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TOTAL CURVATURE OF GRAPHS AFTER MILNOR AND EULER ROBERTGULLIVERANDSUMIOYAMADA 1 1 0 Abstract. Wedefineanewnotionoftotalcurvature,callednettotalcurvature,forfinite 2 graphsembeddedinRn,andinvestigateitsproperties. Twoguidingprinciplesaregiven n byMilnor’swayofmeasuringthelocalcrookednessofaJordancurveviaaCrofton-type a formula,andbyconsideringthedoublecoverofagivengraphasanEuleriancircuit. The J strengthofcombiningtheseideasindefiningthecurvaturefunctionalis(1)itallowsusto interpretthesingular/non-eulideanbehaviorattheverticesofthegraphasasuperposition 2 ofvertices ofa1-dimensional manifold, andthus (2)one cancompute thetotal curva- 1 ture forawiderange ofgraphs bycontrasting local and global properties ofthegraph utilizingtheintegralgeometricrepresentationofthecurvature. Acollectionofresultson ] G upper/lowerboundsofthetotalcurvatureonisotopy/homeomorphismclassesofembed- dingsispresented,whichinturndemonstratestheeffectivenessofnettotalcurvatureasa D newfunctional . measuringcomplexityofspatialgraphsindifferential-geometricterms. h t a m [ 1. INTRODUCTION:CURVATUREOFAGRAPH 1 The celebrated Fa´ry-Milnor theorem states that a curve in Rn of total curvature v atmost4πisunknotted. 5 0 As a key step in his 1950 proof, John Milnor showed that for a smooth Jordan 3 curveΓinR3,thetotalcurvatureequalshalftheintegralovere ∈S2ofthenumber 2 µ(e)oflocalmaximaofthelinear“height”functionhe,·ialongΓ[M]. Thisequality . 1 canbe regarded asaCrofton-type representation formula oftotal curvature where 0 the order of integrations over the curve and the unit tangent sphere (the space of 1 1 directions) are reversed. The Fa´ry-Milnor theorem follows, since total curvature : less than 4π implies there isaunit vector e ∈ S2 sothat he ,·ihas aunique local v 0 0 i maximum, and therefore that this linear function is increasing on an interval of Γ X and decreasing on the complement. Without changing the pointwise value of this ar “height”function,Γcanbetopologically untwistedtoastandardembeddingofS1 intoR3. TheFenchel theorem, that anycurveinR3 hastotal curvature atleast 2π, also follows from Milnor’s key step, since for all e ∈ S2, the linear function he,·i assumes its maximum somewhere along Γ, implying µ(e) ≥ 1. Milnor’s proof is independent of the proof of Istvan Fa´ry, published earlier, which takes a different approach [Fa]. We would like to extend the methods of Milnor’s seminal paper, replacing the simple closed curve by a finite graph Γ in R3. Γ consists of a finite number of points, the vertices, and a finite number of simple arcs, the edges, each of which has as its endpoints one or two of the vertices. We shall assume Γ is connected. Date:December31,2010. SupportedinpartbyJSPSGrant-in-aidforScientificResearchNo.17740030. ThankstotheKoreaInstituteforAdvancedStudyforinvitations. 1 2 ROBERTGULLIVERANDSUMIOYAMADA Thedegree ofavertexqisthenumberd(q)ofedgeswhichhaveqasanendpoint. (Anotherwordfordegreeis“valence”.) Weremarkthatitistechnicallynotneeded that the dimension n of the ambient space equals three. All the arguments can be generalized tohigher dimensions, although inhigher dimensions (n ≥ 4)thereare nonontrivial knots. Moreover, anytwohomeomorphic graphsareisotopic. The key idea in generalizing total curvature for knots to total curvature for graphs is to consider the Euler circuits of the given graph, namely, parameteri- zations by S1, of the double cover of the graph. We note that given a graph of even degree, there can be several Euler circuits, or ways to “trace it without lift- ing the pen.” A topological vertex of a graph of degree d is a singularity, in that thegraph isnotlocally Euclidean. Howeverbyconsidering anEulercircuit ofthe double 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 sum of those measures is well-defined, independent of the choice of the Euler circuit of the double cover. Wedefine(Definition 1)the nettotal curvature (NTC)of apiecewiseC2 graph to bethesumofthetotalcurvature ofthesmootharcsandthecontributions fromthe verticesasdescribed. This notion of net total curvature is substantially different from the total curvature, denoted TC, as defined by Taniyama [T]. (Taniyama writes τ for TC.) See section2below. This is consistent with known results for the vertices of degree d = 2; with vertices of degree three or more, this definition helps facilitate a new Crofton- type representation formula (Theorem 1) for total curvature of graphs, where the total curvature is represented as an integral over the unit sphere. Recall that the vertex is now seen as d distinct points on an Euler circuit. The way we pick up thecontribution ofthetotalcurvatureattheverticesidentifiestheddistinctpoints, and thus the 2d unit tangent spheres on a circuit. As Crofton’s formula in effect reversestheorderofintegrations—oneoverthecircuit,theotheroverthespaceof tangent directions —the sum ofthed exterior angles atthevertex isincorporated intheintegral overtheunitsphere. Ontheother handtheintegrand oftheintegral overtheunit sphere counts the number ofnetlocal maximaofthe height function alonganaxis,wherenetlocalmaximummeansthenumberoflocalmaximaminus thenumber oflocal minimaat these d points of theEuler circuit. Thisestablishes a correspondence between the differential geometric quantity (net total curvature) andthedifferentialtopological quantity(averagenumberofmaxima)ofthegraph, asstatedinTheorem1below. Insection 2,wecompareseveral definitions fortotalcurvature ofgraphs which have appeared in the recent literature. In section 3, we introduce the main tool (Lemma1)whichinasensereducesthecomputationofNTCtocountingintersec- tionswithplanes. Milnor’s treatment [M] of total curvature also contained an important topological extension. Namely, in order to define total curvature, the knot needs only to becontinuous. Thismakesthetotalcurvature ageometricquantity definedonany TOTALCURVATUREOFGRAPHSAFTERMILNORANDEULER 3 homeomorphicimageofS1. Inthisarticle,wefirstdefinenettotalcurvature(Def- inition 1) on piecewise C2 graphs, and then extend the definition to continuous graphs (Definition 3.) In analogy to Milnor, we approximate a given continuous graphbyasequenceofpolygonalgraphs. Inshowingthemonotonicity ofthetotal curvature (Proposition 2) under the refining process of approximating graphs we useourrepresentation formula(Theorem1)applied tothepolygonal graphs. Consequently the Crofton-type representation formula is also extended (Theo- rem2)tocovercontinuous graphs. Additionally, weareabletoshowthatcontinu- ous graphs with finite total curvature (NTC or TC) are tame. We say that a graph istamewhenitisisotopictoanembedded polyhedral graph. In sections 5through 8, wecharacterize NTCwith respect tothe geometry and the topology of the graph. Proposition 5 shows the subadditivity of NTC under the union of graphs which meet in a finite set. In section 6, the concept of bridge numberisextended fromknotstographs, intermsofwhichtheminimumofNTC can 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 width ofanisotopyclass. TheinfimumofNTCiscomputedforspecificgraphtypes: the two-vertexgraphsθ ,the“ladder” L ,the“wheel”W ,thecompletegraphK on m m m m mverticesandthecompletebipartite graph K . m,n Finally weprovearesult (Theorem 7)whichgives aFenchel type lowerbound (≥ 3π) for total curvature of a theta graph (an image of the graph consisting of a circle with an arc connecting a pair of antipodal points), and a Fa´ry-Milnor type upperbound(< 4π)toimplythethetagraphisisotopictothestandardembedding. AsimilarresultwasgivenbyTaniyama[T],referringtoTC.Incontrast,forgraphs of the type of K (m ≥ 4), the infimum of NTC in the isotopy class of a polygon m onmverticesisalsotheinfimumforasequence ofdistinct isotopyclasses. Many of the results in our earlier preprint [GY2] have been incorporated into thepresent paper. Wethank YuyaKodaforhiscomments regarding Proposition 7,andJaigyoung Choe and Rob Kusner for their comments about Theorem 7, especially about the sharpcaseNTC(Γ) = 3πofthelowerboundestimate. 2. DEFINITIONSOFTOTALCURVATURE The first difficulty, in extending the results of Milnor’s classic paper, is to un- derstand thecontribution tototalcurvature atavertexofdegree d(q) ≥ 3. Wefirst considerthewell-knowncase: DefinitionofTotalCurvatureforKnots ForasmoothclosedcurveΓ,thetotalcurvature is C(Γ)= |~k|ds, ZΓ where s denotes arc length along Γ and ~k is the curvature vector. If x(s) ∈ R3 denotes the position of the point measured at arc length s along the curve, then 4 ROBERTGULLIVERANDSUMIOYAMADA ~k = d2x. For a piecewise smooth curve, that is, a graph with vertices q ,...,q ds2 1 N havingalwaysdegreed(q) = 2,thetotalcurvature isreadilygeneralized to i N (1) C(Γ)= c(q)+ |~k|ds, i Xi=1 ZΓreg wheretheintegralistakenovertheseparateC2edgesofΓwithouttheirendpoints; and where c(q) ∈ [0,π] is the exterior angle formed by the two edges of Γ which i meet at q. That is, cos(c(q)) = hT ,−T i, where T = dx(q+) and T = −dx(q−) i i 1 2 1 ds i 2 ds i aretheunittangentvectorsatq pointing intothetwoedgeswhichmeetatq. The i i exterioranglec(q)isthecorrectcontributiontototalcurvature,sinceanysequence i ofsmooth curves converging toΓinC0,withC1 convergence oncompact subsets of each open edge, includes a small arc near q along which the tangent vector i changes from near dx(q−) to near dx(q+). The greatest lower bound of the contri- ds i ds i bution to total curvature of this disappearing arc along the smooth approximating curvesequalsc(q). i NotethatC(Γ)iswelldefinedforanimmersedknotΓ. DefinitionsofTotalCurvatureforGraphs WhenweturnourattentiontoagraphΓ,wefindtheabovedefinitionforcurves (degree d(q) = 2) does not generalize in any obvious way to higher degree (see [G]). The ambiguity of the general formula (1) is resolved if we specify the re- placement for c(0) when Γ is the cone over a finite set {T ,...,T } in the unit 1 d sphereS2. The earliest notion of total curvature of a graph appears in the context of the first variation of length of a graph, which we call variational total curvature, and is called the mean curvature of the graph in [AA]: we shall write VTC. The contribution toVTCatavertexqofdegree2,withunittangent vectorsT andT , 1 2 isvtc(q) = |T +T | = 2sin(c(q)/2). Atanon-straight vertexqofdegree2,vtc(q) 1 2 is less than the exterior angle c(q). For a vertex of degree d, the contribution is vtc(q) = |T +···+T |. 1 d ArathernaturaldefinitionoftotalcurvatureofgraphswasgivenbyTaniyamain [T]. Wehavecalledthismaximaltotalcurvature TC(Γ)in[G]. Thecontribution tototalcurvature atavertexqofdegreed is tc(q) := arccoshT ,−T i. i j 1≤Xi<j≤d In the case d(q) = 2, the sum above has only one term, the exterior angle c(q) at q. SincethelengthoftheGaussimageofacurveinS2 isthetotalcurvatureofthe curve, tc(q) maybe interpreted asadding totheGauss imageinRP2 ofthe edges, a complete great-circle graph on T (q),...,T (q), for each vertex q of degree d. 1 d Notethat theedge between twovertices does notmeasure thedistance inRP2 but itssupplement. TOTALCURVATUREOFGRAPHSAFTERMILNORANDEULER 5 Inourearlierpaper[GY1]onthedensityofanarea-minimizingtwo-dimensional rectifiable set Σ spanning Γ, we found that it was very useful to apply the Gauss- Bonnet formula to the cone over Γ with a point p of Σ as vertex. The relevant notion of total curvature in that context is cone total curvature CTC(Γ), defined usingctc(q)asthereplacement forc(q)inequation(1): d π (2) ctc(q) := sup −arccoshT ,ei .  2 i  e∈S2Xi=1 (cid:18) (cid:19)   Note that in the case d(q) = 2, the supremum above is assumed at vectors elying in the smaller angle between the tangent vectors T and T to Γ, so that ctc(q) 1 2 is then the exterior angle c(q) at q. The main result of [GY1] is that 2π times the area density of Σ at any of its points is at most equal to CTC(Γ). The same resulthadbeen provenbyEckholm, WhiteandWienholtz forthecase ofasimple closed curve [EWW]. Taking Σ to be the branched immersion of the disk given by Douglas [D1] and Rado´ [R], it follows that if C(Γ) ≤ 4π, then Σ is embedded, and therefore Γ is unknotted. Thus [EWW] provided an independent proof of the Fa´ry-Milnor theorem. However, CTC(Γ) may be small for graphs which are far fromthesimplestisotopytypesofagraphΓ. Inthis paper, weintroduce the notion ofnettotal curvature NTC(Γ),which is the appropriate definition for generalizing — to graphs — Milnor’s approach to isotopy andtotal curvature ofcurves. Foreachunit tangent vectorT atq, 1 ≤ i ≤ i d = d(q), let χ : S2 → {−1,+1} be equal to −1 on the hemisphere with center at i T , and +1 on the opposite hemisphere (modulo sets of zero Lebesgue measure). i Wethendefine + d 1 (3) ntc(q) := χ(e) dA (e). 4ZS2Xi=1 i  S2 Wenotethatthefunction d χ(e)isodd,hencethequantityabovecanbewritten i=1 i as P d 1 ntc(q) := χ(e) dA (e). 8ZS2(cid:12)(cid:12)(cid:12)Xi=1 i (cid:12)(cid:12)(cid:12) S2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) as well. In the case d(q) = 2, the integ(cid:12)rand of ((cid:12)3) is positive (and equals 2) only on the set of unit vectors e which have negative inner products with both T and 1 T , ignoring e in sets of measure zero. This set is bounded by semi-great circles 2 orthogonaltoT andtoT ,andhassphericalareaequaltotwicetheexteriorangle. 1 2 So in this case, ntc(q) is the exterior angle. Thus, in the special case where Γ is a piecewise smooth curve, the following quantity NTC(Γ) coincides with total curvature, aswellaswithTC(Γ)andCTC(Γ): 6 ROBERTGULLIVERANDSUMIOYAMADA Definition 1. We define the net total curvature of a piecewise C2 graph Γ with vertices{q ,...,q }as 1 N N (4) NTC(Γ) := ntc(q)+ |~k|ds. i Xi=1 ZΓreg Forthesakeofsimplicity,elsewhereinthispaper,weconsidertheambientspace tobeR3. Howeverthedefinition ofthenettotalcurvature canbegeneralized fora graphinRn bydefiningthevertexcontribution intermsofanaverage overSn−1: + d ntc(q) := π χ(e) dA (e) , ?  i  Sn−1 whichisconsistent withthedefi(cid:16)nitSion−n1(X3i=)1ofntcwhenn = 3.(cid:17) RecallthatMilnor[M]definesthetotalcurvatureofacontinuous simpleclosed curveC asthesupremum ofthetotal curvature ofallpolygons inscribed inC. By analogy, we define net total curvature of a continuous graph Γ to be the supre- mumofthenettotal curvature ofallpolygonal graphs Psuitably inscribed inΓas follows. Definition 2. For a given continuous graph Γ, we say a polygonal graph P ⊂ R3 isΓ-approximating, provided thatitstopological vertices(thoseofdegree, 2)are exactly the topological vertices of Γ, and having the same degrees; and that the arcsof Pbetween twotopological vertices correspond one-to-one tothe edges of Γbetweenthosetwovertices. Note that if P is a Γ-approximating polygonal graph, then P is homeomorphic to Γ. According to the statement of Proposition 2, whose proof will be given in the next section, if P and P are Γ-approximating polygonal graphs, and P is a refinement of P, then NTC(P) ≥ NTC(P). Here P is said to be a refinement of e e P provided the set of vertices of P is a subset of the vertices of P. Assuming e e Proposition2forthemoment,wecangeneralizethedefinitionofthetotalcurvature e tonon-smooth graphs. Definition3. Definethenettotalcurvature ofacontinuous graphΓby NTC(Γ):= supNTC(P) P wherethesupremumistakenoverallΓ-approximating polygonal graphs P. Forapolygonal graph P,applying Definition1, N NTC(P) := ntc(q), i Xi=1 whereq ,...,q arethevertices ofP. 1 N Definition 3 is consistent with Definition 1 in the case of a piecewise C2 graph Γ. Namely,asMilnorshowed,thetotalcurvatureC(Γ )ofasmoothcurveΓ isthe 0 0 supremum of the total curvature ofinscribed polygons ([M], p. 251), which gives therequiredsupremumforeachedge. Atavertexqofthepiecewise-C2 graphΓ,as TOTALCURVATUREOFGRAPHSAFTERMILNORANDEULER 7 a sequence P of Γ-approximating polygons become arbitrarily fine, a vertex q of k P (and ofΓ)has unit tangent vectors converging inS2 tothe unittangent vectors k toΓatq. Itfollowsthatfor1 ≤ i≤ d(q),χPk → χΓ inmeasureonS2,andtherefore i i ntc (q) → ntc (q). Pk Γ 3. CROFTON-TYPEREPRESENTATIONFORMULAFORTOTAL CURVATURE WewouldliketoexplainhowthenettotalcurvatureNTC(Γ)ofagraphisrelated tomorefamiliarnotionsoftotalcurvature. RecallthatagraphΓhasanEulercircuit if and only if its vertices all have even degree, by a theorem of Euler. An Euler circuitisaclosed,connected pathwhichtraverseseachedgeofΓexactlyonce. Of course,wedonothavethehypothesisofevendegree. Wecanattainthathypothesis bypassingtothedoubleΓofΓ: ΓisthegraphwiththesameverticesasΓ,butwith two copies of each edge of Γ. Then at each vertex q, the degree as a vertex of Γ e e is d(q) = 2d(q), which is even. By Euler’s theorem, there is an Euler circuit Γ′ e of Γ, which may be thought of as a closed path which traverses each edge of Γ e exactlytwice. Nowateachofthepoints{q ,...,q }alongΓ′ whicharemappedto 1 d e q ∈ Γ,wemayconsider theexterior angle c(q). Thesum ofthese exterior angles, i however, depends on the choice of the Euler circuit Γ′. For example, if Γ is the unionofthe x-axisandthey-axisinEuclideanspaceR3,thenonemightchooseΓ′ tohavefourrightangles, ortohavefourstraight angles, orsomething inbetween, with completely different values of total curvature. In order to form a version of totalcurvatureatavertexqwhichonlydependsontheoriginalgraphΓandnoton thechoiceofEulercircuitΓ′,itisnecessarytoconsidersomeoftheexteriorangles aspartially balancing others. Intheexamplejust considered, whereΓistheunion of two orthogonal lines, two opposing right angles will be considered to balance each other completely, so that ntc(q) = 0, regardless ofthe choice ofEuler circuit ofthedouble. It will become apparent that the connected character of an Euler circuit of Γ is notrequiredforwhatfollows. Instead,weshallrefertoaparameterizationΓ′ofthe e double Γ, which is a mapping from a 1-dimensional manifold without boundary, notnecessarily connected; themappingisassumedtocovereachedgeofΓonce. e The nature of ntc(q) is clearer when it is localized on S2, analogously to [M]. e Inthecased(q) = 2,Milnorobserved thattheexteriorangle atthevertexqequals halftheareaofthosee∈ S2suchthatthelinearfunctionhe,·i,restrictedtoΓ,hasa localmaximumatq. Inourcontext,wemaydescribentc(q)asone-halftheintegral overthesphere ofthenumber ofnetlocal maxima, whichishalf thedifference of local maxima and local minima. Along the parameterization Γ′ of the double of Γ, the linear function he,·i may have a local maximum at some of the vertices q ,...,q over q, and may have a local minimum at others. In our construction, 1 d each local minimum balances against one local maximum. Ifthere are more local minimathanlocalmaxima,thenumbernlm(e,q),thenetnumberoflocalmaxima, willbenegative;however,ourdefinitionusesonlythepositivepart[nlm(e,q)]+. 8 ROBERTGULLIVERANDSUMIOYAMADA Weneedtoshowthat [nlm(e,q)]+dA (e) S2 ZS2 is independent of the choice of parameterization, and in fact is equal to 2ntc(q); thiswillfollowfromanother wayofcomputing nlm(e,q)(seeCorollary 2below). Definition4. Letaparameterization Γ′ofthedoubleof Γbegiven. Thenavertex qof Γcorresponds to anumber ofvertices q ,...,q of Γ′, whered is thedegree 1 d d(q)ofqasavertexof Γ. Choosee∈ S2. Ifq ∈Γisalocalextremumofhe,·i,then weconsiderqasavertexofdegreed(q) = 2. Letlmax(e,q)bethenumberoflocal maximaofhe,·ialong Γ′atthepointsq ,...,q overq,andsimilarlyletlmin(e,q) 1 d bethenumberoflocalminima. Wedefinethenumber ofnetlocalmaximaofhe,·i atqtobe 1 nlm(e,q) = [lmax(e,q)−lmin(e,q)] 2 . Remark 1. The definition of nlm(e,q) appears to depend not only on Γ but on a choiceoftheparameterizationΓ′ofthedoubleof Γ: lmax(e,q)andlmin(e,q)may depend on the choice of Γ′. However, we shall see in Corollary 1 below that the numberofnetlocalmaximanlm(e,q)isinfactindependent of Γ′. Remark 2. We have included the factor 1 in the definition of nlm(e,q) in order 2 to agree with the difference of the numbers of local maxima and minima along a parameterization of Γitself,ifd(q)iseven. Weshallassumefortherestofthissectionthataunitvectorehasbeenchosen, andthatthelinear“height”functionhe,·ihasonlyafinitenumberofcriticalpoints along Γ; this excludes e belonging to a subset of S2 of measure zero. We shall alsoassumethatthegraphΓissubdividedtoincludeamongtheverticesallcritical points of the linear function he,·i, with degree d(q) = 2 if q is an interior point of oneofthetopological edgesofΓ. Definition 5. Choose a unit vector e. Ata point q ∈ Γ of degree d = d(q), let the up-degreed+ = d+(e,q)bethenumberofedgesof Γwithendpointqonwhichhe,·i isgreater(“higher”) thanhe,qi,the“height” ofq. Similarly, letthedown-degree d−(e,q) be the number of edges along which he,·iis less than its value at q. Note thatd(q) = d+(e,q)+d−(e,q),foralmostalleinS2. Lemma1. (CombinatorialLemma)Forallq∈ Γandfora.a. e∈ S2,nlm(e,q) = 1[d−(e,q)−d+(e,q)]. 2 Proof.LetaparameterizationΓ′ofthedoubleofΓbechosen,withrespecttowhich lmax(e,q)andlmin(e,q)aredefined. Recalltheassumptionabove,thatΓhasbeen subdivided sothatalongeachedge,thelinearfunction he,·iisstrictlymonotone. Consider a vertex q of Γ, of degree d = d(q). Then Γ′ has 2d edges with an endpoint among the points q ,...,q which are mapped to q ∈ Γ. On 2d+, resp. 1 d 2d− of these edges, he,·i is greater resp. less than he,qi. But for each 1 ≤ i ≤ d, theparameterization Γ′ hasexactly twoedges whichmeetatq. Depending onthe i TOTALCURVATUREOFGRAPHSAFTERMILNORANDEULER 9 up/down character of the two edges of Γ′ which meet at q, 1 ≤ i ≤ d, we can i count: (+) If he,·i is greater than he,qi on both edges, then q is a local minimum point; i therearelmin(e,q)oftheseamongq ,...,q . 1 d (-)Ifhe,·iislessthanhe,qionbothedges,thenq isalocalmaximumpoint;there i arelmax(e,q)ofthese. (0) In all remaining cases, the linear function he,·i is greater than he,qi along one edge and less along the other, in which case q is not counted in computing i lmax(e,q)norlmax(e,q);thereared(q)−lmax(e,q)−lmin(e,q)ofthese. Nowcounttheindividual edgesofΓ′: (+)Thereare lmin(e,q)pairsofedges, eachofwhichispartofalocalminimum, both of which are counted among the 2d+(e,q) edges of Γ′ with he,·igreater than he,qi. (-)Thereare lmax(e,q)pairsofedges, eachofwhichispartofalocalmaximum; these are counted among the number 2d−(e,q) of edges of Γ′ with he,·i less than he,qi. Finally, (0)thereared(q)−lmax(e,q)−lmin(e,q)edgesofΓ′ whicharenotpartofalocal maximumorminimum,withhe,·igreaterthanhe,qi;andanequalnumberofedges withhe,·ilessthanhe,qi. Thus,thetotalnumberoftheseedgesofΓ′ withhe,·igreaterthanhe,qiis 2d+ = 2lmin+(d−lmax−lmin) = d+lmin−lmax. Similarly, 2d− = 2lmax+(d−lmax−lmin)= d+lmax−lmin. Subtracting givestheconclusion: lmax(e,q)−lmin(e,q) d−(e,q)−d+(e,q) nlm(e,q) := = . 2 2 Corollary 1. The number of net local maxima nlm(e,q) is independent of the choiceofparameterization Γ′ ofthedoubleofΓ. Proof. Given a direction e ∈ S2, the up-degree and down-degree d±(e,q) at a vertexq∈ Γaredefinedindependently ofthechoiceofΓ′. + Corollary2. Foranyq∈ Γ,wehaventc(q) = 1 nlm(e,q) dA . 2 S2 S2 R h i Proof. Consider e ∈ S2. In the definition (3) of ntc(q), χ(e) = ±1 whenever i ±he,T i < 0. Butthenumberof1 ≤ i≤ d with±he,T i < 0equalsd∓(e,q),sothat i i d χ(e) = d−(e,q)−d+(e,q) = 2nlm(e,q) i Xi=1 byLemma1,foralmostalle ∈ S2. Definition6. ForagraphΓinR3 ande∈ S2,definethemultiplicity ateas µ(e)= µ (e) = {nlm+(e,q) :qavertexof Γoracriticalpointof he,·i}. Γ X 10 ROBERTGULLIVERANDSUMIOYAMADA Note that µ(e) is a half-integer. Note also that in the case when Γ is a knot, or equivalently, whend(q) ≡ 2,µ(e)isexactlytheintegerµ(Γ,e),thenumberoflocal maximaofhe,·ialongΓasdefinedin[M],p. 252. Corollary 3. Foralmost alle ∈ S2 and foranyparameterization Γ′ ofthedouble ofΓ,µΓ(e) ≤ 21µΓ′(e). Proof.WehaveµΓ(e) = 12 q[lmaxΓ′(e,q)−lminΓ′(e,q)],≤ 21 qlmaxΓ′(e,q) = 12µΓ′. P P If, in place of the positive part, we sum nlm(e,q) itself over q located above a planeorthogonal toe,wefindausefulquantity: Corollary4. Foralmostall s ∈ Randalmostalle ∈ S2, 0 2 {nlm(e,q) :he,qi > s } = #(e,s ), 0 0 X thecardinality ofthefiber{p ∈ Γ :he,pi = s }. 0 Proof. If s > max he,pi, then #(e,s ) = 0. Now proceed downward, using 0 p∈Γ 0 Lemma1byinduction. NotethatthefibercardinalityofCorollary4isalsothevalueobtainedforknots, wherethemoregeneralnlmmaybereplaced bythenumberoflocalmaxima[M]. Remark 3. In analogy with Corollary 4, we expect that an appropriate general- izationofNTCtocurvedpolyhedral complexes ofdimension ≥ 2willinthefuture allow computation of the homology of level sets and sub-level sets of a (generalized)Morsefunction intermsofageneralization ofnlm(e,q). Corollary5. Themultiplicityofagraphindirectione∈ S2mayalsobecomputed asµ(e) = 1 |nlm(e,q)|. 2 q∈Γ Proof.ItPfollowsfrom Corollary 4with s < min he,·ithat nlm(e,q) = 0, 0 Γ q∈Γ which is the difference of positive and negative parts. The sum of these parts is P |nlm(e,q)| = 2µ(e). q∈Γ P ItwasshowninTheorem3.1of[M]that,inthecaseofknots,C(Γ)= 1 µ(e)dA , 2 S2 S2 where Milnor refers to Crofton’s formula. We may now extend thisRresult to graphs: Theorem1. Fora(piecewiseC2)graphΓmappedintoR3,thenettotalcurvature hasthefollowing representation: 1 NTC(Γ) = µ(e)dA (e). 2ZS2 S2 Proof. We have NTC(Γ) = N ntc(q ) + |~k|ds, where q ,...,q are j=1 j Γ 1 N reg the vertices of Γ, including locaPl extrema as veRrtices of degree d(q ) = 2, and j + wherentc(q) := 1 d χ(e) dA (e)by the definition (3)of ntc(q). Apply- 4 S2 i=1 i S2 ing Milnor’s resultRtohPeach C2 eidge, we have C(Γ ) = 1 µ (e)dA . But reg 2 S2 Γreg S2 R