Counting realizations of Laman graphs on the sphere
CCOUNTING REALIZATIONS OF LAMAN GRAPHS ON THESPHERE
MATTEO GALLET ∗ , ◦ , (cid:5) , GEORG GRASEGGER ∗ ,. , AND JOSEF SCHICHO ∗ , ◦ Abstract.
We present an algorithm that computes the number of realizationsof a Laman graph on a sphere for a general choice of the angles between thevertices. The algorithm is based on the interpretation of such a realization asa point in the moduli space of stable curves of genus zero with marked points,and on the explicit description, due to Keel, of the Chow ring of this space. Introduction
Maybe the most important open problem in rigidity theory is the characteriza-tion and study of rigid structures in three dimensional space. On the other side,planar structures are reasonably well-understood, in the sense that, for example,we have a characterization of graphs that are generically minimally rigid in theplane. These are the graphs that, once a general assignment for the edge lengthsis prescribed, admit only finitely many ways of realizing them in the plane re-specting the assignment, if we consider equivalent realizations that differ by anisometry. Pollaczek-Geiringer [Pol27] and Laman [Lam70] described these graphsin terms of their combinatorics, and so they also go under the name of
Lamangraphs . In [CGG + +
19] that Laman graphs are generically minimally rigid also when we considerrealizations on the sphere, so as before one can ask in how many different ways onecan realize a Laman graph on the sphere. In this paper, we provide a recursivealgorithm that computes this number (again, under the assumption that complexcoordinates for the points are allowed) based on a completely different techniquefrom the one used in the planar case. We hope that, although we still work on asurface, moving from the plane to the sphere could be a first step towards deter-mining the number of realizations for generically minimally rigid graphs in threedimensions. For a related work on this topic, discussing real realizations of graphson the sphere (in addition to the plane and the space), see the recent paper by ∗ Supported by the Austrian Science Fund (FWF): W1214-N15, Project DK9. ◦ Supported by the Austrian Science Fund (FWF): P31061. (cid:5)
Supported by the Austrian Science Fund (FWF): Erwin Schrödinger Fellowship J4253. . Supported by the Austrian Science Fund (FWF): P31888. a r X i v : . [ m a t h . C O ] M a r M. GALLET, G. GRASEGGER, AND J. SCHICHO
Bartzos et al. [BELT18]. Among other things, the latter paper proves that forsome graphs one can achieve all possible complex realizations via real instances.
Figure 1.
Realizations of graphs on the sphere.
Our result.
The main result of this paper is an algorithm that computes thenumber of realizations of a Laman graph on the (complex) sphere, up to the actionof the group SO ( C ) (recall that the real analogue of this group, SO ( R ), is thegroup of isometries of the real sphere). The key idea is to interpret realizationson the sphere up to SO ( C ) as elements of a moduli space, the so-called modulispace of rational curves with marked points , where each point of the realizationcorresponds to two marked points. In this interpretation, assigning the distancebetween two points on the sphere corresponds to prescribing the cross-ratio of the 4related marked points. By using the properties of the moduli space, in particular thedescription of its Chow ring and the geometry of some of its divisors, we compute thecardinality of those elements for which the cross-ratios of the points correspondingto the edges of a Laman graph are assigned; this provides the answer to our originalproblem. Remarkably, we have been informed by Gaiane Panina that the modulispace of rational curves with marked points appears also in investigations of flexiblepolygons, see [NP18]. Structure of the paper.
Section 2 describes the problem we want to addressin this paper. Section 3 provides the translation from realizations of a graph onthe sphere to points of a moduli space of points on the projective line. Section 4describes, following [Kee92], the moduli space of points on the projective line, andits compactification as moduli space of stable curves of genus zero with markedpoints; in particular, we recall the description by Keel of the Chow rings of thelatter, which plays a key role in the algorithm. Section 5 describes the algorithm.Section 6 reports some computational data obtained using the algorithm.
Acknowledgments.
We thank the Erwin Schrödinger Institute (ESI) of the Uni-versity of Vienna for the hospitality during the workshop “Rigidity and Flexibilityof Geometric Structures”, when this project started. We thank Anthony Nixon forproviding us with useful references concerning rigidity of graphs on the sphere. Wethank Jan Legerský for helpful discussions on the topic of this paper.
OUNTING REALIZATIONS OF LAMAN GRAPHS ON THE SPHERE 3 Realizations of graphs A Laman graph is a graph G = ( V, E ) such that | E | = 2 | V | − | E | ≤ | V | − G = ( V , E ). Geiringer [Pol27] and Laman [Lam70] provedthat these graphs are generically infinitesimally rigid in the plane. This meansthe following. A realization of G is a tuple ( P v ) v ∈ V of points in the plane indexedby the vertices of G . By applying a Euclidean isometry, we can always supposethat one of the points is the origin O of the plane, and another one lies on the x -axis. The set of all possible realizations satisfying the previous two requirementsis then given by { O } × A R × (cid:0) A R ) n − , where n is the number of vertices. We cannow consider the function that computes, for every edge { a, b } ∈ E , the distance d A R ( P a , P b ) of the corresponding points in a realization. In this way we get a map:Ψ A R : { O } × A R × (cid:0) A R ) n − −→ R | E | ( P v ) v ∈ V (cid:0) d A R ( P a , P b ) (cid:1) { a,b }∈ E Notice that the map Ψ A R is a smooth map (better, an algebraic one) between smoothmanifolds (better, algebraic varieties) of the same dimension. Laman proved that,if we pick a general point ~P of the domain (namely, if we remove a finite number of“bad” subvarieties from the domain), then the Jacobian of Ψ A R at ~P is invertible,i.e., the map Ψ A R is an isomorphism locally around P . Notice that the fibers of Ψ A R ,namely the sets Ψ − A R ( λ ) for some λ ∈ R | E | , are the sets of realizations of G wherethe distances between points, whose corresponding vertices are connected by anedge, are prescribed by λ . We give a name to these sets: Definition 2.1.
Let G = ( V, E ) be a Laman graph and let λ : E −→ R . Arealization of G compatible with λ is a function ρ : V −→ A R such that d A R (cid:0) ρ ( a ) , ρ ( b ) (cid:1) = λ ( { a, b } ) for every { a, b } ∈ E .
When a fiber of Ψ A R is finite—namely when the number of realizations of G com-patible with given edge lengths is finite—one may be interested in counting itscardinality, namely the number of ways of realizing a Laman graph with a spec-ified assignment of edge lengths; however, over the real numbers, the cardinal-ity of such fibers may depend on the point in the codomain. Since the resultof Laman proves also that the map Ψ A R is dominant, if we pass to the complexnumbers we have that, for a general element λ in the codomain, the fiber Ψ − A C ( λ )is finite and its cardinality does not depend on the point. In recent years, therehas been some interest in providing lower and upper bounds for this number (see[BS04, ST10, ETV13, GKT18, JO19, BELT18]), and in [CGG +
18] the authors pro-vide an iterative formula to compute it, based on tropical geometry. A fully tropicalproof of Laman theorem has recently been provided in [BK18].One can consider the notion of being generically infinitesimally rigid also on thesphere. In [EJN +
19] it is proven that, also on the sphere, the class of genericallyinfinitesimally rigid graphs coincides with the class of Laman graphs. On the sphere,the distance between two points can be taken to be the angle they form (viewed fromthe origin of the sphere). In this paper we adopt a slightly different definition, which
M. GALLET, G. GRASEGGER, AND J. SCHICHO involves the cosine of that angle, because it fits better in the algebraic framework weare going to use. Adopting this definition has no impact as the matter of computingthe number of realizations of a graph is concerned. The advantage of this choice isthat it provides an algebraic function, which hence allows extensions of fields (inparticular, from the real to the complex numbers).
Definition 2.2.
Given two points
P, Q ∈ S , we define their spherical distance as d S ( P, Q ) := 1 − h
P, Q i , where h P, Q i = P i =1 P i Q i . In particular, if P and Q are antipodal, their sphericaldistance is 1.In this context, we can repeat the same considerations as before: given a configu-ration ~P on the sphere of a Laman graph G = ( V, E ), we can always suppose, byapplying rotations, that one of the points is (1 , ,
0) and another lies on a greatcircle through (1 , , G is a Laman graph, then the mapΨ S C : { (1 , , } × S C × (cid:0) S C ) n − −→ C | E | ( P v ) v ∈ V (cid:0) d S ( P a , P b ) (cid:1) { a,b }∈ E is dominant and its fibers over general points are finite and of constant cardinality.Here, we denoted by S C the set { ( x, y, z ) ∈ C : x + y + z = 1 } , namely thecomplexification of the sphere, and similarly for the circle S C . As we remarked,since the function d S describing the spherical distance is algebraic, we can applyit also to pairs of complex points in S C . Notice that in the real setting we considerrealizations of the graph up to rotations, namely elements of SO ( R ); when we passto the complex numbers, we consider realizations up to SO ( C ), whereSO ( C ) := (cid:8) R ∈ C × : RR t = R t R = id , det( R ) = 1 (cid:9) . In this paper, the elements of SO ( C ) will also be (improperly) referred as rotations,or isometries of S C .This is the main goal of our paper: Goal.
Compute the cardinality of a general fiber of the map Ψ S C . In other words,compute the number of realizations of a Laman graph on the sphere compatiblewith a general assignment of spherical distances for its edges, up to SO ( C ).We are going to achieve this goal by interpreting realizations up to SO ( C ) ascollections of points on the projective complex line P C , up to the action of P GL ( C ),namely the group of automorphisms of P C . These objects can be interpreted aspoints in a moduli space, and the explicit description of the intersection theory onthat moduli space provides the answer to our question.3. Realizations on the sphere as points on a moduli space
The aim of this section is to show how we can interpret a realization of a graphon the sphere, up to sphere isometries, as a point of the moduli space of stable
OUNTING REALIZATIONS OF LAMAN GRAPHS ON THE SPHERE 5 curves of genus zero with marked points. This provides the theoretical backgroundon which the algorithm presented in Section 5 is based.We would like to express the spherical distance between two points in S C as thecross-ratio of four points in P C . To do so, we associate to each point in S C twopoints in P C via the following construction. Definition 3.1.
Let S C = (cid:8) ( x : y : z : w ) ∈ P C : x + y + z − w = 0 (cid:9) be the projective closure of S C in P C . Let A be the intersection of S C with the planeat infinity { w = 0 } . The conic A is called the absolute conic . Since S C is a smoothquadric in P C there are exactly two families of lines on S C ; we denote them by F and F . Every point P ∈ S C is contained in exactly one line L of F and exactlyone line L of F . The union of these two lines can be obtained by intersecting S C with the tangent plane of S C at P . We define the left and the right lifts of P as theintersections of L and L with A , respectively. We denote them by P l and P r ,respectively. Remark 3.2.
Notice that the absolute conic A is a rational curve. This meansthat, given four points on A (for example, the left and right lifts of two pointsin S C ), we can speak about their cross-ratio. Lemma 3.3.
Let
P, Q ∈ S C . Let P l , P r be the left and right lifts of P , and Q l , Q r be the left and right lifts of Q . Then d S ( P, Q ) = cr (cid:0) P l , P r , Q l , Q r (cid:1) cr (cid:0) P l , P r , Q l , Q r (cid:1) − (cid:0) P l , Q r , Q l , P r (cid:1) , where cr stands for cross-ratio.Proof. Recall that isometries of S C are projective automorphisms of P C leaving theabsolute conic A invariant. Hence applying an isometry to P and Q correspondsto applying an automorphism of P C to their lifts, so the cross-ratio of the latterdoes not change. Hence we can suppose that P = (1 , ,
0) and Q = ( c, s, c + s = 1. With this choice, we have d S ( P, Q ) = 1 − c . A direct computation shows that, since the tangent planes at P and Q have equa-tions x − w = 0 and c x + s y − w = 0, respectively: P l = (0 : i : 1 : 0) , P r = (0 : − i : 1 : 0) ,Q l = ( − is : ic : 1 : 0) , Q r = ( is : − ic : 1 : 0) . In order to compute the cross-ratio, we take an isomorphism between A and P C ,for example the one given by the projection of A from the point ( i : 0 : 1 : 0) to theline { z = w = 0 } . The projections of the previous four points are( − , (1 : 1 : 0 : 0) , ( − − s : c : 0 : 0) , (1 − s : c : 0) . M. GALLET, G. GRASEGGER, AND J. SCHICHO
Their cross-ratio is (cid:18) − − sc + 1 (cid:19) (cid:18) − sc − (cid:19) (cid:30) (cid:18)(cid:18) − − sc − (cid:19) (cid:18) − sc + 1 (cid:19)(cid:19) . A direct computation then proves the statement. (cid:3)
Proposition 3.4.
Let ~P = ( P , . . . , P n ) and ~Q = ( Q , . . . , Q n ) be two n -tuples ofpoints in S C . Denote by P li , P ri and Q li , Q ri the left and right lifts of P i and Q i ,respectively, for all i ∈ { , . . . , n } . Then ~P and ~Q differ by an isometry of S C ifand only if ( ~P l , ~P r ) and ( ~Q l , ~Q r ) differ by an element of P GL ( C ) .Proof. Every isometry of S C is a projective automorphism of P C leaving the ab-solute conic A invariant. This means that every isometry of S C determines anautomorphism of A . In this way we get a mapSO ( C ) −→ P GL ( C ) , which is a homomorphism of Lie groups. Our statement is proven if we can showthat this is an isomorphism. Suppose that we have an isometry of S C that inducesthe identity on A . Then the corresponding projective automorphism of P C fixesthe whole plane at infinity, and also the center of S C : the only element in SO ( C )satisfying these requirements is the identity. Hence, the homomorphism is injective.Since the two Lie groups have the same dimension and are both connected, thenthe homomorphism is also surjective. This concludes the proof. (cid:3) Remark 3.5.
Notice that, in general, it is not true that n distinct points on S C determine 2 n -tuples of distinct points on the absolute conic via the lift operation.In fact, this fails precisely when two points on S C belong to the same line. If P, Q ∈ S C with P = ( α, β, γ ) and Q = ( α , β , γ ), then P and Q belong to thesame line when, considered as points in P C , they are orthogonal with respect to thequadratic form determined by the equation of S C , namely if: (cid:16) α β γ (cid:17) − α β γ = 0 ⇔ * αβγ , α β γ + − . Hence, if we suppose that ~P is a realization of a Laman graph G compatible with ageneral assignment of spherical distances for its edges, then ~P determines a 2 n -tupleof distinct points ( ~P l , ~P r ) on the absolute conic A .The combination of Lemma 3.3, Proposition 3.4, and Remark 3.5 shows that, in-stead of considering n -tuples ~P that are realizations of a Laman graph compatiblewith a general assignment of spherical distances for its edges, up to the actionof SO ( C ), we can consider 2 n -tuples ( ~P l , ~P r ) of points on P C for which somecross-ratios are assigned, up to the action of P GL ( C ). The latter are elements ofa so-called moduli space of curves of genus zero with marked points. In the nextsection we describe this object and its properties concerning intersection theory.Afterwards, we come back to our original problem and cast it into this theoreticalframework. OUNTING REALIZATIONS OF LAMAN GRAPHS ON THE SPHERE 7 The moduli space of stable curves of genus zero with markedpoints
In this section we describe for the reader’s convenience the well-known modulispace of stable curves of genus zero with marked points and its Chow ring, follow-ing [Kee92]. No new results are presented in this section.Let us start by recalling a basic and fundamental result in projective geometry:every triple of distinct points P , Q , and R in P C can be mapped to the triple(1 : 0), (0 , P C , namely an elementof P GL ( C ). Hence, any triple of distinct points in P C is projectively equivalentto any other one, namely there always exists an automorphism mapping one tothe other. If we consider a 4-tuple of distinct points P , Q , R , S , then there is aunique element that takes it to the 4-tuple (1 : 0), (0 : 1), (1 : 1), and (1 : λ ): thenumber λ is called the cross-ratio of the tuple ( P, Q, R, S ). Two 4-tuples of distinctpoints are then projectively equivalent if and only if their cross-ratios are the same.Hence, each equivalence class of 4-tuples of distinct points modulo P GL ( C ) isrepresented by its cross-ratio, namely by an element in C \ { , } , or equivalentlyin P C \ { (1 : 0) , (0 : 1) , (1 : 1) } . We then say that P C \ { (1 : 0) , (0 : 1) , (1 : 1) } is the moduli space of 4-tuples of distinct points in P C .We can consider, for every m ≥
4, the space of equivalence classes of m -tuples ofdistinct points under the action of P GL ( C ): each such equivalence class is uniquelydetermined by an element in the quasi-projective variety (cid:0) P C \ { (1 : 0) , (0 : 1) , (1 : 1) } (cid:1) m − \ ( ( m − ) . This space is called the moduli space of m -tuples of distinct points in P C , and isdenoted by M ,m . One may notice that this space is not compact under the Eu-clidean topology (in more algebro-geometric terms, it is not complete), and thismay be a problem when dealing with enumerative questions, as the one we addressin this work. Because of this, researchers focused on finding compactifications ofthese moduli spaces. A possible smooth compactification of the space M ,m , de-noted M ,m , was constructed by Knudsen [Knu83] (see [Mum65, Mum77, Gie82]for a more general account on the topic). This construction introduces a boundary for M ,m , constituted of particular curves, called stable curves , which are essentiallyreducible curves whose irreducible components are rational curves, intersecting innodes. More precisely, we have: Definition 4.1 ([Kee92, Introduction]) . A stable curve of genus zero with m marked points is a reduced, possibly reducible, curve C with at worst node sin-gularities, together with m distinct marked points p , . . . , p m on it such that: . the points { p i } mi =1 lie on the smooth locus of C ; . each irreducible component of C is isomorphic to P C , and altogether allirreducible components form a tree; . for each irreducible component of C , the sum of the numbers of singularpoints and of marked points on that component is at least 3. M. GALLET, G. GRASEGGER, AND J. SCHICHO
The geometry of M ,m is rich and well-studied: we refer to [Kee92, Introduction],[Kap93], and [KV07, Chapter 1] for a discussion.As we are going to see in Section 5, our algorithm relies on the understanding ofhow subvarieties of M ,m intersect each other. This piece of information is encodedin the so-called Chow ring , which is a standard object in intersection theory. Forits definition and properties we refer to the introduction [Ful84], or to the standardbook [Ful98].For the reader’s convenience, we briefly provide in Theorem 4.4 the descriptionby Keel of the Chow ring of M ,m . First, we need to introduce some particulardivisors that Keel calls “vital”. Definition 4.2 ([Kee92, Introduction]) . Let (
I, J ) be a partition of { , . . . , m } where | I | ≥ | J | ≥
2. We define the divisor D I,J in M ,m to be the divisorwhose general point is a stable curve with two irreducible components such that themarked points labeled by I lie on one component, while the marked points labeledby J lie on the other component. Proposition 4.3 (Knudsen, [Kee92, Introduction and Fact 2]) . Any divisor D I,J asin Definition 4.2 is smooth and it is isomorphic to the product M , | I | +1 × M , | J | +1 ;in the previous isomorphism, the point of intersection of the two components of ageneral stable curve in D I,J counts as an extra marked point in each of the twofactors of the product.
Theorem 4.4 ([Kee92, Introduction and Theorem 1]) . The Chow ring of M ,m admits the following description: Z (cid:2) D I,J : (
I, J ) is a partition of { , . . . , m } where | I | ≥ | J | ≥ (cid:3) . ∼ , where the equivalence ∼ is given by the relations: . D I,J = D J,I , . for any four distinct elements a, b, c, d ∈ { , . . . , m } we have X a,b ∈ Ic,d ∈ J D I,J = X a,c ∈ Ib,d ∈ J D I,J = X a,d ∈ Ib,c ∈ J D I,J , ( ) . D I,J · D K,L = 0 unless one of the following holds: I ⊆ K, or K ⊆ I, or J ⊆ L, or L ⊆ J .
Moreover, the three sums in Equation ( ) are the pullbacks of the respective divisors D { a,b } , { c,d } , D { a,c } , { b,d } , and D { a,d } , { b,c } under the map π a,b,c,d : M ,m −→ M , ∼ = P C . Here, the map π a,b,c,d is the map that forgets all the marked points except for theones labeled by a , b , c , and d . OUNTING REALIZATIONS OF LAMAN GRAPHS ON THE SPHERE 9 The algorithm
At the end of Section 2 we understood that realizations on the sphere up to SO ( C )can be considered as elements of a moduli space. In particular, configurations of n points on the complex sphere up to SO ( C ) correspond to configurations of 2 n points on a rational curve up to P GL ( C ), i.e., elements in the moduli space ofrational curves with marked points. Moreover, assigning angles between two pointson a sphere corresponds to assigning the cross-ratio of the 4-tuple constituted ofthe left and right lifts of those two points. The elements of the moduli space, forwhich the cross-ratio of 4 marked points is prescribed, are fibers of the map M , n −→ M , ∼ = P C that forgets all but the 4 considered marked points (see Theorem 4.4). Hence, theelements of the moduli space we are interested in, namely the ones for which thecross-ratios are specified for the 4-tuples arising from edges of a graph, are fibersof products of these maps. We give a name to these maps: Definition 5.1.
Let G = ( V, E ) be a graph, and suppose that V = { , . . . , n } . Wedefine the morphism Φ G : M , n −→ Y { a,b }∈ E M a,b,a + n,b + n , whose components are the maps π a,b,a + n,b + n : M , n −→ M , forgetting all but4 marked points and defined in Theorem 4.4. The choice for the indices reflectsthe labeling of the marked points on the rational curves by ( P l , . . . , P ln , P r , . . . , P rn ),putting first the points corresponding to left lifts, and then the points correspondingto right lifts of points on the sphere. The translation we operated in Section 2 tells usthat Φ G is a dominant morphism between smooth varieties of the same dimension,so its general fibers are constituted of finitely many points. Remark 5.2.
For any Laman graph G with n vertices, the image of the boundaryof M , n under Φ G is a proper subvariety of Q { a,b }∈ E M a,b,a + n,b + n , . This meansthat a general fiber of Φ G will not intersect the boundary, and so it is constitutedof classes of rational curves with 2 n distinct marked points. Each of such ratio-nal curves is then isomorphic to the absolute conic A , and the 2 n marked pointsdetermine a realization of G on the sphere. Moreover, when G is a Laman graph,a general fiber of Φ G is a complete intersection in M , n and it is constituted byreduced points.The discussion so far proves the following theorem. Theorem 5.3.
Given a Laman graph G , then the number of realizations of G in S C for a general assignment of spherical distances for its edges, up to SO ( C ) , equalsthe cardinality of a general fiber of the map Φ G as in Definition 5.1. In the light of Theorem 5.3, our goal becomes, given a Laman graph G , to computethe cardinality of a general fiber of Φ G . Remark 5.4.
Let us focus on how the class in the Chow ring of a fiber of Φ G looks like. Since the codomain of Φ G is a product, we can express a fiber as theintersection of the fibers of the maps to each component of the product. In otherwords, if we fix Λ ∈ Q { a,b }∈ E M a,b,a + n,b + n , and we denote by Φ a,b the compositionof Φ G with the projection to the factor M a,b,a + n,b + n , , then we have thatΦ − G (Λ) = \ { a,b }∈ E Φ − a,b (Λ a,b ) . It follows then, using the notation of Theorem 4.4, that the class of Φ − G (Λ) in theChow ring of M , n is given by Y { a,b }∈ E X a,b ∈ Ia + n,b + n ∈ J D I,J . Since a general fiber of Φ G is a reduced complete intersection in M , n when G isa Laman graph, its cardinality is the degree of its Chow class. Hence we get: Proposition 5.5.
Given a Laman graph G = ( V, E ) with V = { , . . . , n } , thenumber of realizations of G in S C for a general assignment of spherical distancesfor its edges, up to SO ( C ) , equals deg Y { a,b }∈ E X a,b ∈ Ia + n,b + n ∈ J D I,J . Algorithm
CountRealizations computes the degree in Proposition 5.5 using thedescription of vital divisors D I,J provided by Proposition 4.3 and the relations inthe Chow ring stated in Theorem 4.4. In fact, if we fix an edge { a , b } ∈ E of aLaman graph G = ( V, E ), then the class we want to compute is X a ,b ∈ I a + n,b + n ∈ J D I ,J · Y { a,b }∈ E \{ a ,b } X a,b ∈ Ia + n,b + n ∈ J D I,J | {z } =: F a ,b . (1)For every ( I , J ) such that a , b ∈ I and a + n, b + n ∈ J , the product D I ,J · F a ,b can be computed by restricting F a ,b to D I ,J and using the isomorphism D I ,J ∼ = M ,I ∪{∗} × M ,J ∪{∗} . We show that both the restrictions of F a ,b to M ,I ∪{∗} and to M ,J ∪{∗} have the same structure of the initial class wewanted to compute, and this determines a recursive procedure to solve our task.To clarify the recursive procedure, let us start by noticing that the class Y { a,b }∈ E X a,b ∈ Ia + n,b + n ∈ J D I,J (2)from Proposition 5.5 is a particular instance of a more general construction. Theconstruction works as follows: we start from a set N , and a set Q of 4-tuples ofdistinct elements of N . Then we form the following class in the Chow ring of M , | N | : A N,Q := Y q ∈ Qq =( a,b,c,d ) X a,b ∈ Ic,d ∈ J D I,J . (3) OUNTING REALIZATIONS OF LAMAN GRAPHS ON THE SPHERE 11
Notice that if we set N := { , . . . , n } and Q := { ( a, b, a + n, b + n ) : { a, b } ∈ E } ,then the class from Equation (2) equals A N ,Q . In this perspective, also theclass F a ,b from Equation (1) can be seen as a particular case of a general con-struction: what we do here is to select some ¯ q ∈ Q , with ¯ q = (¯ a, ¯ b, ¯ c, ¯ d ), and todefine the class G ¯ q := Y q ∈ Q \{ ¯ q } q =( a,b,c,d ) X a,b ∈ Ic,d ∈ J D I,J , so that we get the factorization A N,Q = X ¯ a, ¯ b ∈ ¯ I ¯ c, ¯ d ∈ J D ¯ I, ¯ J · G ¯ q . (4)With this notation, the class F a ,b equals G q , where q = ( a , b , a + n, b + n ).We show now that we can set up an iterative procedure for the computation of thedegree of classes of type A N,Q . Notice that, taking into account Equation (4), thiscan be achieved once we are able to compute the degree of a product D ¯ I, ¯ J · G ¯ q . Wethen fix ¯ q ∈ Q with ¯ q = (¯ a, ¯ b, ¯ c, ¯ d ) and we select a pair ( ¯ I, ¯ J ) such that ¯ a, ¯ b ∈ ¯ I and ¯ c, ¯ d ∈ ¯ J . If there exists q ∈ Q \ { ¯ q } such that | q ∩ ¯ I | = | q ∩ ¯ J | = 2, then therestriction of G ¯ q to D ¯ I, ¯ J is zero by [Kee92, Fact 2]. Otherwise, the restriction of G ¯ q to D ¯ I, ¯ J ∼ = M , ¯ I ∪{∗} × M , ¯ J ∪{∗} is the product of two classes G ¯ I ¯ q and G ¯ J ¯ q . Recall, in fact,that the Chow ring of M , ¯ I ∪{∗} × M , ¯ J ∪{∗} is the tensor product of the Chow ringsof M , ¯ I ∪{∗} and M , ¯ J ∪{∗} by [Kee92, Theorem 2]. Analyzing the isomorphismmaking D ¯ I, ¯ J into a product (see [Kee92, Fact 2] and [Knu83, Theorem 3.7]), one seesthat the two classes G ¯ I ¯ q and G ¯ J ¯ q admit the following description. For k ∈ { , . . . , } ,define the sets: Q k, − k := (cid:8) q ∈ Q \ { ¯ q } : | q ∩ ¯ I | = k (cid:9) = (cid:8) q ∈ Q \ { ¯ q } : | q ∩ ¯ J | = 4 − k (cid:9) . Notice that, by definition, all tuples in Q , have exactly one element in ¯ J . Define Q , to be set obtained by substituting in all 4-tuples of Q , their element in ¯ J bythe new symbol ∗ . Analogously, define Q , . Then the classes G ¯ I ¯ q and G ¯ J ¯ q are Y q ∈ Q , ∪ Q , q =( a,b,c,d ) X a,b ∈ Ic,d ∈ J D I,J and Y q ∈ Q , ∪ Q , q =( a,b,c,d ) X a,b ∈ Ic,d ∈ J D I,J , where the D I,J are divisors in the appropriate moduli spaces, namely for G ¯ I ¯ q theyare divisors in M , | ¯ I ∪{∗}| , while for G ¯ J ¯ q they are divisors in M , | ¯ J ∪{∗}| .Hence, in the notation of Equation (3), if we set N ¯ I := ¯ I ∪ {∗} , Q ¯ I := Q , ∪ Q , ,N ¯ J := ¯ J ∪ {∗} , Q ¯ J := Q , ∪ Q , , we have G ¯ I ¯ q = A N ¯ I ,Q ¯ I and G ¯ J ¯ q = A N ¯ J ,Q ¯ J . Therefore, the degree of a class A N,Q can be computed by fixing an element ¯ q ∈ Q ,¯ q = (¯ a, ¯ b, ¯ c, ¯ d ), and using Equation (4), thus obtaining the formuladeg A N,Q = X ¯ a, ¯ b ∈ ¯ I ¯ c, ¯ d ∈ ¯ J (cid:0) deg A N ¯ I ,Q ¯ I · deg A N ¯ J ,Q ¯ J (cid:1) , where it is intended that a summand is zero if the corresponding set Q , is notempty. This allows one to set up a recursive procedure for the computation of thedegree of a class A N,Q —so in particular of the class of a fiber of Φ G . The recursionstops if we reach one of these situations: . The set Q , is not empty: in this case we can skip the contribution givenby this class, since its degree is zero. . The set N is composed of four elements, and Q consists of a single tuple:in this case the degree of the class is 1. . The cardinality of Q , ∪ Q , is different from | ¯ I ∪{∗}|− Q , ∪ Q , is different from | ¯ J ∪ {∗}| −
3: in this case either G ¯ I ¯ q or G ¯ J ¯ q iszero, and so this contribution can be skipped.The discussion so far proves the correctness of Algorithm CountRealizations .Termination is implied by the fact that the size of the sets always decreases andtherefore, the base cases are reached.6.
Computed Data
Using Algorithm
CountRealizations we computed the number of realizations onthe sphere of all Laman graphs with up to 10 vertices. Table 1 lists those graphsthat have the maximal number of realizations on the sphere within the class ofgraphs with the same number of vertices.
Remark 6.1.
The paper [BELT18] shows that the number of real spherical real-izations matches the number of complex ones for some graphs in Table 1 (all graphswith 6 and 7 vertices, and one of the graphs with 8 vertices).Note that up till 8 vertices the graph with maximal Laman number (i.e., numberof realizations in the plane) is also in the list of graphs with maximal number ofrealizations on the sphere. However, the graph with maximal Laman number with9 vertices is different from the one with maximal number of realizations on thesphere. The latter has a very particular structure (see last row of Table 1).Our recursive algorithm gives a significant improvement over the naive approach,which is to determine the number of solutions via a Gröbner basis computation.Furthermore, the Gröbner basis approach needs randomly fixed edge lengths where
CountRealizations computes the numbers symbolically. For a graph with 9 ver-tices and maximal number of realizations (see Table 1) our algorithm needs 5.66s inMathematica and 3.57s in Python. The Gröbner basis computation needed 5850sin Mathematica and 27s in Maple.
OUNTING REALIZATIONS OF LAMAN GRAPHS ON THE SPHERE 13
Algorithm
CountRealizations
Input:
A pair (
N, Q ), where N is a set and Q is a list of 4-tuples of elements of N . Output:
A natural number. When, for a Laman graph G = ( V, E ), with V = { , . . . , n } , we have N = { , . . . , n } and Q = { ( a, b, a + n, b + n ) : { a, b } ∈ E } ,then this natural number represents the number of realizations of G on thecomplex sphere, up to SO ( C ). If ( | N | = 4 and | Q | = 1) or ( | N | = 3 and | Q | = 0) Then Return End If Select any element ¯ q ∈ Q and write ¯ q = (¯ a, ¯ b, ¯ c, ¯ d ). Set Q := Q \ { ¯ q } and N := N \ { ¯ a, ¯ b, ¯ c, ¯ d } . Compute L := { subsets of N } . Set S := ∅ . For each subset L ∈ L Do Set ¯ I := { ¯ a, ¯ b } ∪ L and ¯ J := complement of ¯ I in N . Append ( ¯ I, ¯ J ) to S . End For
Set sum := 0.
For each pair ( ¯ I, ¯ J ) in S Do Compute the following five lists: Q , := (cid:8) q ∈ Q : | q ∩ ¯ I | = 4 (cid:9) = (cid:8) q ∈ Q : | q ∩ ¯ J | = 0 (cid:9) ,Q , := (cid:8) q ∈ Q : | q ∩ ¯ I | = 3 (cid:9) = (cid:8) q ∈ Q : | q ∩ ¯ J | = 1 (cid:9) ,Q , := (cid:8) q ∈ Q : | q ∩ ¯ I | = 2 (cid:9) = (cid:8) q ∈ Q : | q ∩ ¯ J | = 2 (cid:9) ,Q , := (cid:8) q ∈ Q : | q ∩ ¯ I | = 1 (cid:9) = (cid:8) q ∈ Q : | q ∩ ¯ J | = 3 (cid:9) ,Q , := (cid:8) q ∈ Q : | q ∩ ¯ I | = 0 (cid:9) = (cid:8) q ∈ Q : | q ∩ ¯ J | = 4 (cid:9) . If | Q , | > Then
Continue
End If
Let ∗ be a new symbol, not belonging to N . Set Q , := ∅ and Q , := ∅ . For each element q ∈ Q , Do Substitute in q the element q ∩ ¯ J with ∗ . Append the resulting tuple to Q , . End For
Apply the analogous procedure to the elements of Q , , obtaining Q , . If | Q , ∪ Q , | 6 = | ¯ I ∪ {∗}| − | Q , ∪ Q , | 6 = | ¯ J ∪ {∗}| − Then
Continue
End If
Update (here CR stands for CountRealizations ) sum := sum + CR (cid:0) ¯ I ∪ {∗} , Q , ∪ Q , (cid:1) · CR (cid:0) ¯ J ∪ {∗} , Q , ∪ Q , (cid:1) End For
Return sum . Table 1.
Graphs with maximal number of complex realizations onthe sphere within graphs of n vertices. Lam denotes the numberof complex realizations in the plane.n Graph(s) References [BELT18] Evangelos Bartzos, Ioannis Z. Emiris, Jan Legerský, and Elias Tsigaridas,
On themaximal number of real embeddings of minimally rigid graphs in R , R and S ,Available at https://arxiv.org/abs/1811.12800.[BK18] Daniel I. Bernstein and Robert Krone, The tropical Cayley-Menger variety , Availableat https://arxiv.org/abs/1812.09370.[BS04] Ciprian Borcea and Ileana Streinu,
The number of embeddings of minimally rigidgraphs , Discrete & Computational Geometry (2004), 287–303.[CGG +
18] Jose Capco, Matteo Gallet, Georg Grasegger, Christoph Koutschan, Niels Lubbes,and Josef Schicho,
The number of realizations of a Laman graph , SIAM Journal onApplied Algebra and Geometry (2018), no. 1, 94–125.[EJN +
19] Yaser Eftekhari, Bill Jackson, Anthony Nixon, Bernd Schulze, Shin-ichi Tanigawa, andWalter Whiteley,
Point-hyperplane frameworks, slider joints, and rigidity preservingtransformations , Journal of Combinatorial Theory, Series B (2019), 48–74.[ETV13] Ioannis Z. Emiris, Elias P. Tsigaridas, and Antonios Varvitsiotis,
Mixed Volume andDistance Geometry Techniques for Counting Euclidean Embeddings of Rigid Graphs , OUNTING REALIZATIONS OF LAMAN GRAPHS ON THE SPHERE 15
Distance Geometry: Theory, Methods, and Applications, pp. 23–45, Springer NewYork, New York, NY, 2013.[Ful84] William Fulton,
Introduction to intersection theory in algebraic geometry , CBMS Re-gional Conference Series in Mathematics, vol. 54, Published for the Conference Boardof the Mathematical Sciences, Washington, DC; by the American Mathematical Soci-ety, Providence, RI, 1984. MR 735435[Ful98] ,
Intersection theory , second ed., Ergebnisse der Mathematik und ihrer Gren-zgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematicsand Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2,Springer-Verlag, Berlin, 1998. MR 1644323[Gie82] David Gieseker,
Lectures on moduli of curves , Tata Institute of Fundamental ResearchLectures on Mathematics and Physics, vol. 69, Published for the Tata Institute ofFundamental Research, Bombay; Springer-Verlag, Berlin-New York, 1982.[GKT18] Georg Grasegger, Christoph Koutschan, and Elias Tsigaridas,
Lower bounds on thenumber of realizations of rigid graphs , Experimental Mathematics (2018), no. 0,1–12, online first.[JO19] Bill Jackson and John C. Owen, Equivalent realisations of a rigid graph , DiscreteApplied Mathematics (2019), 42–58.[Kap93] Mikhail M. Kapranov,
Veronese curves and Grothendieck-Knudsen moduli space M ,n , Journal of Algebraic Geometry (1993), no. 2, 239–262. MR 1203685[Kee92] Sean Keel, Intersection theory of moduli space of stable n -pointed curves of genuszero , Transactions of the American Mathematical Society (1992), no. 2, 545–574.[Knu83] Finn F. Knudsen, The projectivity of the moduli space of stable curves. II. The stacks M g,n , Mathematica Scandinavica (1983), no. 2, 161–199.[KV07] Joachim Kock and Israel Vainsencher, An invitation to quantum cohomology , Progressin Mathematics, vol. 249, Birkhäuser Boston, Inc., Boston, MA, 2007, Kontsevich’sformula for rational plane curves. MR 2262630[Lam70] Gerard Laman,
On graphs and rigidity of plane skeletal structures , Journal of Engi-neering Mathematics (1970), 331–340.[Mum65] David Mumford, Geometric invariant theory , Ergebnisse der Mathematik und ihrerGrenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin-New York, 1965.[Mum77] ,
Stability of projective varieties , L’Enseignement Mathématique. Revue Inter-nationale. IIe Série (1977), no. 1–2, 39–110.[NP18] Ilia Nekrasov and Gaiane Panina, Compactifications of M ,n associated with Alexan-der self-dual complexes: Chow ring, ψ -classes and intersection numbers , Available athttps://arxiv.org/abs/1808.08600.[Pol27] Hilda Pollaczek-Geiringer, Über die Gliederung ebener Fachwerke , Zeitschrift fürAngewandte Mathematik und Mechanik (ZAMM) (1927), 58–72.[ST10] Reinhard Steffens and Thorsten Theobald, Mixed volume techniques for embeddingsof Laman graphs , Computational Geometry (2010), no. 2, 84 – 93, Special Issueon the 24th European Workshop on Computational Geometry (EuroCG’08).(MG) International School for Advanced Studies/Scuola Internazionale Superiore diStudi Avanzati (ISAS/SISSA), Via Bonomea 265, 34136 Trieste, Italy
E-mail address : [email protected] (JS) Research Institute for Symbolic Computation (RISC), Johannes Kepler University
E-mail address : [email protected] (GG) Johann Radon Institute for Computation and Applied Mathematics (RICAM), Aus-trian Academy of Sciences
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