be forced to immedi-

ately repeat any edge which ends in a vertex of degree 1; thus,we may assume thatΓ has only vertices of degree 2 or 3. SinceΓ′ covers each edge ofΓ twice, we needonly show, for every vertexq of Γ, having degreed = d(q) ∈ {2, 3}, that

(8) 2 ntcΓ(q) =d∑

i=1

cΓ′(qi),

whereq1, . . . , qd are the vertices ofΓ′ over q. If d = 2, sinceΓ′ does not imme-diately repeat any edge ofΓ, we have ntcΓ(q) = cΓ′(q1) = cΓ′(q2), so equation (8)clearly holds. Ford = 3, write both sides of equation (8) as integrals overS2, usingthe definition (3) of ntcΓ(q). SinceΓ′ does not immediately repeat any edge, thethree pairs of tangent vectors{TΓ

′

1 (q j ),TΓ′

2 (q j)}, 1≤ j ≤ 3, comprise all three pairs


taken from the triple{TΓ1 (q),TΓ2 (q),TΓ3 (q)}. We need to show that

2∫

S2

[χ1 + χ2 + χ3

]+ dAS2 =

∫

S2

[χ1 + χ2

]+ dAS2 +

+

∫

S2

[χ2 + χ3

]+ dAS2 +

∫

S2

[χ3 + χ1

]+ dAS2,

where at each directione ∈ S2, χ j(e) = ±1 is the sign of〈−e,TΓj (q)〉. But the

integrands are equal at almost every pointeof S2:

2[χ1 + χ2 + χ3

]+=[χ1 + χ2

]++[χ2 + χ3

]++[χ3 + χ1

]+ ,as may be confirmed by cases: 6= 6 if χ1 = χ2 = χ3 = +1; 2= 2 if exactly one oftheχi equals−1, and 0= 0 in the remaining cases.

6.2. Simple description of net total curvature fails,d ≥ 4.

Observation 5. We have seen in Corollary 3 that for graphs with vertices of degree≤ 3, if a parameterizationΓ′ of the doubleΓ of Γ does not immediately repeat anyedge ofΓ, thenNTC(Γ) = 1

2C(Γ′), the total curvature in the usual sense of the linkΓ′. A natural suggestion would be that for general graphsΓ, NTC(Γ) might behalf the infimum of total curvature of all such parameterizations Γ′ of the double.However, in some cases, we have thestrict inequality NTC(Γ) < infΓ′ 1

2NTC(Γ′).

In light of Proposition 7, we choose an example of a vertexq of degree four, andconsider the local contributions to NTC forΓ = K1,4 and forΓ′, which is the unionof four arcs.

Suppose that for a small positive angleα, (α ≤ 1 radian would suffice) the fourunit tangent vectors atq areT1 = (1, 0, 0); T2 = (0, 1, 0); T3 = (− cosα, 0, sinα);andT4 = (0,− cosα,− sinα). Write the exterior angles asθi j = π − arccos〈Ti ,T j〉.

Then infΓ′ 12C(Γ′) = θ13 + θ24 = 2α. However, ntc(q) is strictly less than 2α. This

may be seen by writing ntc(q) as an integral overS2, according to the definition (3),and noting that cancellation occurs between two of the four lune-shaped sectors.

6.3. Minimum NTC for trivalent graphs. Using the relation NTC(Γ) = 12NTC(Γ′)

between the net total curvature of a given trivalent graphΓ and the total curvaturefor a non-reversing double coverΓ′ of the graph, we can determine the minimumnet total curvature of a trivalent graph embedded inRn, whose value is then relatedto the Euler characteristic of the graphχ(Γ) = −k/2.

First we introduce the following definition.

Definition 11. For a given graphΓ and a mapping f: Γ → R, let theextendedbridge numberB( f ) be one-half the number of local extrema. Write B({Γ}) for theminimum of B( f ) among all mappings f: Γ → R. For a given isotopy type[Γ] ofembeddings intoR3, let B([Γ]) be one-half the minimum number of local extremafor a mapping f: Γ→ R in the closure of the isotopy class[Γ].


For an integerm ≥ 3, let θm be the graph with two verticesq+, q− andm edges,each of which hasq+ andq− as its two endpoints. Thenθ = θ3 has the form of thelower-case Greek letterθ.

Remark 5. For a knot, the number of local maxima equals the number of localminima. The minimum number of local maxima is called thebridge number, andequals the number of local minima. This is consistent with our Definition 11 of theextended bridge number. Of course, for knots, the minimum bridge number amongall isotopy classes B({S1}) = 1, and only B([S1]) is of interest for a specific iso-topy class[S1]. For certain graphs, the minimum numbers of local maxima andlocal minima may not occur at the same time for any mapping: see the example ofObservation 6 below. For isotopy classes ofθ-graphs, Goda[Go] has given a def-inition of an integer-valued bridge index which is similar in spirit to the definitionabove.

Theorem 4. If Γ is a trivalent graph, and if f0 : Γ→ R is monotone on topologicaledges and has the minimum number2B({Γ}) of local extrema, thenNTC( f0) =NTC({Γ}) = π

(2B({Γ})+ k

2

), where k is the number of topological vertices ofΓ. For

a given isotopy class[Γ], NTC([Γ]) = π(2B([Γ]) + k

2

).

Proof.Recall that NTC({Γ}) denotes the infimum of NTC(f ) amongf : Γ→ R3

or amongf : Γ→ R, as may be seen from Corollary 7.We first consider a mappingf1 : Γ → R with the property that any local

maximum or local minimum points off1 are interior points of topological edges.Then all topological verticesv, since they have degreed(v) = 3 andd±(v) , 0,have nlm(v) = ±1/2, by Proposition 4. LetΛ be the number of local maximumpoints of f1, V the number of local minimum points,λ the number of vertices withnlm = +1/2, andy the number of vertices with nlm= −1/2. Thenλ + y = k, thetotal number of vertices, andΛ + V ≥ 2B({Γ}). Hence applying Corollary 5,

(9) µ =12

∑

v

|nlm(v)| =12

[Λ + V +λ + y

2] ≥ B({Γ}) + k/4,

with equality iff Λ + V = 2B({Γ}).We next consider any mappingf0 : Γ→ R in general position: in particular, the

critical valuess off0 are isolated. In a similar fashion to the proof of Proposition1, we shall replacef0 with a mapping whose local extrema are not topologicalvertices. Specifically, iff0 assumes a local maximum at any topological vertexv,then, sinced(v) = 3, nlmf0(v) = 3/2. f0 may be isotoped in a small neighborhoodof v to f1 : Γ → R so that nearv, the local maximum occurs at an interior pointq of one of the three edges with endpointv, and thus nlmf1(q) = 1; while the up-degreed+f1(v) = 1 and the down-degreed−f1(v) = 2, so that nlmf1(v) is now 1

2. Thus,µ f1(e) = µ f0(e). Similarly, if f0 assumes a local minimum at a topological vertexw, then f0 may be isotoped in a neighborhood ofw to f1 : Γ → R so that the localminimum of f1 nearw occurs at an interior point of any of the three edges withendpointw, andµ f1(e) = µ f0(e). Then any local extreme points off1 are interiorpoints of topological edges. Thus, we have shown thatµ f0(e) ≥ B({Γ}) + k/4, with


equality if f1 has exactly 2B({Γ}) as its number of local extrema, which holds iff f0has the minimum number 2B({Γ}) of local extrema.

Thus NTC({Γ}) = 2πµ f0(e) = 2π(B({Γ}) + k/4

)= π(2B({Γ}) + k/2

).

Similarly, for a given isotopy class [Γ] of embeddings intoR3, we may choosef0 : Γ → R in the closure of the isotopy class, deformf0 to a mappingf1 in theclosure of [Γ] having no topological vertices as local extrema and countµ f0(e) =µ f1(e) ≥ B([Γ]) +k/4, with equality if f0 has the minimum number 2B([Γ]) of local

extrema. This shows that NTC([Γ]) = π(2B([Γ]) + k/2

).

Remark 6. An example geometrically illustrating the lower bound is given by thedual graphΓ∗ of the one-skeletonΓ of a triangulation of S2, with the {∞} notcoinciding with any of the vertices ofΓ∗. The Koebe-Andreev-Thurston theoremsays that there is a circle packing which realizes the vertexset ofΓ∗ as the setof centers of the circles (see[S]). The so realizedΓ∗, stereographically projectedto R2 ⊂ R3, attains the lower bound of Theorem 4 with B({Γ∗}) = 1, namelyNTC([Γ]) = π(2+ k

2) = π(2− χ(Γ∗)), where k is the number of vertices.

Corollary 10. If Γ is a trivalent graph with k topological vertices, and f0 : Γ→ Ris a mapping in general position, havingΛ local maximum points and V localminimum points, then

µ f0(e) =12

(Λ + V) +k4≥ B({Γ}) +

k4.

Proof. Follows immediately from the proof of Theorem 4:f0 and f1 have thesame number of local maximum or minimum points.

An interesting trivalent graph isLm, the “ladder ofm rungs” obtained from twounit circles in parallel planes by addingm line segments (“rungs”) perpendicularto the planes, each joining one vertex on the first circle to another vertex on thesecond circle. For example,L4 is the 1-skeleton of the cube inR3. Note thatLm

may be embedded inR2, and that the bridge numberB({Lm}) = 1. SinceLm has2m trivalent vertices, we may apply Theorem 4 to compute the minimum NTC forthe type ofLm:

Corollary 11. The minimum net total curvatureNTC({Lm}) for graphs of the typeof Lm equalsπ(2+m).

Observation 6. For certain connected trivalent graphsΓ containing cut points,the minimum extended bridge number B({Γ}) may be greater than1.

Example:Let Γ be the union of three disjoint circlesC1,C2,C3 with three edgesEi connecting a pointpi ∈ Ci with a fourth vertexp0, which is not in any of theCi, and which is acut pointof Γ: the number of connected components ofΓ\p0

is greater than forΓ. Given f : Γ → R, after a permutation of{1, 2, 3}, we mayassume there is a minimum pointq1 ∈ C1∪E1 and a maximum pointq3 ∈ C3∪E3.If q1 andq3 are both inC1 ∪ E1, we may chooseC2 arbitrarily in what follows.Restricted to the closed setC2 ∪ E2, f assumes either a maximum or a minimumat a pointq2 , p0. Sinceq2 , p0, q2 is also a local maximum or a local minimum


for f on Γ. That is,q1, q2, q3 are all local extrema. In the notation of the proof ofTheorem 4, we have the number of local extremaV+Λ ≥ 3. ThereforeB({Γ}) ≥ 3

2,and NTC({Γ}) ≥ π(3+ k/2) = 5π.

The reader will be able to construct similar trivalent examples withB({Γ}) arbi-trarily large.

In contrast to the results of Theorem 4 and of Theorem 5, below, for trivalentor nearly trivalent graphs, the minimum of NTC for a given graph type cannot becomputed merely by counting vertices, but depends in a more subtle way on thetopology of the graph:

Observation 7. WhenΓ is not trivalent, the minimumNTC({Γ}) of net total cur-vature for a connected graphΓ with B({Γ}) = 1 is not determined by the number ofvertices and their degrees.

Example: We shall construct two planar graphsSm and Rm having the samenumber of vertices, all of degree 4.

Choose an integerm ≥ 3 and take the image of the embeddingfε of the “sinewave”Sm to be the union of the polar-coordinate graphsC± ⊂ R2 of two functions:r = 1± ε sin(mθ). Sm has 4m edges; and 2m vertices, all of degree 4, atr = 1 andθ = π/m, 2π/m, . . . , 2π. For 0 < ε < 1, fε(Sm) = C+ ∪ C− is the union of twosmooth cycles. For small positiveε, C+ andC− are convex. The 2m vertices allhave nlm(q) = 0, so NTC(fε) = NTC(C+) + NTC(C−) = 2π + 2π. ThereforeNTC({Sm}) ≤ NTC( fε) = 4π.

For the other graph type, let the “ring graph”Rm ⊂ R2 be constructed by adding

mdisjoint small circlesCi, each crossing one large circleC at two pointsv2i−1, v2i ,1 ≤ i ≤ m. ThenRm has 4m edges. We constructRm so that the 2m verticesv1, v2, . . . , v2m, appear in cyclic order aroundC. ThenRm has the same number 2mof vertices as doesSm, all of degree 4. At each vertexv j , we have nlm(v j) = 0,so in this embedding, NTC(Rm) = 2π(m+ 1). We shall show that NTC(f1) ≥ 2πmfor any f1 : Rm → R

3. According to Corollary 7, it is enough to show for everyf : Rm→ R thatµ f ≥ m. We may assumef is monotone on each topological edge,according to Proposition 1. Depending on the order off (v2i−2), f (v2i−1) and f (v2i),nlm(v2i−1) might equal±1 or±2, but cannot be 0, as follows from Lemma 1, sincethe unordered pair{d−(v2i−1), d+(v2i−1)} may only be{1, 3} or {0, 4}. Similarly, v2i

is connected by three edges tov2i−1 and by one edge tov2i+1. For the same reasons,nlm(v2i ) might equal±1 or±2, and cannot= 0. So|nlm(v j)| ≥ 1, 1≤ j ≤ 2m, andthus by Corollary 5,µ = 1

2

∑j |nlm(v j )| ≥ m. Therefore the minimum of net total

curvature NTC({Rm}) ≥ 2mπ, which is greater than NTC({Sm}) ≤ 4π, sincem≥ 3.(A more detailed analysis shows that NTC({Sm}) = 4π and NTC({Rm}) = 2π(m+

1).)

Finally, we may extend the methods of proof for Theorem 4 to allow onevertexof higher degree:

Theorem 5. If Γ is a graph with one vertex w of degree d(w) = m ≥ 3, all othervertices being trivalent, and if w shares edges with m distinct trivalent vertices,


thenNTC({Γ}) = π(2B({Γ})+ k

2

), where k is the number of vertices ofΓ having odd

degree. For a given isotopy class[Γ], NTC([Γ]) ≥ π(2B([Γ]) + k

2

).

Proof. Consider any mappingg : Γ → R in general position. Ifm is even,then |nlmg(w)| ≥ 0; if m is odd, then|nlmg(w)| ≥ 1

2, by Proposition 4. If sometopological vertex is a local extreme point, then as in the proof of Theorem 4,gmay be modified without changing NTC(g) so that allΛ + V local extreme pointsare interior points of edges, with nlm= ±1. By Corollary 5, we haveµg(e) =12

∑|nlm(v)| ≥ 1

2

(Λ + V + k

2

)≥ B({Γ}) + k

4. This shows that

NTC({Γ}) ≥ π(2B({Γ}) +

k2

).

Now let f0 : Γ → R be monotone on topological edges and have the minimumnumber 2B({Γ}) of local extreme points (see Corollary 1). As in the proof ofThe-orem 4, f0 may be modified without changing NTC(f0) so that all 2B({Γ}) localextreme points are interior points of edges.f0 may be further modified so that thedistinct verticesv1, . . . , vm which share edges withw are balanced:f (v j) < f (w)for half of the j = 1, . . . ,m, if m is even, or for half ofm+ 1, if m is odd. Havingchosenf (v j), we definef along the (unique) edge fromw to v j to be monotone,for j = 1, . . . ,m. Therefore ifm is even, then nlmf (w) = 0; and ifm is odd, thennlmf (w) = 1

2, by Lemma 1. We computeµ f (e) = 12

∑|nlm(v)| = 1

2(Λ + V + k2) =

B({Γ}) + k4. We conclude that NTC({Γ}) = π

(2B({Γ}) + k

2

).

For a given isotopy class [Γ], the proof is analogous to the above. Choose amappingg : Γ→ R in the closure of [Γ], and modifyg without leaving the closureof the isotopy class. Choosef : Γ → R which has the minimum number 2B([Γ])of local extreme points, and modify it so that topological vertices are not localextreme points. In contrast to the proof of Theorem 4, a balanced arrangement ofvertices may not be possible in the given isotopy class. In any case, ifm is even,then|nlm f (w)| ≥ 0; and ifm is odd,|nlm f (w)| ≥ 1

2, by Proposition 4. Thus applying

Corollary 5, we find NTC([Γ]) ≥ π(2B([Γ]) + k

2

).

Observation 8. When all vertices ofΓ are trivalent except w, d(w) ≥ 4, andwhen w shares more than one edge with another vertex ofΓ, then in certain cases,NTC({Γ}) > π

(2B({Γ}) + k

2

), where k is the number of vertices of odd degree.

Example:ChooseΓ to be the one-point union ofΓ1, Γ2 andΓ3, whereΓi = θ =θ3, i = 1, 2, 3, and the pointwi chosen fromΓi is one of its two verticesvi ,wi. Thenthe identified pointw = w1 = w2 = w3 of Γ hasd(w) = 9, and each of the otherthree verticesv1, v2, v3 has degree 3.

Choose a flat mapf : Γ → R. We may assume thatf is monotone on eachedge, applying Proposition 1. Iff (v1) < f (v2) < f (w) < f (v3), thend+(w) = 3,d−(w) = 6, so nlm(w) = 3

2, while vi is a local extreme point, so nlm(vi ) = ±32,

1 = 1, 2, 3. This givesµ = 3. The case wheref (v1) < f (w) < f (v2) < f (v3)is similar. If w is an extreme point off , then nlm(w) = ±9

2 andµ ≥ 92 > 3,

contradicting flatness off . This shows that NTC({Γ}) = NTC( f ) = 6π.


On the other hand, we may show as in Observation 6 thatB({Γ}) = 32. All four

vertices have odd degree, sok = 4, andπ(2B({Γ}) + k

2

)= 5π.

Let Wm denote the “wheel” ofm spokes, consisting of a cycleC containingmverticesv1, . . . , vm (the “rim”), a central vertexw (the “hub”) not onC, and edgesEi (the “spokes”) connectingw to vi , 1 ≤ i ≤ m.

Corollary 12. The minimum net total curvatureNTC({Wm}) for graphs inR3

homeomorphic to Wm equalsπ(2+ ⌈m2 ⌉).

Proof. We have one “hub” vertexw with d(w) = m, and all other vertices havedegree 3. Observe that the bridge numberB({Wm}) = 1. According to Theorem5, we have NTC({Wm}) = π

(2B({Wm}) + k

2

), wherek is the number of vertices of

odd degree:k = m if m is even, ork = m + 1 if m is odd: k = 2⌈m2 ⌉. Thus

NTC({Wm}) = π(2+ ⌈m2 ⌉

).

7. LOWER BOUNDS OF NET TOTAL CURVATURE

Thewidthof an isotopy class [Γ] of embeddings of a graphΓ intoR3 is the min-imum among representatives of the class of the maximum number of points of thegraph meeting a family of parallel planes. More precisely, we write width([Γ]) :=minf :Γ→R3| f∈[Γ] mine∈S2 maxs∈R #(e, s). For any homeomorphism type{Γ} definewidth({Γ}) to be the minimum over isotopy types.

Theorem 6. Let Γ be a graph, and consider an isotopy class[Γ] of embeddingsf : Γ→ R3. Then

NTC([Γ]) ≥ π width([Γ]).

As a consequence,NTC({Γ}) ≥ π width({Γ}). Moreover, if for some e∈ S2, anembedding f: Γ → R3 and s0 ∈ R, the integers#(e, s) are increasing in s fors< s0 and decreasing for s> s0, thenNTC([Γ]) = #(e, s0) π.

Proof. Choose an embeddingg : Γ → R3 in the given isotopy class, withmaxs∈R #(e, s) = width([Γ]). There existe ∈ S2 and s0 ∈ R with #(e, s0) =maxs∈R #(e, s) = width([Γ]). Replacee if necessary by a nearby point inS2 sothat the valuesg(vi), i = 1, . . . ,m are distinct. Next do cylindrical shrinking: with-out changing #(e, s) for s ∈ R, shrink the image ofg in directions orthogonalto e by a factorδ > 0 to obtain a family{gδ} from the same isotopy class [Γ],with NTC(gδ) → NTC(g0), where we may identifyg0 : Γ → Re ⊂ R3 withpe ◦ g = pe ◦ gδ : Γ→ R. But

NTC(pe ◦ g) =12

∫

S2µ(u) dAS2(u) = 2π µ(e),

since forpu ◦ pe ◦ g, the local maximum and minimum points are the same as forpe ◦ g if 〈e, u〉 > 0 and reversed if〈e, u〉 < 0 (recall thatµ(−e) = µ(e)).

We write the topological vertices and the local extrema ofg0 asv1, . . . , vm. Letthe indexing be chosen so thatg0(vi ) < g0(vi+1), i = 1, . . . ,m− 1. Now estimate


µ(e) from below: using Lemma 1,

(10) µ(e) =m∑

i=1

nlm+g0(vi) ≥

m∑

i=k+1

nlmg0(vi) =12

#(e, s)

for anys, g0(vk) < s< g0(vk+1). This shows thatµ(e) ≥ 12width([Γ]), and therefore

NTC(g) ≥ NTC(g0) = 2π µ(e) ≥ πwidth([Γ]).

Now suppose that the integers #(e, s) are increasing ins for s< s0 and decreas-ing for s> s0. Then forg0(vi) > s0, we have nlm(g0(vi )) ≥ 0 by Lemma 1, and theinequality (10) becomes equality ats= s0.

Lemma 5. For an integerℓ, the minimum width of the complete graph K2ℓ on 2ℓvertices iswidth({K2ℓ}) = ℓ2; for 2ℓ + 1 vertices,width({K2ℓ+1}) = ℓ(ℓ + 1).

Proof.Write Ei j for the edge ofKm joining vi to v j , 1 ≤ i < j ≤ m, and supposeg : Km→ R has distinct values at the vertices:g(v1) < g(v2) < · · · < g(vm).

Then for anyg(vk) < s< g(vk + 1), there arek(m− k) edgesEi j with i ≤ k < j;each of these edges has at least one interior point mapping tos, which shows that#(e, s) ≥ k(m− k). If m is even:m = 2ℓ, these lower bounds have the maximumvalueℓ2 whenk = ℓ. If m is odd:m= 2ℓ+1, these lower bounds have the maximumvalueℓ(ℓ + 1) whenk = ℓ or k = ℓ + 1. This shows that the width ofK2ℓ ≥ ℓ

2 andthe width ofK2ℓ+1 ≥ ℓ(ℓ + 1). On the other hand, equality holds for the piecewiselinear embedding ofKm into R with vertices in general position and straight edgesEi j , which shows that width({K2ℓ}) = ℓ2 and width({K2ℓ+1}) = ℓ(ℓ + 1).

Proposition 8. For all g : Km → R, NTC(g) ≥ π ℓ2 if m = 2ℓ is even; andNTC(g) ≥ π ℓ(ℓ + 1) if m = 2ℓ + 1 is odd. Equality holds for an embedding ofKm into R with vertices in general position and monotone on each edge;thereforeNTC({K2ℓ}) = π ℓ2, andNTC({K2ℓ+1}) = π ℓ(ℓ + 1).

Proof.The lower bound on NTC({Km}) follows from Theorem 6 and Lemma 5.Now supposeg : Km → R is monotone on each edge, and number the ver-

tices of Km so that for alli, g(vi ) < g(vi+1). Then as in the proof of Lemma 5,#(e, s) = k(m− k) for g(vk) < s < g(vk+1). These cardinalities are increasing for0 ≤ k ≤ ℓ and decreasing forℓ + 1 < k < m. Thus, ifg(vℓ) < s0 < g(vℓ+1), then byTheorem 6, NTC([Γ]) = #(e, s0) π = ℓ(m− ℓ) π, as claimed.

Let Km,n be the complete bipartite graph withm+ n vertices divided into twosets:vi , 1 ≤ i ≤ mandw j , 1 ≤ j ≤ n, having one edgeEi j joining vi to w j , for each1 ≤ i ≤ mand 1≤ j ≤ n.

Proposition 9. NTC({Km,n}) = ⌈mn2 ⌉ π.

Proof. Km,n has verticesv1, . . . , vm of degreed(vi ) = n and verticesw1, . . . ,wn

of degreed(w j) = m. Consider a mappingg : Km,n→ R in general position, so thatthem+n vertices ofKm,n have distinct images. We wish to showµ(e) = µg(e) ≥ mn

4 ,

if mor n is even, ormn+14 , if both mandn are odd.


For this purpose, according to Proposition 1, we may first reduceµ(e) or leaveit unchanged by replacingg with a mapping (also calledg) which is monotone oneach edgeEi j of Km,n. The values of nlm(w j) and of nlm(vi ) are now determinedby the order of the vertex imagesg(v1), . . . , g(vm), g(w1), . . . , g(wn). SinceKm,n

is symmetric under permutations of{v1, . . . , vm} and permutations of{w1, . . . ,wn},we shall assume thatg(vi ) < g(vi+1), i = 1, . . . ,m− 1 andg(w j) < g(w j+1), j =1, . . . , n− 1. For i = 1, . . . ,m we writeki for the largest indexj such thatg(w j) <g(vi). Then 0≤ k1 ≤ · · · ≤ km ≤ n, and these integers determineµ(e). Accordingto Lemma 1, nlm(vi) = ki −

n2 , i = 1, . . . ,m. For j ≤ k1 and for j ≥ km+ 1, we have

nlm(w j) = ±m2 ; for k1 < j ≤ k2 and forkm−1 < j ≤ km, we find nlm(w j) = ±

(m2 −1);

and so on until we find nlm(w j) = 0 on the middle intervalkp < j ≤ kp+1, ifm = 2p is even; or, ifm = 2p + 1 is odd, nlm(w j) = −1

2 for kp < j ≤ kp+1 andnlm(w j) = +1

2 for the other middle intervalkp+1 < j ≤ kp+2. Thus according toLemma 1 and Corollary 5, ifm= 2p is even,

2µ(e) =m∑

i=1

|nlm(vi)| +n∑

j=1

|nlm(w j)| =m∑

i=1

|ki −n2| + (k1 + n− km)

m2

+ (k2 − k1 + km − km−1)[m

2− 1]+ . . .

+ (kp − kp−1 + kp+2 − kp+1)[m

2− (p− 1)

]+ (kp+1 − kp)

[0]

(11)

=

m∑

i=1

∣∣∣∣ki −n2

∣∣∣∣ +mn2+

p∑

i=1

ki −

m∑

i=p+1

ki

=mn2+

p∑

i=1

[|ki −

n2| + (ki −

n2

)]+

m∑

i=p+1

[|ki −

n2| − (ki −

n2

)].

Note that formula (11) assumes its minimum value 2µ(e) = mn2 whenk1 ≤ · · · ≤

kp ≤n2 ≤ kp+1 ≤ · · · ≤ km.

If m= 2p+ 1 isodd, then

2µ(e) =m∑

i=1

|ki −n2| + (k1 + n− km)

m2+ (k2 − k1 + km − km−1)

[m2− 1]+ . . .

+ (kp+3 − kp+2)[m

2− (p− 1)

]+ (kp+2 − kp)

[12

]=

=

m∑

i=1

|ki −n2| +

mn2+

p∑

i=1

ki −

m∑

i=p+2

ki(12)

=mn2+

p∑

i=1

[|ki −

n2| + (ki −

n2

)]+

m∑

i=p+2

[|ki −

n2| − (ki −

n2

)]+ |kp+1 −

n2|.

Observe that formula (12) has the minimum value 2µ(e) = mn2 when n is even

andk1 ≤ · · · ≤ kp ≤n2 = kp+1 ≤ · · · ≤ km. If n as well asm is odd, then the


last term|kp+1 −n2 | ≥

12, and the minimum value of 2µ(e) is mn+1

2 , attained iffk1 ≤ · · · ≤ kp ≤

n2 ≤ kp+2 ≤ · · · ≤ km.

This shows that for either parity ofmor of n, µ(e) ≥ mn4 . If n andmare both odd,

we have the stronger inequalityµ(e) ≥ mn+14 . We may summarize these conclusions

as 2µ(e) ≥ ⌈mn2 ⌉, and therefore as in the proof of Corollary 7, NTC({Km,n}) ≥

⌈mn2 ⌉ π, as we wished to show.By abuse of notation, write the formula (11) or (12) asµ(k1, . . . , km).To show the inequality in the opposite direction, we need to find a mapping

f : Km,n → R with NTC( f ) = mnπ2 (m or n even) or NTC(f ) = (mn+1)π

2 (mand n odd). The above computation suggests choosingf with f (v1), . . . , f (vm)together in the middle of the images of thew j. Write n = 2ℓ if n is even, orn = 2ℓ + 1 if n is odd. Choose valuesf (w1) < · · · < f (wℓ) < f (v1) < · · · < f (vm) <f (wℓ+1) < · · · < f (wn), and extendf monotonically to each of themnedgesEi j .From formulas (11) and (12), we haveµ f (e) = µ(ℓ, . . . , ℓ) = mn

4 , if m or n is even;or µ f (e) = µ(ℓ, . . . , ℓ) = mn+1

4 , if mandn are odd.

Recall thatθm is the graph with two verticesq+, q− andmedges.

Corollary 13. NTC({θm}) = mπ.

Proof. θm is homeomorphic to the complete bipartite graphKm,2, and by theproof of Proposition 9, we findµ(e) ≥ m

2 for a.a.e ∈ S2, and hence NTC({Km,2}) =mπ.

8. FARY-MILNOR TYPE ISOTOPY CLASSIFICATION

Recall the Fary-Milnor theorem, which states that if the total curvature of aJordan curveΓ in R3 is less than or equal to 4π, thenΓ is unknotted. As we havedemonstrated above, there are a collection of graphs whose values of the minimumtotal net curvatures are known. It is natural to hope when thenet total curvature issmall, in the sense of being in a specific interval to the rightof the minimal value,that the isotopy type of the graph is restricted, as is the case for knots:Γ = S1.The following proposition and corollaries, however, tell us that results of the Fary-Milnor type cannot be expected to hold for more general graphs.

Proposition 10. If Γ is a graph inR3 and if C ⊂ Γ is a cycle, such that for somee∈ S2, pe◦C has at least two local maximum points, then for each positive integerq, there is a nonisotopic embeddingΓq of Γ in which C is replaced by a knot notisotopic to C, withNTC(Γq) as close as desired toNTC(pe ◦ Γ).

Proof. It follows from Corollary 7 that the one-dimensional graphpe ◦ Γ maybe replaced by an embeddingΓ into a small neighborhood of the lineRe in R3,with arbitrarily small change in its net total curvature. Since pe ◦ C has at leasttwo local maximum points, there is an interval ofR over whichpe ◦ C containsan interval which is the image of four oriented intervalsJ1, J2, J3, J4 appearing inthat cyclic order around the oriented cycleC. Consider a plane presentation ofΓby orthogonal projection into a generic plane containing the lineRe. Choose an


integerq ∈ Z, |q| ≥ 3. We modify Γ by wrapping its intervalJ1 q times aroundJ3 and returning, passing over any other edges ofΓ, including J2 andJ4, which itencounters along the way. The new graph inR3 is calledΓq. Then, ifC was theunknot, the cycleCq which has replaced it is a (2, q)-torus knot (see [L]). In anycase,Cq is not isotopic toC, and thereforeΓq is not isotopic toΓ.

As in the proof of Theorem 6, letgδ : R3 → R3 be defined by cylindricalshrinking, so thatg1 is the identity andg0 = pe. Thenpe ◦ Γq = g0(Γq), and forδ > 0, gδ(Γq) is isotopic toΓq. But NTC(gδ)→ NTC(g0) asδ→ 0.

Corollary 14. If e = e0 ∈ Sn−1 minimizesNTC(pe ◦ Γ), and there is a cycle C⊂ Γso that pe0 ◦ C has two (or more) local maximum points, then there is a sequenceof nonisotopic embeddingsΓq of Γ with NTC(Γq) less than, or as close as desired,to NTC(Γ), in which C is replaced by a(2, q)-torus knot.

Corollary 15. If Γ is an embedding of Km into R3, linear on each topologicaledge of Km, m ≥ 4, then there is a sequence of nonisotopic embeddingsΓq of Γwith NTC([Γq]) as close as desired toNTC([Γ]), in which an unknotted cycle C ofΓ is replaced by a(2, q)-torus knot.

Proof. According to Corollary 14, we only need to construct an isotopy of Km

with the minimum value of NTC, such that there is a cycleC so thatpe◦C has twolocal maximum points, whereµ(e) is a minimum amonge ∈ S2.

Chooseg : Km→ Rwhich is monotone on each edge ofKm, and has distinct val-ues at vertices. Then according to Proposition 8, we have NTC(g) = NTC({Km}).Number the verticesv1, . . . , vm so thatg(v1) < g(v2) < · · · < g(vm). Write E ji

for the edgeEi j with the reverse orientation,i , j. Then the cycleC formed insequence fromE13,E32,E24 andE41 has local maximum points atv3 andv4, andcovers the interval

(g(v2), g(v3)

)⊂ R four times. SinceC is formed out of four

straight edges, it is unknotted. The procedure of Corollary14 replacesC with a(2, q)-torus knot, with an arbitrarily small increase in NTC.

Note that Corollary 14 gives a set of conditions for those graph types where aFary-Milnor type isotopy classification might hold. In particular, we consider oneof the simpler homeomorphism types of graphs, thetheta graph, θ = θ3 = K3,2

(cf. description following Definition 11). Thestandard theta graph is the isotopyclass inR3 of a plane circle plus a diameter. We have seen in Corollary 13that theminimum of net total curvature for a theta graph is 3π. On the other hand note thatin the range 3π ≤ NTC(Γ) < 4π, for e in a set of positive measure ofS2, pe(Γ)cannot have two local maximum points. In Theorem 7 below, we shall show that atheta graphΓ with NTC(Γ) < 4π is isotopically standard.

We may observe that there are nonstandard theta graphs inR3. For example, the

union of two edges might form a knot. Moreover, as S. Kinoshita has shown, thereareθ-graphs inR3, not isotopic to a planar graph, such that each of the three cyclesformed by deleting one edge is unknotted [Ki].


We begin with a well-known property of knots, whose proof we give for the sakeof completeness.

Lemma 6. Let C ⊂ R3 be homeomorphic to S1, andnot a convex planar curve.Then there is a nonempty open set of planes P⊂ R3 which each meet C in at leastfour points.

Proof.For e∈ S2 andt ∈ R write the planePet = {x ∈ R

3 : 〈e, x〉 = t}.If C is not planar, then there exist four non-coplanar pointsp1, p2, p3, p4, num-

bered in order aroundC. Note that no three of the points can be collinear. Let anoriented planeP0 be chosen to containp1 andp3 and rotated until bothp2 andp4

are aboveP0 strictly. Write e1 for the unit normal vector toP0 on the side wherep2 andp3 lie, so thatP0 = Pe1

t0=0. .Then the setPt ∩C contains at least four points,for t0 = 0 < t < δ1, with someδ1 > 0, since each planePt = Pe1

t meets each ofthe four open arcs between the pointsp1, p2, p3, p4. This conclusion remains true,for some 0< δ < δ1, when the normal vectore1 to P0 is replaced by any nearbye∈ S2, andt is replaced by any 0< t < δ.

If C is planar but nonconvex, then there exists a planeP0 = Pe10 , transverse to

the plane containingC, which supportsC and touchesC at two distinct points, butdoes not include the arc ofC between these two points. Consider disjoint openarcs ofC on either side of these two points and including points not inP0. Thenfor 0 < t < δ ≪ 1, the setPt ∩ C contains at least four points, since the planesPt = Pe1

t meet each of the four disjoint arcs. Here once againe1 may be replacedby any nearby unit vectore, and the planePe

t will meetC in at least four points, fort in a nonempty open intervalt1 < t < t1 + δ.

Using the notion of net total curvature, we may extend the theorems of Fenchel[Fen] as well as of Fary-Milnor ([Fa],[M]), for curves homeomorpic toS1, tographs homeomorphic to the theta graph. An analogous resultis given by Taniyamain [T], who showed that the minimum of TC for polygonalθ-graphs is 4π, and thatanyθ-graphΓ with TC(Γ) < 5π is isotopically standard,

Theorem 7. Suppose f: θ → R3 is a continuous embedding,Γ = f (θ). ThenNTC(Γ) ≥ 3π. If NTC(Γ) < 4π, thenΓ is isotopic inR3 to the planar theta graph.Moreover,NTC(Γ) = 3π iff the graph is a planar convex curve plus a straightchord.

Proof.We consider first the case whenf : θ → R3 is piecewiseC2.(1) We have shown thelower bound 3π for NTC( f ), where f : θ → Rn is any

piecewiseC2 mapping, sinceθ = θ3 is one case of Corollary 13, withm= 3.(2) We show next that if there is a cycleC in a graphΓ (a subgraph home-

omorphic toS1) which satisfies the conclusion of Lemma 6, thenµ(e) ≥ 2 fore in a nonempty open set ofS2. Namely, for t0 < t < t0 + δ, a family ofplanesPe

t meetsC, and therefore meetsΓ, in at least four points. This is equiv-alent to saying that the cardinality #(e, t) ≥ 4. This implies, by Corollary 4, that∑{nlm(e, q) : pe(q) > t0} ≥ 2. Thus, since nlm+(e, q) ≥ nlm(e, q), using Definition

6, we haveµ(e) ≥ 2.


Now consider theequality case of a theta graphΓ with NTC(Γ) = 3π. Aswe have seen in the proof of Proposition 9 withm = 3 andn = 2, the multiplicityµ(e) ≥ 3

2 =mn4 for a.a.e ∈ S2, while the integral ofµ(e) overS2 equals 2 NTC(Γ) =

6π by Theorem 1, implyingµ(e) = 3/2 a.e. onS2. Thus, the conclusion of Lemma6 is impossible for any cycleC in Γ. By Lemma 6, all cyclesC of Γmust be planarand convex.

Now Γ consists of three arcsa1, a2 anda3, with common endpointsq+ andq−.As we have just shown, the three Jordan curvesΓ1 := a2 ∪ a3, Γ2 := a3 ∪ a1 andΓ3 := a1 ∪ a2 are each planar and convex. It follows thatΓ1, Γ2 andΓ3 lie in acommon plane. In terms of the topology of this plane, one of the three arcsa1, a2

anda3 lies in the middle between the other two. But the middle arc, saya2, must bea line segment, as it needs to be a shared piece of two curvesΓ1 andΓ3 boundingdisjoint convex open sets in the plane. The conclusion is that Γ is a planar, convexJordan curveΓ2, plus a straight chorda2, whenever NTC(Γ) = 3π.

(3) We next turn our attention to theupper bound of NTC, to imply that aθ-graph is isotopically standard: we shall assume thatg : θ → R3 is an embeddingin general position with NTC(g) < 4π, and writeΓ = g(θ). By Theorem 1, sinceS2 has area 4π, the average ofµ(e) overS2 is less than 2, and it follows that thereexists a set of positive measure ofe0 ∈ S2 with µ(e0) < 2. Sinceµ(e0) is a half-integer, and sinceµ(e) ≥ 3/2, as we have shown in part(1) of this proof, we haveµ(e0) = 3/2 exactly.

From Corollary 10 applied tope0 ◦ g : θ → R, we findµg(e0) = 12(Λ + V) + k

4,whereΛ is the number of local maximum points,V is the number of local minimumpoints andk = 2 is the number of vertices, both of degree 3. Thus,3

2 =12(Λ+V)+ 1

2,so thatΛ + V = 2. This implies that the local maximum/minimum points areunique, and must be the unique global maximum/minimum pointspmax and pmin

(which may be one of the two verticesq±). Thenpe0 ◦ g is monotone along edgesexcept at the pointspmax, pmin andq±.

Introduce Euclidean coordinates (x, y, z) for R3 so thate0 is in the increasingz-direction. Writetmax = pe0 ◦ g(pmax) = 〈e0, pmax〉 and tmin = 〈e0, pmin〉 for themaximum and minimum values ofzalongg(θ). Write t± for the value ofzatg(q±),where we may assumetmin ≤ t− < t+ ≤ tmax.

We construct a “model” standardθ-curveΓ in the (x, z)-plane, as follows.Γ willconsist of a circleC plus the straight chord ofC, joining q− to q+ (points to bechosen). ChooseC so that the maximum and minimum values ofzonC equaltmax

andtmin. Write pmax resp. pmin for the maximum and minimum points ofz alongC. Chooseq+ as a point onC wherez = t+. There may be two nonequivalentchoices forq− as a point onC wherez= t−: we choose so thatpmax and pmin are inthe same or different topological edge ofΓ, wherepmax andpmin are in the same ordifferent topological edge, resp., ofΓ. Note that there is a homeomorphism fromg(θ) to Γ which preservesz.

We now proceed to extend this homeomorphism to an isotopy. For t ∈ R, writePt for the plane{z = t}. As in the proof of Proposition 3, there is a continuous 1-parameter family of homeomorphismsΦt : Pt → Pt such thatΦt(Γ ∩ Pt) = Γ∩ Pt;


Φt is the identity outside a compact subset ofPt; andΦt is isotopic to the identityof Pt, uniformly with respect tot. DefiningΦ : R3 → R3 byΦ(x, y, z) := Φz(x, y),we have an isotopy ofΓ with the model graphΓ.

(4) Finally, consider an embeddingg : θ → R3 which is onlycontinuous, andwrite Γ = g(θ).

It follows from Theorem 3 that for anyθ-graphΓ of finite net total curvature,there is aΓ-approximating polygonalθ-graph P isotopic toΓ, with NTC(P) ≤NTC(Γ) and as close as desired to NTC(Γ).

If a θ-graphΓ would have NTC(Γ) < 3π, then theΓ-approximating polygonalgraphP would also have NTC(P) < 3π, in contradiction to what we have shownfor piecewiseC2 theta graphs in part(1) above. This shows that NTC(Γ) ≥ 3π.

If equality NTC(Γ) = 3π holds, then NTC(P) ≤ NTC(Γ) = 3π, so that bythe equality case part(2) above, NTC(P) must equal 3π, andP must be a convexplanar curve plus a chord. But this holds forall Γ-approximating polygonal graphsP, implying thatΓ itself must be a convex planar curve plus a chord.

Finally, If NTC(Γ) < 4π, then NTC(P) < 4π, implying by part(3) above thatPis isotopic to the standardθ-graph. ButΓ is isotopic toP, and hence is isotopicallystandard.

References

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Math. Z.253(2006), 315–331.[GY2] R. Gulliver and S. Yamada,Total Curvature and isotopy of graphs in R3, arXiv:0806.0406.[Go] H. Goda,Bridge index for theta curves in the 3-sphere, Topology Applic.79 (1997), 177-196.[Ki] S. Kinoshita,On elementary ideals of polyhedra in the3-sphere, Pacific J. Math.42 (1972), 89-98.[L] Lickorish, W.B.R., An Introduction to Knot Theory.Graduate Texts in Mathematics175. Springer, 1997.

Springer, 1971.[M] J. Milnor, On the total curvature of knots, Annals of Math.52 (1950), 248–257.[R] T. Rado,On the Problem of Plateau. Springer, 1971.[vR] A. C. M. van Rooij,The total curvature of curves, Duke Math. J.32, 313–324 (1965).[S] K. Stephenson, Circle packing: a mathematical tale Notices AMS,50 No.11, 1376–1388 (2003).[T] K. Taniyama,Total curvature of graphs in Euclidean spaces. Differential Geom. Appl.8 (1998), 135–155.

Robert Gulliver Sumio YamadaSchool of Mathematics Mathematical InstituteUniversity of Minnesota Tohoku UniversityMinneapolis MN 55414 Aoba, Sendai, Japan 980-8578

http://arxiv.org/abs/0806.0406


[email protected] [email protected]

www.math.umn.edu/˜gulliver www.math.tohoku.ac.jp/˜yamada

arxiv:1101.2305v1 [math.dg] 12 jan 2011

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