# Total Curvature of Graphs after Milnor and Euler

###### Abstract.

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in , and investigate its properties. Two guiding principles are given by Milnor’s way of measuring the local crookedness of a Jordan curve via a Crofton-type formula, and by considering the double cover of a given graph as an Eulerian circuit. The strength of combining these ideas in defining the curvature functional is (1) it allows us to interpret the singular/non-eulidean behavior at the vertices of the graph as a superposition of vertices of a -dimensional manifold, and thus (2) one can compute the total curvature for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the curvature. A collection of results on upper/lower bounds of the total curvature on isotopy/homeomorphism classes of embeddings is presented, which in turn demonstrates the effectiveness of net total curvature as a new functional

measuring complexity of spatial graphs in differential-geometric terms.

## 1. Introduction: Curvature of a Graph

The celebrated Fáry-Milnor theorem states that a curve in of total curvature at most is unknotted.

As a key step in his 1950 proof, John Milnor showed that for a smooth Jordan curve in , the total curvature equals half the integral over of the number of local maxima of the linear “height” function along [M]. This equality can be regarded as a Crofton-type representation formula of total curvature where the order of integrations over the curve and the unit tangent sphere (the space of directions) are reversed. The Fáry-Milnor theorem follows, since total curvature less than implies there is a unit vector so that has a unique local maximum, and therefore that this linear function is increasing on an interval of and decreasing on the complement. Without changing the pointwise value of this “height” function, can be topologically untwisted to a standard embedding of into . The Fenchel theorem, that any curve in has total curvature at least , also follows from Milnor’s key step, since for all , the linear function assumes its maximum somewhere along , implying . Milnor’s proof is independent of the proof of Istvan Fáry, 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 in . 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 is connected. The degree of a vertex is the number of edges which have as an endpoint. (Another word for degree is “valence”.) We remark that it is technically not needed that the dimension of the ambient space equals three. All the arguments can be generalized to higher dimensions, although in higher dimensions there are no nontrivial knots. Moreover, any two homeomorphic graphs are isotopic.

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, parameterizations by , 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 lifting the pen.” A topological vertex of a graph of degree is a singularity, in that the graph is not locally Euclidean. However by considering an Euler circuit of the double of the graph, the vertex becomes locally the intersection point of 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. We define (Definition 1) the net total curvature (NTC) of a piecewise graph to be the sum of the total curvature of the smooth arcs and the contributions from the vertices as described.

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

This is consistent with known results for the vertices of degree ; 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 distinct points on an Euler circuit. The way we pick up the contribution of the total curvature at the vertices identifies the distinct points, and thus the unit tangent spheres on a circuit. As Crofton’s formula in effect reverses the order of integrations — one over the circuit, the other over the space of tangent directions — the sum of the exterior angles at the vertex is incorporated in the integral over the unit sphere. On the other hand the integrand of the integral over the unit sphere counts the number of net local maxima of the height function along an axis, where net local maximum means the number of local maxima minus the number of local minima at these points of the Euler circuit. This establishes a 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 which have 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 intersections with planes.

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 be continuous. This makes the total curvature a geometric quantity defined on any homeomorphic image of . In this article, we first define net total curvature (Definition 1) on piecewise graphs, and then extend the definition to continuous graphs (Definition 3.) In analogy to Milnor, we approximate a given continuous graph by a sequence of polygonal graphs. In showing the monotonicity of the total curvature (Proposition 2) under the refining process of approximating graphs we use our representation formula (Theorem 1) applied to the polygonal graphs.

Consequently the Crofton-type representation formula is also extended (Theorem 2) to cover continuous graphs. Additionally, we are able to show that continuous graphs with finite total curvature (NTC or TC) are tame. We say that a graph is tame when it is isotopic to an embedded polyhedral graph.

In sections 5 through 8, we characterize NTC with respect to the 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 number is extended from knots to graphs, in terms of which the minimum of NTC can be explicitly computed, provided the graph has at most one vertex of degree . In section 7, Theorem 6 gives a lower bound for NTC in terms of the width of an isotopy class. The infimum of NTC is computed for specific graph types: the two-vertex graphs , the “ladder” , the “wheel” , the complete graph on vertices and the complete bipartite graph .

Finally we prove a result (Theorem 7) which gives a Fenchel type lower bound 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 Fáry-Milnor type upper bound 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 graphs of the type of , the infimum of NTC in the isotopy class of a polygon on vertices is also the infimum for a sequence of distinct isotopy classes.

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

## 2. Definitions of Total Curvature

The first difficulty, in extending the results of Milnor’s classic paper, is to understand the contribution to total curvature at a vertex of degree . We first consider the well-known case:

Definition of Total Curvature for Knots

For a smooth closed curve , the total curvature is

where denotes arc length along and is the curvature vector. If denotes the position of the point measured at arc length along the curve, then . For a piecewise smooth curve, that is, a graph with vertices having always degree , the total curvature is readily generalized to

(1) |

where the integral is taken over the separate edges of without their endpoints; and where is the exterior angle formed by the two edges of which meet at . That is, where and are the unit tangent vectors at pointing into the two edges which meet at . The exterior angle is the correct contribution to total curvature, since any sequence of smooth curves converging to in , with convergence on compact subsets of each open edge, includes a small arc near along which the tangent vector changes from near to near . The greatest lower bound of the contribution to total curvature of this disappearing arc along the smooth approximating curves equals .

Note that is well defined for an immersed knot .

Definitions of Total Curvature for Graphs

When we turn our attention to a graph , we find the above definition for curves (degree ) 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 replacement for when is the cone over a finite set in the unit sphere .

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 to VTC at a vertex of degree , with unit tangent vectors and , is . At a non-straight vertex of degree , is less than the exterior angle . For a vertex of degree , the contribution is .

A rather natural definition of total curvature of graphs was given by Taniyama in [T]. We have called this maximal total curvature in [G]. The contribution to total curvature at a vertex of degree is

In the case , the sum above has only one term, the exterior angle at . Since the length of the Gauss image of a curve in is the total curvature of the curve, may be interpreted as adding to the Gauss image in of the edges, a complete great-circle graph on , for each vertex of degree . Note that the edge between two vertices does not measure the distance in but its supplement.

In our earlier paper [GY1] on the density of an area-minimizing two-dimensional rectifiable set spanning , we found that it was very useful to apply the Gauss-Bonnet formula to the cone over with a point of as vertex. The relevant notion of total curvature in that context is cone total curvature , defined using as the replacement for in equation (1):

(2) |

Note that in the case , the supremum above is assumed at vectors lying in the smaller angle between the tangent vectors and to , so that is then the exterior angle at . The main result of [GY1] is that times the area density of at any of its points is at most equal to . The same result had been proven by Eckholm, White and Wienholtz for the case of a simple closed curve [EWW]. Taking to be the branched immersion of the disk given by Douglas [D1] and Radó [R], it follows that if , then is embedded, and therefore is unknotted. Thus [EWW] provided an independent proof of the Fáry-Milnor theorem. However, may be small for graphs which are far from the simplest isotopy types of a graph .

In this paper, we introduce the notion of net total curvature , which is the appropriate definition for generalizing — to graphs — Milnor’s approach to isotopy and total curvature of curves. For each unit tangent vector at , , let be equal to on the hemisphere with center at , and on the opposite hemisphere (modulo sets of zero Lebesgue measure). We then define

(3) |

We note that the function is odd, hence the quantity above can be written as

as well. In the case , the integrand of (3) is positive (and equals 2) only on the set of unit vectors which have negative inner products with both and , ignoring in sets of measure zero. This set is bounded by semi-great circles orthogonal to and to , and has spherical area equal to twice the exterior angle. So in this case, is the exterior angle. Thus, in the special case where is a piecewise smooth curve, the following quantity coincides with total curvature, as well as with and :

###### Definition 1.

We define the net total curvature of a piecewise graph with vertices as

(4) |

For the sake of simplicity, elsewhere in this paper, we consider the ambient space to be . However the definition of the net total curvature can be generalized for a graph in by defining the vertex contribution in terms of an average over :

which is consistent with the definition (3) of when .

Recall that Milnor [M] defines the total curvature of a continuous simple closed curve as the supremum of the total curvature of all polygons inscribed in . By analogy, we define net total curvature of a continuous graph to be the supremum of the net total curvature of all polygonal graphs suitably inscribed in as follows.

###### Definition 2.

For a given continuous graph , we say a polygonal graph is -approximating, provided that its topological vertices (those of degree ) are exactly the topological vertices of , and having the same degrees; and that the arcs of between two topological vertices correspond one-to-one to the edges of between those two vertices.

Note that if is a -approximating polygonal graph, then is homeomorphic to . According to the statement of Proposition 2, whose proof will be given in the next section, if and are -approximating polygonal graphs, and is a refinement of , then . Here is said to be a refinement of provided the set of vertices of is a subset of the vertices of . Assuming Proposition 2 for the moment, we can generalize the definition of the total curvature to non-smooth graphs.

###### Definition 3.

Define the net total curvature of a continuous graph by

where the supremum is taken over all -approximating polygonal graphs .

Definition 3 is consistent with Definition 1 in the case of a piecewise graph . Namely, as Milnor showed, the total curvature of a smooth curve is the supremum of the total curvature of inscribed polygons ([M], p. 251), which gives the required supremum for each edge. At a vertex of the piecewise- graph , as a sequence of -approximating polygons become arbitrarily fine, a vertex of (and of ) has unit tangent vectors converging in to the unit tangent vectors to at . It follows that for , in measure on , and therefore .

## 3. Crofton-Type Representation Formula for Total Curvature

We would like to explain how the net total curvature of a graph is related to more familiar notions of total curvature. Recall that a graph has an Euler circuit if and only if its vertices all have even degree, by a theorem of Euler. An Euler circuit is a closed, connected path which traverses each edge of exactly once. Of course, we do not have the hypothesis of even degree. We can attain that hypothesis by passing to the double of : is the graph with the same vertices as , but with two copies of each edge of . Then at each vertex , the degree as a vertex of is , which is even. By Euler’s theorem, there is an Euler circuit of , which may be thought of as a closed path which traverses each edge of exactly twice. Now at each of the points along which are mapped to , we may consider the exterior angle . The sum of these exterior angles, however, depends on the choice of the Euler circuit . For example, if is the union of the -axis and the -axis in Euclidean space , then one might choose to have four right angles, or to have four straight angles, or something in between, with completely different values of total curvature. In order to form a version of total curvature at a vertex which only depends on the original graph and not on the choice of Euler circuit , it is necessary to consider some of the exterior angles as partially balancing others. In the example just considered, where is the union of two orthogonal lines, two opposing right angles will be considered to balance each other completely, so that , regardless of the choice of Euler circuit of the double.

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

The nature of is clearer when it is localized on , analogously to [M]. In the case , Milnor observed that the exterior angle at the vertex equals half the area of those such that the linear function , restricted to , has a local maximum at . In our context, we may describe as one-half the integral over the sphere of the number of net local maxima, which is half the difference of local maxima and local minima. Along the parameterization of the double of , the linear function may have a local maximum at some of the vertices over , and may have a local minimum at others. In our construction, each local minimum balances against one local maximum. If there are more local minima than local maxima, the number , the net number of local maxima, will be negative; however, our definition uses only the positive part .

We need to show that

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

###### Definition 4.

Let a parameterization of the double of be given. Then a vertex of corresponds to a number of vertices of , where is the degree of as a vertex of . Choose . If is a local extremum of , then we consider as a vertex of degree . Let be the number of local maxima of along at the points over , and similarly let be the number of local minima. We define the number of net local maxima of at to be

.

###### Remark 1.

The definition of appears to depend not only on but on a choice of the parameterization of the double of : and may depend on the choice of . However, we shall see in Corollary 1 below that the number of net local maxima is in fact independent of .

###### Remark 2.

We have included the factor in the definition of in order to agree with the difference of the numbers of local maxima and minima along a parameterization of itself, if is even.

We shall assume for the rest of this section that a unit vector has been chosen, and that the linear “height” function has only a finite number of critical points along ; this excludes belonging to a subset of of measure zero. We shall also assume that the graph is subdivided to include among the vertices all critical points of the linear function , with degree if is an interior point of one of the topological edges of .

###### Definition 5.

Choose a unit vector . At a point of degree , let the up-degree be the number of edges of with endpoint on which is greater (“higher”) than , the “height” of . Similarly, let the down-degree be the number of edges along which is less than its value at . Note that , for almost all in .

###### Lemma 1.

(Combinatorial Lemma) For all and for a.a. , .

Proof. Let a parameterization of the double of be chosen, with respect to which and are defined. Recall the assumption above, that has been subdivided so that along each edge, the linear function is strictly monotone.

Consider a vertex of , of degree . Then
has edges with an endpoint among the points
which are mapped to .
On , resp. of these edges, is greater resp. less than .
But for each , the parameterization has
exactly two edges which meet at . Depending on the up/down
character of the two edges of which meet at ,
, we can count:

(+) If is greater than on both edges, then is a local minimum point;
there are of these among .

(-) If is less than on both edges, then is a local maximum point;
there are of these.

(0) In all remaining cases, the linear function is greater than along one edge and
less along the other, in which case is not counted in
computing nor ; there are
of these.

Now count the individual edges of :

(+) There are pairs of edges, each of which is
part of a local minimum, both of which are counted among the
edges of with
greater than .

(-) There are pairs of edges, each of which is
part of a local
maximum; these are counted among the number of edges
of with
less than .
Finally,

(0) there are edges of
which are not part of a local
maximum or minimum, with
greater than ; and an equal number of edges
with less than .

Thus, the total number of these edges of with greater than is

Similarly,

Subtracting gives the conclusion:

###### Corollary 1.

The number of net local maxima is independent of the choice of parameterization of the double of .

Proof. Given a direction , the up-degree and down-degree at a vertex are defined independently of the choice of .

###### Corollary 2.

For any , we have

Proof. Consider . In the definition (3) of whenever . But the number of with equals , so that

by Lemma 1, for almost all .

###### Definition 6.

For a graph in and , define the multiplicity at as

Note that is a half-integer. Note also that in the case when is a knot, or equivalently, when , is exactly the integer , the number of local maxima of along as defined in [M], p. 252.

###### Corollary 3.

For almost all and for any parameterization of the double of ,

Proof. We have

If, in place of the positive part, we sum itself over located above a plane orthogonal to , we find a useful quantity:

###### Corollary 4.

For almost all and almost all ,

the cardinality of the fiber .

Proof. If , then
. Now proceed downward, using Lemma 1 by
induction.

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

###### Remark 3.

In analogy with Corollary 4, we expect that an appropriate generalization of to curved polyhedral complexes of dimension will in the future allow computation of the homology of level sets and sub-level sets of a (generalized) Morse function in terms of a generalization of .

###### Corollary 5.

The multiplicity of a graph in direction may also be computed as .

Proof. It follows from Corollary 4 with that , which is the difference of positive and negative parts. The sum of these parts is

It was shown in Theorem 3.1 of [M] that, in the case of knots, , where Milnor refers to Crofton’s formula. We may now extend this result to graphs:

###### Theorem 1.

For a (piecewise ) graph mapped into the net total curvature has the following representation:

Proof. We have
where are the vertices of , including
local extrema as vertices of degree , and where
by the definition (3) of . Applying Milnor’s
result to each edge, we have
. But
, and the theorem follows.

###### Corollary 6.

If is piecewise but is not an embedding, then the net total curvature is well defined, using the right-hand side of the conclusion of Theorem 1. Moreover, has the same value when points of self-intersection of are redefined as vertices.

For , we shall use the notation for the orthogonal projection . We shall sometimes identify with the one-dimensional subspace of .

###### Corollary 7.

For any homeomorphism type of graphs, the infimum of net total curvature among mappings is assumed by a mapping .

For any isotopy class of embeddings , the infimum of net total curvature is assumed by a mapping in the closure of the given isotopy class.

Conversely, if is in the closure of a given isotopy class of embeddings into , then for all there is an embedding in that isotopy class with .

Proof. Let be any piecewise smooth mapping. By Corollary 6 and Corollary 4, the net total curvature of the projection of onto the line in the direction of almost any is given by It follows from Theorem 1 that is the average of over in . But the half-integer-valued function is lower semi-continuous almost everywhere, as may be seen using Definition 4. Let be a point where attains its essential infimum. Then But is the limit as of the map whose projection in the direction is the same as that of and is multiplied by in all orthogonal directions. Since is isotopic to , is in the closure of the isotopy class of .

Conversely, given in the closure of a given isotopy class, let be an embedding in that isotopy class uniformly close to ; as constructed above converges uniformly to as , and .

###### Definition 7.

We call a mapping flat (or -flat) if , the minimum value for the topological type of , among all ambient dimensions .

In particular, Corollary 7 above shows that for any , there is a flat mapping .

###### Proposition 1.

Consider a piecewise mapping . There is a mapping which is monotonic along the topological edges of , has values at topological vertices of arbitrarily close to those of , and has

Proof. Any piecewise mapping may be approximated uniformly by mappings with a finite set of local extreme points, using the compactness of . Thus, we may assume without loss of generality that has only finitely many local extreme points. Note that for a mapping , : hence, we only need to compare with .

If is not monotonic on a topological edge , then it has a local extremum at a point in the interior of . For concreteness, we shall assume is a local maximum point; the case of a local minimum is similar. Write for the endpoints of . Let be the closest local minimum point to on the interval of from to (or if there is no local minimum point between), and let be the closest local minimum point to on the interval from to (or ). Let denote the interval between and . Then is an interval of a topological edge of , having end points and and containing an interior point , such that is monotone increasing on the interval from to , and monotone decreasing on the interval from to . By switching and if needed, we may assume that .

Let be equal to except on the interior of the interval , and map monotonically to the interval of between and . Then for , the cardinality . For in all other intervals of , this cardinality is unchanged. Therefore, , by Lemma 1. This implies that . Meanwhile, , a term which does not appear in the formula for (see Definition 6).Thus and .

Proceeding inductively, we remove each local extremum in the interior of any edge of , without increasing .

## 4. Representation Formula for Nowhere-Smooth Graphs

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

###### Proposition 2.

Let and be polygonal graphs in , having the same topological vertices, and homeomorphic to each other. Suppose that every vertex of is also a vertex of : is a refinement of . Then for almost all , the multiplicity As a consequence, .

Proof. We may assume, as an induction step, that is obtained from by replacing the edge having endpoints , with two edges, one having endpoints , and the other having endpoints , . Choose . We consider various cases:

If the new vertex satisfies , then for and , hence .

If , then
and
. The vertex
requires more careful counting: the up- and down-degree
, so that by
Lemma 1,
.
Meanwhile, for each of the polygonal graphs, is the sum
over of , so the change from to
depends on the value of
:

(a) if , then
;

(b) if then
;

(c) if , then
.

Since the new vertex does not appear in , recalling
that , we have
or in the
respective cases (a), (b) or (c). In any case,
.

The reverse inequality may be reduced to the case just above by replacing with , since for any polhedral graph . Then, depending whether is , or , we find that , , or . In any case, .

These arguments are unchanged if is switched with
. This covers all cases except those in which equality
occurs between and
(). The set of such unit vectors form a set of
measure zero in . The conclusion
now follows from Theorem
1.

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

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

###### Definition 8.

We say a point is critical relative to when is a topological vertex of or when is not monotone in any open interval of containing .

Note that at some points of a differentiable curve, may have derivative zero but still not be considered a critical point relative to by our definition. This is appropriate to the category. For a continuous graph , when is finite, we shall show that the number of critical points is finite for almost all in (see Lemma 4 below).

###### Lemma 2.

Let be a continuous, finite graph in , and choose a sequence of -approximating polygonal graphs with Then for each , there is a refinement of such that exists in .

Proof. First, for each in sequence, we refine to include all vertices of . Then for all , , by Proposition 2. Second, we refine so that the arc of corresponding to each edge of has diameter . Third, given a particular , for each edge of , we add or points from as vertices of so that where is the closed arc of corresponding to ; and similarly so that . Write for the result of this three-step refinement. Note that all vertices of