Abstract
We prove an equilibrium stressability criterion for trivalent multidimensional frameworks. The criterion appears in different languages: (1) in terms of stress monodromies, (2) in terms of surgeries, (3) in terms of exact discrete 1forms, and (4) in Cayley algebra terms.
1 Introduction
In the previous century, Fuller [9] coined the term tensegrity, a combination of ‘tension’ and ‘integrity’, to describe networks of rods and cables, such as those created by artist Kenneth Snelson, in which the tension of the cables and the compression in the rods combine to yield structural integrity to the whole. More generally, the word tensegrity is used to describe a variety of practical and abstract structures, e.g. bicycle tires and tents, whose rigidity follows from the balance of tension and compression, in the mathematical literature the stress, on the members.
Practically, structures exhibiting equilibrium stressability may be generated and analyzed using conventional techniques of structural engineering [19]. Theoretically, tensegrity is often considered as part of the study of geometric constraint systems, [2, 17]. The classical tensegrity model consists of a set of vertices V, two graphs, (V, S) and (V, C), the graph of struts and the graph of cables, together with a placement function \(\mathbf {p}:V \rightarrow \mathbb {R}^d\). One looks for a motion of the placed vertices such that the distances between pairs of vertices connected by struts do not fall below their initial values, and such that the distances between pairs of vertices connected by cables do not expand beyond their initial value. If no such motion exists, apart from the rigid motions of the space itself, the system is said to be rigid. A stress is a function \(s:S \cup C \rightarrow \mathbb {R}\), with \(s(t) \geqslant 0\) for all struts \(t \in S\) and \(s(c) \leqslant 0\) for all cables \(c \in C\). A stress is an equilibrium stress if for each vertex v
There are two avenues in which the existence of a proper, i.e. nowhere zero, equilibrium stress may allow one to establish structural integrity. A result of Roth and Whiteley [13] states that if a tensegrity has a proper equilibrium stress, and if the placement for \((V,C \,{\cup }\, S)\) is statically rigid as a bar and joint framework, then that tensegrity must be first order rigid and hence rigid. More delicately, if the proper equilibrium stress passes the secondorder stress test of Connelley and Whiteley [3], then the tensegrity structure is second order rigid, hence rigid.
Our goal in this paper is to generalize such tensegrity structures as they appear in Roth and Whiteley [13] to higher multidimensional frameworks and show several different equilibrium stressability criteria. Our investigation was inspired by real world tensile structures which are supported by structural elements bearing compression forces (often used to span large areas). Such structures can be seen as physical realizations of multidimensional stressable frameworks. Typical examples are camping tents, in which the fabric bears tension and the poles and the ground the compression, or, in the large scale, roof structures like in the stadium in the Olympiapark in Munich. We can also think of the multilayer freeform structures in architecture [11] which are realized in static equilibrium. Also soap films as minimal surfaces represent particular smooth limits of frameworks with tensile stresses in equilibrium.
Another important aspect of proper equilibrium stresses is the connection between the existence of an equilibrium stress and the lifting of embedded graphs into higher dimensions, as provided by the theory of Maxwell–Cremona. It was understood by Lee, Ryshkov, Rybnikov, and some others that the connection between equilibrium stresses and lifts extends in certain cases to arbitrary CWcomplexes M realized in any dimension d, also not necessarily embedded. Although the subject can be traced earlier (see [8, 20]), the first systematic study of multidimensional stresses, liftings and reciprocal diagrams was undertaken by Rybnikov in [14, 15].
In the present paper, we investigate structures which are polyhedral CW complexes where all faces in which forces appear are of the same dimension. This is the simplest practical case. We introduce dframeworks and their equilibrium stresses, i.e. selfstresses, in a slightly broader way than it was done by Rybnikov (Sect. 2). We introduce face paths and stress transition along such face paths in dframeworks in Sect. 3. We also give (Sect. 4) necessary and sufficient conditions for the equilibrium stressability of trivalent frameworks. This result appears as a generalization of equilibrium stressability criteria for classical tensegrities [7]. The conditions are equivalently expressed in terms of exact discrete multiplicative 1forms, Cayley algebra, or, in terms of some surgeries introduced in Sect. 4.
In Sect. 5 we complete our treatment by linking our model with the already existing constructions of Rybnikov [14, 15]. We will explain how Rybnikov’s frameworks (Rframeworks) constitute a subclass of our d–frameworks introduced by Definition 2.1 and why the space of d frameworks properly contains the space of Rframeworks, so our stressability results apply to his objects as well. All the examples in Sect. 5 are valid for both models.
2 Main definitions and constructions
Our model for selfstressable frameworks in this paper will be based on the following structure. Let \(D>d\geqslant 1\) be two integers. The term plane will refer to an affine subspace in \(\mathbb {R}^D\).
Definition 2.1
A dframework \(\mathscr {F}=(E,F,I,\mathbf {n})\) consists of E, a collection of \((d\,{}\,1)\)dimensional planes in \(\mathbb {R}^D\); F, a collection of ddimensional planes in \(\mathbb {R}^D\); a subset \(I \subset \{(p,q) \,{\in }\, (E {\times }F) \,{}\, p\,{\subset }\, q\}\); a function \(\mathbf {n}\) assigning to each pair (e, f) with \(e\in E\) and \((e,f)\in I\), a unit vector \(\mathbf {n}(e,f)\) which is contained in f and which is normal to e. We call planes from F faces and planes from E edges. The set I is called the set of incidences.
A dframework is called generic if, for every \(e\in E\), all the planes f with \((e,f)\in I\) are distinct.
Let \(\mathscr {F}=(E,F,I,\mathbf {n})\) be a dframework. A stress s on \(\mathscr {F}\) is any function \(s:F\rightarrow \mathbb {R}\). A framework \(\mathscr {F}\) together with a stress s is said to be in equilibrium if for every \(e\in E\) we have
Such a stress is called an equilibrium stress, a selfstress or sometimes a prestress for \(\mathscr {F}\).
Definition 2.2
A dframework is said to be selfstressable or a tensegrity if there exists a nonzero selfstress on it.
Example 2.3
The simplest nontrivial example here is the classical case of graphs in the plane (\(D=2\), \(d=1\)).
We say that a dframework is trivalent if each element of E is incident (i.e., contained in a pair in I) to precisely three elements of F.
Example 2.4
Let \(d = 2\) and \(D = 3\). Consider a 2framework whose edges, E, and faces, F, correspond to the edges and triangles of the graph \(K_5\), embedded in \(\mathbb {R}^3\). If four of the vertices, \(\{1,2,3,4\}\), of \(K_5\) are placed as vertices of a regular tetrahedron and the fifth one as their centroid, then the resulting 2framework is generic in our sense. For each edgeface pair choose the normal to be a unit vector pointing into the interior of the facetriangle. Note that the chosen normals sum to the zero vector around the lines corresponding to edges \(\{ i,5\}\), so choosing equal stress on these interior triangles leaves those edges equilibrated. Then, it is easy to see that choosing stresses on the exterior and interior triangles in the ratio \(\sqrt{6}/4\) yields an equilibrium stress.
In this example one may imagine the interior expanding triangles exerting an outward force balanced by the contracting “skin” of the exterior triangles.
Example 2.5
Again, let \(d = 2\) and \(D = 3\). We may create a different 2framework based on the graph \(K_5\) in \(\mathbb {R}^3\) by keeping E as before, and associating the faces F to the \(K_4\) subgraphs of \(K_5\), with the usual incidence relation. Since any two \(K_4\)’s intersect in three edges, their face planes must be identical, and the five vertices of the embedded \(K_5\) must be coplanar.
Since the normal vectors all lie in the plane of the \(K_5\), it is no loss of generality to assume that the vertices lie on a regular pentagon, and it is quickly checked that only the zero stress satisfies equation (1).
Example 2.6
Let \(d = 2\) and \(D = 3\). Consider the vertices of a regular cube and set E to be the set of all lines joining a pair of nonantipodal vertices (see Fig. 1). The face planes F consist of all six planes containing the faces of the cube, together with the six planes containing antipodal pairs of cube edges, as well as the eight planes of the dual tetrahedra. Let incidences be induced by containment. Since each plane contains a polygon of edges supported by incident lines of the structure, we may take the normals to be inwardly pointing unit vectors. It is easy to check that the selfstresses on the three types of faces are in the ratio \(1\,{:}\,{}\sqrt{2}\,{:}\,\sqrt{3}/4\).
Example 2.7
This example has two versions, both with \(d = 2\) and \(D = 3\). Consider the vertices of an octahedron, regularly embedded in \(\mathbb {R}^3\) (Fig. 2).
The set of 12 edge lines E lie along the edges of the octahedron, and the set F of 11 face planes will consist of those eight supporting triangles of the octahedron, together with the three planes which pass through four coplanar vertices. Let the incidences be all those induced by containment and, as before, let all normals be chosen inwardly pointing with respect to the triangle or square to which they belong. Then this is easily computed to be selfstressable. In fact, the selfstress is unique, since each each line is incident to three distinct planes.
As an alternative, we can take each of the three planes containing four vertices to have multiplicity 2, with each one incident to a different pair of opposite lines, and with the same choice of normals. This structure consisting of 12 lines and 14 planes is also a selfstressable framework.
3 Main definitions of Cayley algebra
Let us conclude this introductory section with a few words about Cayley algebra. Cayley algebra is an algebra whose elements are affine subspaces of a given Euclidean space. Cayley algebra has two operations: the intersection and the sum.
where \(\langle L_1,L_2\rangle \) is the plane spanned by \(L_1\) and \(L_2\); these two operations are sometimes referred to as meet and join operations.
4 Selfstressability of frameworks
From now on we study selfstressability of trivalent dframeworks in \(\mathbb {R}^{d+1}\), namely we assume that \(D=d+1\).
4.1 Selfstressability of facepaths and facecycles
Let us start with the following general definition.
4.1.1 Facepath and facecycle
Let \(E = (e_1, \ldots , e_k)\) be a sequence of distinct \((d\,{}\,1)\)dimensional planes in \(\mathbb {R}^D\); \(F = (f_{0,1}, f_{1,2}, \ldots , f_{k,k+1})\) be a sequence of ddimensional planes in \(\mathbb {R}^D\); \(\widehat{F} = (\hat{f}_{1}, \ldots , \hat{f}_k)\) be a sequence of ddimensional planes in \(\mathbb {R}^D\). Then the collection \((E,F,\widehat{F})\) is said to be a facepath if for \(i=1,\ldots ,k\) we have
For example, removal of two antipodal triangles of an octahedron leaves a facecycle with six faces.
Denote the set of pairs defined by these inclusions by I. For a particular choice of normals \(\mathbf {n}\) we obtain a dframework \(\mathscr {F} = (E,F\,{\cup }\,\widehat{F}, I, \mathbf {n})\) which is called a facepath dframework, see Fig. 3. It is called a facecycle dframework if \(f_{0,1}=f_{k,k+1}\). In this case it is denoted by \(C(E,F,\widehat{F},\mathbf {n})\), see Fig. 4.
4.1.2 Selfstressability of facepath dframeworks
Proposition 3.1
Any generic facepath dframework which contains no facecycle has a onedimensional space of selfstresses. All the stresses for all the planes of F and \(\widehat{F}\) are either simultaneously zero, or simultaneously nonzero.
Proof
Setting \(s(f_{0,1})=1\) we inductively define all stresses for all other planes using equation (1). Therefore a nonzero selfstress exists. By construction the obtained stress is nonzero at all planes of F and \(\widehat{F}\).
Once we know any of the stresses at one of the planes of F and \(\widehat{F}\), we reconstruct the remaining stresses uniquely using equation (1). Hence the space of stresses is at most onedimensional. Therefore, all selfstresses are proportional to a selfstress that is nonzero at all planes of F and \(\widehat{F}\). \(\square \)
4.1.3 Edgeorientation transition
Suppose we have a face path whose edges are \(e_1, e_2, \ldots \), and we are given an orientation of \(e_i\) by declaring a frame spanning \(e_i\) as positive. We pass over this positive orientation on \(e_i\) to a positive orientation on \(e_{i+1}\) by requiring that the given frame of \(e_i\) together with \(\mathbf {n}(e_i,f_{i,i+1})\) and a new (chosen to be positive) frame of \(e_{i+1}\) together with \(\mathbf {n}(e_{i+1},f_{i,i+1})\) differ by an orientation reversing automorphism on \(f_{i,i+1}\). We call this the edgeorientation transition.
A facecycle dframework \(C(E,F, \widehat{F},\mathbf {n})\) is said to be edgeorientable if the edgeorientation transition around the cycle returns to the starting edge in its initial orientation. See Fig. 5 for an example of a facecycle which is not edgeorientable.
Nonorientable facecycles are a usual phenomenon in frameworks. Indeed, we see them even in small examples like Example 2.4. Figure 5 depicts such a facecycle. Here the first and the last edges coincide, but are oppositely oriented.
Note that the edgeorientability of a cycle depends neither on the choice of the first element \(e_1\in E\), nor the choice of direction in the cycle. We have the following proposition.
Proposition 3.2
A facecycle dframework \(C=C(E,F,\widehat{F},\mathbf {n})\) has the following properties:

(i)
Reversing simultaneously all the normals at a single \(e_i\in E\) (namely \(\mathbf {n}(e_i,f_{i1,i})\), \(\mathbf {n}(e_i,f_{i,i+1})\), and \(\mathbf {n}(e_i,\hat{f}_{i})\)) does not change selfstressability or orientability of C.

(ii)
Reversing simultaneously the normals at \(f_{i,i+1}\in F\) (namely reversing \(\mathbf {n}(e_i,f_{i,i+1})\) and \(\mathbf {n}(e_{i+1},f_{i,i+1})\)) does not change the selfstressability or orientability of C.

(iii)
Reversing the normal \(\mathbf {n}(e_i,\hat{f}_{i})\) does not change the selfstressability or orientability of C.
Proof
In all the items we change altogether an even number of normals for all the faces in F. Therefore, orientability is preserved.
The change in (i) does not change the equations of selfstressability, so it preserves selfstressability. For (ii) and (iii) the change of the signs of stresses \(s(f_{i,i+1})\) and \(s(\hat{f}_i)\) respectively delivers the equivalence of the selfstressability conditions. \(\square \)
4.1.4 Selfstressability of facecycles of length 3
Let us consider a trivalent cycle \(C(E,F,\widehat{F},\mathbf {n})\) of length 3 with
(for a schematic sketch see Fig. 6). Create a new plane \(\hat{f}'_3\) by the following Cayley algebra algorithm (see Fig. 7):

(i)
\(g_1 = (e_1 {\vee }e_2) {\wedge }\hat{f}_3\),

(ii)
\(g_2 = f_{2, 3} {\wedge }\hat{f}_1\),

(iii)
\(g_3 = (g_1 {\vee }g_2) {\wedge }f_{3, 1}\),

(iv)
\(g_4 = (g_2 {\vee }e_1) {\wedge }(g_3 {\vee }e_2)\),

(v)
\(\hat{f}'_3 = e_3 {\vee }g_4\).
In this notation, the stressability conditions are given by the following.
Proposition 3.3
A generic facecycle dframework
is selfstressable if and only if
where \(\hat{f}'_3\) is constructed as above in step (v) (see also Fig. 7).
In the proof we will use spherical frameworks, so let us continue with the following remark and definition.
Remark 3.4
Consider a twodimensional sphere \(S^2\) and let G(P) be a mapping of a graph G to \(S^2\) such that edges are mapped to geodesics (i.e., the arcs of great circles) and vertices are mapped to their endpoints. Let us assign a stress for every edge of G(P). Then all the definitions of stressability, selfstressability, cycles, etc. may be literally translated to spherical frameworks.
Proof of Proposition 3.3 Let us first examine the case where \(d=1\) and \(D=2\), with stresses at all edges of the triangles equal to 1. The triangle in Fig. 8 (left) corresponds to an edgeoriented dframework. In this case we have a classical definition of selfstressable framework (a simple barjoint framework) and therefore \(\hat{f}_1\cap \hat{f}_2 \cap \hat{f}_3\) is not empty.
Note that this intersection is empty for the triangle in Fig. 8 (right), which corresponds to a nonedgeoriented choice of normals. The condition for nonedgeoriented selfstreassable frameworks is more complicated as we will see below.
Let us consider a selfstressed trivalent cycle \(C(E,F,\widehat{F},\mathbf {n})\) of length 3 (for a schematic sketch see Fig. 6). Then let us change the direction of the normal \(\mathbf {n}(e_3, f_{2,3})\). Consequently, its orientability changes and the old stress for this new cycle is not a selfstress. However, by changing the dplane \(\hat{f}_3\) to a new dplane \(\hat{f}'_3\) we can resolve the stresses around \(e_3\) again to reobtain a selfstressed framework.
Let us consider the following two cases in the planar situation as depicted by Fig. 6. The classical selfstressable framework (Fig. 6), which corresponds to the orientable case, has the property that the lines \(\hat{f}_1, \hat{f}_2, \hat{f}_3\) meet in a point (see, e.g., [6]). Now changing the orientation of \(\mathbf{n}(e_3, f_{2,3})\) yields \(\hat{f}'_3\) as the new dplane (see Fig. 6 right). From the parallelogram in Fig. 6 (right) we derive the condition for the nonorientable case.
Consider four straight lines of a line pencil. A pencil consists of all lines in a plane passing through a point. In projective geometry a pair of lines of a pencil is said to separate harmonically another pair of lines from the same pencil if their crossratio is equal to \(1\) (see, e.g., [12]). Such a harmonic position is characterized also by a simple incidence relation that is depicted in Fig. 7 where the harmonic position is fulfilled by the pairs \((f_{2, 3}, f_{3, 1})\) and \((\hat{f}_3, \hat{f}'_3)\). Since this property is characterized by incidence relations of points and lines it is therefore expressible in terms of Cayley algebra.
Our next goal is to prove the statement for the case \(d=2\). We do this in the following three steps: In Step 1 we reformulate the problem to the stressability of a spherical cycle; Step 2 reduces that to selfstressability in the plane; Step 3 then corresponds to the planar case (\(d=1\)) above.
Step 1. Consider the point
and let S be the unit sphere centered at v (if v is at “infinity” we can take a plane orthogonal to the three edges \(e_1, e_2, e_3\) instead of S). Let also \(C_1=S\cap C\): here we intersect all edges and faces of C with S, the normals are uniquely defined for C as they are orthogonal to \(e_1\), \(e_2\), and \(e_3\), respectively. It is clear that C is selfstressable if and only if \(C_1\) is selfstressable in the sphere S (since all the equilibrium conditions for edges remain the same).
Step 2. Let now \(C_2\) be the gnomic projection (i.e., the projection from the center or the sphere) of \(C_1\) to some 2plane. By [17, Chapter 17] (for the original manuscript see [16]) cycle \(C_2\) is selfstressable if and only if \(C_1\) is selfstressable.
Step 3. Finally the selfstressability in the plane was studied above.
Next we describe how to reduce any dimension d to the 2dimensional case. Let us consider a 3plane \(\pi \) orthogonal to the \((d\,{}\,2)\)plane \(f_{1,2}\cap f_{2,3}\cap f_{3,1}\). Let \(\widehat{C}=\pi \cap C\) (here we intersect all edges and faces of C with \(\pi \), the normals are uniquely defined for \(\widehat{C}\) as they are orthogonal to \(f_{1,2}\cap f_{2,3}\cap f_{3,1}\)). It is clear that C is selfstressable if and only if \(\widehat{C}\) is selfstressable in \(\pi \) (since all the equilibrium conditions for edges remain the same). The case of \(\widehat{C}\) corresponds to \(d=2\) which it was studied above.
The nonedgeorientable case is reduced to the edgeorientable in the following way. Let us make our threecycle orientable by changing the last normal \(n(e_3,f_3)\) (denote the resulting set of normals by \(\mathbf{n}'\)). In this case, in order to preserve the property of selfstressability condition at the edge \(e_3\), we should also change the sign of one of the coordinates for the plane \(\hat{f}_3\). The resulting plane is the plane \(\hat{f}_3'\), whose Cayley algebra expression is described above (see step (v)). Now the stressability of the original nonedgeorientable cycle is equivalent to the stressability of an edgeoriented cycle
This concludes the proof. \(\square \)
4.2 \(\mathbf{H} \)surgeries
In this section we discuss Hsurgeries and elementary surgeryflips on facepaths dframeworks and facecycle dframeworks which preserve selfstressability in \(\mathbb {R}^{d+1}\). Such operations reduce the combinatorial complexity by simple and local operations which preserve the rigidity of a framework. In fact they are frequently used, for example, the “DeltaY” operation replaces triangles by trivalent vertex stars (see, e.g., [2]).
Definition 3.5
Let \(1 \leqslant i\leqslant n\) be positive integers (\(n\geqslant 4\)), and let
be a facepath (or a facecycle if \(f_{n,1}=f_{0,1}\)) dframework. Denote
We say that the \(\mathrm{H}_i\)surgery of C is the following facepath (facecycle) dframework (see Fig. 9):
The normals \(\mathbf {n}'\) coincide with the normals of \(\mathbf {n}\) for the same adjacent pairs. We have three extra normals in \(\mathbf {n}'\) to the new element in E:
The first two are defined by the fact that the cycle:
of length 3 is edgeorientable. The direction of the third normal does not play any role here (and hence can be chosen arbitrarily).
Alternatively, in terms of Cayley algebra \(\hat{f}'_i\) reads as
In the planar case we have precisely \(\mathrm{H}\)surgeries on framed cycles (i.e., facecycle 1frameworks) that were used for the conditions of planar selfstressable frameworks (for further details see [6]).
In order to have a welldefined \(\mathrm{H}\)surgery, one should consider several simple conditions on the elements of F and \(\widehat{F}\).
Definition 3.6
We say that an \(\mathrm{H}_i(C)\)surgery is admissible if

(i)
the planes \(f_{i1,i}\) and \(f_{i,i+1}\) do not coincide;

(ii)
the planes \(\hat{f}_i\) and \(\hat{f}_{i+1}\) do not coincide;

(iii)
the planes \(f_{i1,i}{\wedge }f_{i,i+1}\) and \(\hat{f}_i{\wedge }\hat{f}_{i+1}\) do not coincide.
For an admissible \(\mathrm{H}_i(C)\)surgery we have
where the last follows from Items (i) and (ii) above. Then by Item (iii) we have \(\dim \hat{f}'_i\geqslant d\). Since the four dplanes \(f_{i1,i}\), \(f_{i+1,i+2}\), \(\hat{f}_{i}\), and \(\hat{f}_{i+1}\) share the affine subspace \(e_i \cap e_{i+1}\), which is \((d\,{}\,2)\)dimensional, we obtain (by the dimension formula)
hence \(\dim \hat{f}'_i \leqslant d\) and therefore \(\dim \hat{f}'_i = d\).
Let us distinguish the following elementary surgeryflips.
Definition 3.7
An elementary surgeryflip is one of the following surgeries:

An admissible \(\mathrm{H}_i\)surgery or its inverse.

Removing or adding consecutive duplicates at position i. Here we say that we have a duplicate at position i if
$$\begin{aligned} e_i=e_{i+1}, \quad f_{i,i+1}=f_{i+1,i+2}, \quad \hbox {and} \quad \hat{f}_i=\hat{f}_{i+1}. \end{aligned}$$ 
Removing or adding a loop of length 2. Here we say that we have a simple loop of length 2 at position i if
$$\begin{aligned} e_i=e_{i+2}, \quad f_{i,i+1}=f_{i+2,i+3}, \quad \hbox {and} \quad \hat{f}_i=\hat{f}_{i+2}. \end{aligned}$$
Proposition 3.8
Assuming that a surgery is admissible, a facecycle dframework C is selfstressable if and only if the facecycle dframework \(\mathrm{H}_i(C)\) is selfstressable.
Proof
Assume that C has a nonzero selfstress s. Let us show that \(\mathrm{H}_i(C)\) has a selfstress.
Consider the facecycle dframework
where
and \(\mathbf {n}\) is constructed according to Definition 3.5. This facecycle dframework admits a selfstress by Proposition 3.3 since three dplanes of \((f_i,f'_{i},f_{i+1})\) intersect in a plane of dimension \(d2\). Now let us add \(C_i\) to C taking the selfstress \(s_i\) which negates the stress at \(e_{i,i+1}\). Then the stresses at \(\hat{f}_i\) for C and \(C_i\) negate each other; and the stresses at \(f_{i1,i}\) for C and \(C_i\) coincide. For the same reason the stresses at \(f_{i+1,i+2}\) for C and \(C_i\) coincide. Therefore, the constructed selfstress is in fact a nonzero selfstress on \(\mathrm{H}_i(C)\).
The same reasoning works for the converse statement. In fact adding the \(C_i\) to C provides an isomorphism between the space of selfstresses on C and the space of selfstresses on \(\mathrm{H}_i(C)\). \(\square \)
Remark 3.9
It is possible to describe one \(\mathrm{H}\)surgery in terms of Cayley algebra. Consider a facecycle dframework C with admissible \(\mathrm{H}_i(C)\)surgery. We have only one new plane \(\hat{f}'_i\) in this case, and its Cayley expression is
4.3 Stress transition and stress monodromy
We will now adapt to our setting the notion of “quality transfer” of Rybnikov [14]. Let us start with the following definition of path genericity which is a relaxed version of genericity from before.
Definition 3.10
A facepath dframework \((E,F,\widehat{F}, \mathbf {n})\) is said to be pathgeneric if dplanes of F do not coincide with the adjacent dplanes of \(\widehat{F}\).
The difference to just generic dframeworks is that in the above definition the adjacent dplanes of F are allowed to coincide.
Proposition 3.11
Any pathgeneric facepath dframework which contains no facecycle has a onedimensional space of selfstresses. All the stresses for all the planes of F are either simultaneously zero, or simultaneously nonzero.
Proof
Setting stress \(s(f_{0,1})=1\) we inductively define all stresses for all other planes using equation (1). Therefore a nonzero selfstress exists. By construction the obtained stress function is nonzero at all planes of F.
Once we know any of the stresses at one of the planes \(f_{i,i+1}\), we reconstruct the remaining stresses uniquely using equation (1). Hence the space of stresses is at most onedimensional. Therefore, all selfstresses are proportional to a selfstress that is nonzero at all planes of F. \(\square \)
Definition 3.12
Let \(\Gamma \) be a pathgeneric facepath dframework with starting plane \(f_a\in F\) and ending plane \(f_{z}\in F\). Assign some stress s to the first plane. Due to genericity, it uniquely defines the stress on the second face. The stress on the second face uniquely defines the stress on the third face, and so on. So the stress on \(f_a\) uniquely defines the stress on \(f_z\). This is called the stress transition along the face path.
If \(f_a=f_z\), that is, we have a facecycle, we arrive eventually at some stress \(s'\) assigned to \(f_a\) again. The ratio \(s(f_{a})/s(f_{z})\) is called the stressmonodromy along C. A stress monodromy of 1 is called trivial.
It is clear that:
Lemma 3.13

1.
A pathgeneric facecycle is selfstressable if and only if the stress monodromy is trivial.

2.
The monodromy does not depend on the choice of the first face.

3.
Reversal of the direction of the cycle takes monodromy m to 1/m.

4.
Monodromy behaves multiplicatively with respect to homological addition: the monodromy of the homological sum is the product of monodromies.
4.4 Facepath equivalence
Let us now introduce the notion of equivalent facepath dframeworks.
Definition 3.14
Two pathgeneric facepath (facecycle) dframeworks \(\Gamma _1\) and \(\Gamma _2\) starting from the plane \(f_a\) and ending at the plane \(f_z\) are equivalent if there exists a sequence of elementary surgeryflips taking \(\Gamma _1\) to \(\Gamma _2\).
It turns out that equivalent facepath dframeworks have equivalent stresstransitions.
Proposition 3.15
The stresstransition of two pathgeneric equivalent facepath dframeworks coincides.
Proof
It is enough to prove this statement for any elementary surgeryflip. In case of Hsurgeries we must show that the facepath dframeworks
and
have the same stresstransition (see Fig. 10).
This is equivalent to the fact that the facecycle dframework
(where \(\mathbf {n}''\) is as in the cycle of Definition 3.5) has a unit stresstransition (i.e. trivial monodromy or, equivalently, is selfstressable).
By the construction of Definition 3.5 we get that this cycle is edgeorientable, and that the intersection
Therefore, by Proposition 3.3 it is selfstressable.
The cases of removing duplicates or loops of length 2 are straightforward. \(\square \)
4.5 Facepath dframeworks in dframeworks
In this subsection we briefly discuss facepath dframeworks and facecycle dframeworks that are parts of a larger dframework.
Definition 3.16
Let \(\mathscr {F}\) be a trivalent dframework. Then for every (cyclic) sequence of adjacent dplanes \(\gamma \) we naturally associate a facepath dframework (facecycle dframework) \(\Gamma ((E,F,\widehat{F}, \mathbf {n}),\gamma )\) with

F is the sequence of the planes spanned by the corresponding dplanes of \(\gamma \);

E is the sequence of the intersections of the dplanes of the above F;

\(\widehat{F}\) is the sequence of planes of \(\mathscr {F}\) that are adjacent to the planes of E and distinct to the faces already considered in F;

\(\mathbf {n}\) is the corresponding sequence of normals defined by the normals of \(\mathscr {F}\).
We say that a facepath dframework (facecycle dframework) \(\Gamma ((E,F,\widehat{F}, \mathbf {n}),\gamma )\) is induced by \(\gamma \) on \(\mathscr {F}\).
Induced facepath and facecycle dframeworks have a natural homotopy relation, which is defined as follows.
Definition 3.17

Two induced facepath dframeworks \(\Gamma _1\) and \(\Gamma _2\) for \(\mathscr {F}\) starting from the plane \(f_a\) and ending at the plane \(f_z\) are facehomotopic if there exists a sequence of elementary surgeryflips taking \(\Gamma _1\) to \(\Gamma _2\) and such that after each surgeryflip we have an induced facepath dframework for \(\mathscr {F}\).

Two facecycle dframeworks \(\Gamma _1\) and \(\Gamma _2\) are facehomotopic if there exists a sequence of elementary surgeryflips taking \(\Gamma _1\) to \(\Gamma _2\) and such that after each surgeryflip we have an induced facepath dframework for \(\mathscr {F}\).
Finally we formulate the following important property of facehomotopic facepath and facecycle dframeworks. Its proof directly follows from Proposition 3.15.
Proposition 3.18
Facehomotopic facepath (facecycle) dframeworks have the same stresstransition (stressmonodromy).
5 Geometric characterizations of selfstressability for trivalent dframeworks
In this section we discuss the practical question of writing geometric conditions for cycles. We characterize selfstressable trivalent dframeworks in terms of exact discrete multiplicative 1forms and in terms of resolvable cycles. Before that we show that a trivalent dframework is selfstressable if and only if every path and every loop is selfstressable.
Definition 4.1
A facecycle dframework is called a faceloop dframework if it contains no repeating planes. A dframework where any two faces are connected by a facepath is called faceconnected.
Let us formulate the following general theorem.
Theorem 4.2
Consider a generic faceconnected trivalent dframework. Then the following three statements are equivalent.

(i)
\(\mathscr {F}\) has a nonzero selfstress (which is in fact nonzero at any dplane).

(ii)
For every two dplanes \(f_a,f_z\) in \(\mathscr {F}\) the stresstransition does not depend on the choice of an induced facepath dframework on \(\mathscr {F}\).

(iii)
Every induced faceloop dframework on \(\mathscr {F}\) is selfstressable.
Proof
(ii) \(\Leftrightarrow \) (i): Item (i) tautologically implies Item (ii). Let us show that Item (ii) implies Item (i). Fix a starting face \(f_a\) and put a stress \(s(f_a)=1\) on it. Expand the stress to all the other faces. By assumption this can be done uniquely. Therefore, this stress is a selfstress. (Indeed, if we do not have the equilibrium condition at some plane e, then at the planes incident to e we have more than one possible stresstransition.)
(ii) \(\Rightarrow \) (iii): Indeed any simple facecycle dframework on \(\mathscr {F}\) can be considered as a path–generic facepath dframework with \(f_a=f_z\). By condition of Items (ii) the stresstransition equals 1, and therefore this facecycle dframework is selfstressable. The last is equivalent to Item (iii).
(iii) \(\Rightarrow \) (ii): Let us use reductio ad absurdum. Suppose Item (iii) is true while Item (ii) is false. If Item (ii) is false then there exist at least two pathgeneric facepath dframeworks with the same \(f_a\) and \(f_z\) where the stresstransitions fail to be the same. Now the union C of the first facepath dframework and the inverse second is an induced facecycle dframework on G(M) with nonunit stresstransition. Let us split C into consecutive loops \(C_1,\ldots , C_k\). At least one of them should have a nonunit translation. Therefore, Item (iii) is false as well, a contradiction.
For completeness of the last proof we should add the following two observations regarding cycles of small length. Firstly, the stress transition remains constant at planes that repeat successively two or more times. This happens due to genericity of \(\mathscr {F}\): there are zero contributions from \(\hat{f}_i\) in case if \(f_{i1,i}=f_{i,i+1}\). And secondly, if it happens that \(f_{i1,i}=f_{i+1,i+2}\) then we immediately have \(e_i=e_{i+1}\) and therefore again the stresstransitions at \(f_{i1,i}\) and at \(f_{i+1,i+2}\) coincide. \(\square \)
5.1 Ratio condition for selfstressable multidimensional trivalent frameworks
In this section we characterize generic trivalent dframeworks \(\mathscr {F}\) with respect to their selfstressability in terms of specific products of ratios. More precisely, we equip each dframework with a socalled discrete multiplicative 1form which turns out to be exact if and only if the dframework is selfstressable. Let us start with the definition of discrete multiplicative 1forms (see, e.g., [1]).
Definition 4.3
A real valued function \(q :\overrightarrow{E}(G) \rightarrow \mathbb {R}{\setminus }\{0\}\) (where \(\overrightarrow{E}(G)\) denotes the set of oriented edges of the graph G) is called a discrete multiplicative 1form, if \(q(a) = 1/q(a)\) for every \(a \in \overrightarrow{E}(G)\). It is called exact if for every cycle \(a_1, \ldots , a_k\) of directed edges the values of the 1form multiply to 1, i.e.,
Now, as a next step we will equip any general trivalent dframework with a discrete multiplicative 1form q. However, we will not define q directly on the dframework but on what we call its face graph. The notion “face graph” is based on the wellknown notion “line graph” from classical graph theory. The vertices of the line graph are in a onetoone correspondence with the edges of a given graph. The edges of the line graph connect two vertices if and only if the two respective edges of the given graph are emanating from the same vertex.
Definition 4.4
The face graph of a dframework is a graph whose vertices are the ddimensional planes and two ddimensional planes are connected by an edge if they share a \((d\,{}\,1)\)dimensional plane.
Consequently, the edges of the face graph of \(\mathscr {F}\) can be identified with triples of successive \((d\,{}\,1)\)planes \(a_i\, {:}{=}\, (e_{i  1}, e_i, e_{i + 1})\) (where \(e_i \in E\)). So let us now equip the face graph of \(\mathscr {F}\) with a discrete multiplicative 1form. For an illustration see Fig. 11.
As our dframework is trivalent the \((d\,{}\,1)\)plane \(e_i\) is contained in three dplanes \(f_{i  1, i}, f_{i, i + 1}, \hat{f}_i\) hence the corresponding normals \(\mathbf {n}(e_i, f_{i  1, i})\), \(\mathbf {n}(e_i, f_{i, i + 1})\), \(\mathbf {n}(e_i, \hat{f}_i)\) are linearly dependent, i.e., lie in a 2plane. This together with the fact that the dframework is generic implies that there are \(\lambda _{i  1, i}, \lambda _{i, i + 1}, \hat{\lambda }_i \in \mathbb {R}\setminus \{0\}\) such that
Now we are in position to define our discrete multiplicative 1form on the oriented face graph by
since clearly \(q(a_i) = 1/q(a_i)\) is fulfilled. The geometric meaning of \(q(a_i)\) is the following (cf. Fig. 11 right). Denote by \(\mathbf {r_i}\) the intersection point of the straight line with direction \(\mathbf {n}(e_i, \hat{f}_i)\) and intersect it with the line through \(\mathbf {n}(e_i, f_{i  1, i})\) and \(\mathbf {n}(e_i, f_{i, i + 1})\). A simple computation shows
Thus \(q(a_i)\) is the affine ratio of the three points \(\mathbf {n}(e_i, f_{i  1, i}), \mathbf {r_i}, \mathbf {n}(e_i, f_{i, i + 1})\), i.e., \(q(a_i) = (\mathbf {n}(e_i, f_{i  1, i})  \mathbf {r_i}) \,{:}\, (\mathbf {r_i}  \mathbf {n}(e_i, f_{i, i + 1}))\).
With that definition of a discrete multiplicative 1form we can now characterize selfstressable dframeworks.
Theorem 4.5
A generic trivalent dframework is selfstressable if and only if the discrete multiplicative 1form defined by (3) is exact.
Proof
Suppose the dframework has a selfstress s. Therefore equation (1) implies
Comparison with equation (2) implies that the coefficients in both equations are just a multiple of each other, i.e.,
Consequently, the value of the discrete multiplicative 1form is the ratio of neighboring stresses:
Therefore it is easy to see that the product of values \(q(a_i)\) along any closed loop in the face graph multiplies to 1:
Now conversely, let us assume that the discrete multiplicative 1form q is exact. By Theorem 4.2 it is sufficient to show that each loop of the form \(e_1, \ldots , e_k\) of \((d \,{}\, 1)\)planes is selfstressable. Choose an arbitrary stress \(s(f_{1, 2}) \in \mathbb {R}{\setminus }\{0\}\) for the first dplane. Equation (1) and the dframework being generic then uniquely determines the stresses of the two other dplanes incident to \(e_1\), that is, \(s(f_{k, 1})\) and \(s(\hat{f}_1)\). Continuing, determining stresses this way defines all stresses along the loop including the last stress that we now denote by \(\tilde{s}(f_{k, 1})\) because it was defined before. However, the exactness of q gives
so \(\tilde{s}(f_{k, 1}) = s(f_{k, 1})\). Consequently, we can consistently define a nonzero stress. \(\square \)
5.2 Cayley algebra conditions
Let us start with the following important definition.
Definition 4.6

A facecycle dframework of length 3 is in general position if all six planes in the sequences F and \(\widehat{F}\) are pairwise distinct.

A facecycle dframework is resolvable if there exists a sequence of \(\mathrm{H}\)surgeries transforming it to a facecycle dframework of length 3 in general position.

A dframework is resolvable if all its simple induced facecycle dframeworks are resolvable.
We continue with the following definition about the Cayley algebra condition for a single facecycle resolvable dframework.
Definition 4.7
Consider a resolvable facecycle dframework
and any sequence of \(\mathrm{H}\)surgeries transforming it to a facecycle dframework
of length 3 in general position.

Let us write all elements of \(F'\) and \(\widehat{F}'\) in \(C'\) as Cayley algebra expressions of the elements of F and \(\widehat{F}\) in C. The resulting expressions are compositions of expressions of Remark 3.9.

Finally, we use the dimension condition of Proposition 3.3 for \(C'\) to determine if \(C'\) is stressable or not.
The composition of the above two items gives an existence condition for nonzero selfstresses on C. We call this condition a Cayley algebra geometric condition for C to admit a nonzero selfstress.
Note that one can write distinct Cayley algebra geometric conditions for C using different sequences of \(\mathrm{H}\)surgeries transforming C to a facecycle dframework of length 3 in general position. As is clear from the above definition, the Cayley algebra geometric conditions detect selfstressability on cycles. So we may use algebra to express selfstressability for a general trivalent dframework. Namely we have the following theorem.
Theorem 4.8
Let \(\mathscr {F}\) be a trivalent resolvable dframework and let further \(C_1,\ldots , C_n\) be all pairwise nonfacehomotopic faceloop dframeworks on \(\mathscr {F}\). Then \(\mathscr {F}\) has a selfstress if and only if it fulfills Cayley algebra geometric conditions for cycles \(C_1,\ldots , C_n\) as in Definition 4.7.
Proof
First of all the selfstressability of \(C_1,\ldots , C_n\) is equivalent to selfstressability of all faceloops on \(\mathscr {F}\). It follows directly from definition of facehomotopic paths. Hence by Theorem 4.2 the selfstressability of all \(C_1,\ldots , C_n\) is equivalent to selfstressability of \(\mathscr {F}\) itself.
Finally the geometric conditions for faceloop dframeworks \(C_i\), \(i=1,\ldots , n\), are described in Definition 4.7. \(\square \)
Corollary 4.9
Each realization of \(K_5\) in \(\mathbb {R}^3\) is selfstressable. Here we mean a realization of a 2framework associated with \(K_5\), in the spirit of Example 2.4. Namely, the edges are all the edges of \(K_5\), and faces are all the associated triangles.
Indeed, each faceloop in \(K_5\) is facehomotopic to a faceloop of length three. For them, the stressability condition is automatic. Another argument of stressability of \(K_5\) will appear later in Remark 5.11 as a consequence of Theorem 5.10.
6 Rframeworks and their selfstressability. Examples
Now we turn to an alternative notion of stressable frameworks, which is borrowed from Rybnikov’s papers [14, 15]. It represents a special case of Definition 2.1, as we will show in Proposition 5.8. The main differences between our definition of dframeworks (Definition 2.1) and Rybnikov’s setting (Definitions 5.3, 5.6, and 5.7) are the following:

In Rybnikov’s setting the faces of a framework are represented by some polytopes (possibly nonconvex and selfintersecting), whereas in our model (Definition 2.1) faces are affine planes. The polytopes of a Rybnikovframework define affine planes as their affine hulls, but not vice versa. So whenever we have a Rybnikovframework, we automatically have a framework in the sense of Definition 2.1.

In Rybnikov’s setting the choice of normal vectors \(\mathbf {n}\) is dictated by the Rframework, whereas in Definition 2.1 the collection of normal vectors \(\mathbf {n}\) can be chosen without any restrictions.
6.1 Rframeworks
Informally, Rframeworks are PL (piecewise linear) realizations of CWcomplexes in \(\mathbb {R}^{d+1}\). To make this precise, let us start with a reminder about CWcomplexes.
A finite CWcomplex is constructed inductively by defining its skeleta (for details see, e.g., [5]). The zero skeleton \({{\,\mathrm{\textit{sk}}\,}}_0\) is a finite set of points called vertices. By induction we construct \({{\,\mathrm{\textit{sk}}\,}}_k\) from a given \((k{}1)\)skeleton \({{\,\mathrm{\textit{sk}}\,}}_{k1}\), by attaching a finite collection of closed kballs \(B_i\) (called cells) by some continuous mappings \(\phi _i:\partial B_i \rightarrow {{\,\mathrm{\textit{sk}}\,}}_{k1}\). The images of \(B_i\) in the complex are called closed cells.
Definition 5.1
A regular CWcomplex is a CWcomplex such that

(i)
For each kcell \(B_i\), the mapping \(\phi _i \) is a homeomorphism between \(\partial B_i\) and a subcomplex of the skeleton \({{\,\mathrm{\textit{sk}}\,}}_{k1}\).

(ii)
The intersection of two closed cells is either empty or some single closed cell of this CWcomplex.
Let M be a regular finite CWcomplex with no cells of dimension greater than d. Its faces of dimension d will be called dfaces. The \((d\,{}\,1)\)faces are called the dedges. The \((d\,{}\,2)\)faces are called the dvertices.
Example 5.2
Let \(\overline{M}\) be a regular finite CWcomplex whose support^{Footnote 1}\(\overline{M}\) is a connected \((d{+}1)\)manifold, either closed or with boundary. Let M be its dskeleton.
In this setting we also have cells of \(\overline{M}\) of dimension \(d+1\). These will be called chambers.
Definition 5.3
Assume that a mapping \(p:\mathrm {Vert}(M)\rightarrow \mathbb {R}^{d+1}\) is such that the image of the vertex set of each kcell spans some affine kplane.
We say that p realizes M in \(\mathbb {R}^{d+1}\); we also say that the pair (M, p) is a realization of M, or a Rybnikovframework, or Rframework, for short.
Notation: given a cell \(f\in M\), we abbreviate the image of the vertex set \(p(\mathrm {Vert}(f))\) as p(f) and denote by \(\langle p(f) \rangle \) its affine span.
Definition 5.4
An Rframework is generic if, whenever two dfaces \(f_1\) and \(f_2\) share a dedge, then \(\langle p(f_1)\rangle \ne \langle p(f_2)\rangle \).
From now on we assume that all Rframeworks we deal with are generic.
As before, we say that an Rframework is trivalent if each dedge is incident to exactly three dfaces.
An Rframework is (3, 4)valent if each dvertex is incident to exactly four dedges (and therefore, to six dfaces).
By construction, vertices are mapped to points. One may also imagine that the 1cells of the complex are mapped to line segments. Therefore 2cells are mapped to some closed planar broken lines (polygons). Here selfintersections may occur. As the dimension of faces grows, the complexity of the associated geometrical object increases. However, there exist nice examples with convex polyhedra as images of the faces. In particular, if a face is a (combinatorial) simplex, one may think that its image is a simplex lying in \(\mathbb {R}^d\).
Example 5.5

(i)
The Schlegel diagram [21] of a convex \((d\,{+}\,2)\)polytope K is a realisation of the boundary complex of K. If K is a simple polytope, we arrive at a (3, 4)valent complex.

(ii)
More generally, the projection of a \((d\,{+}\,2)\)dimensional polyhedral body K (that is, of a body with piecewise linear boundary) to \(\mathbb {R}^{d+1}\) yields a realization (M, p) where M is homeomorphic to \(\partial K\).
6.2 Selfstresses and liftings
Now we turn to a particular notion of stresses, which is borrowed from Rybnikov’s paper [14] and represents a special case of Definition 2.1. The principal difference is that in Rybnikov’s setting the choice of normal vectors \(\mathbf {n}\) is dictated by the Rframework.
As Examples 2.4 and 5.5 show, in certain cases, a realization (M, p) represents all the faces as convex polytopes. Let us call such a realization noncrossing. Otherwise, we say that the realization is selfcrossing.
Let us start by introducing stresses for the noncrossing version (cf. [15]):
Definition 5.6
Assume that a noncrossing realization (M, p) is fixed. Let us assign to each pair (f, e), where e is a dedge contained in a dface f, a unit normal \(\mathbf {n}(e,f)\) to p(e) pointing inside the convex polytope p(f).
A realvalued function s defined on the set of dfaces is called a selfstress if at each dedge e of the complex M,
To relax the noncrossing condition, let us make some preparation following [15]. The informal idea is to triangulate the faces of the complex, since the representation of a simplex is never selfcrossing.
Pick a (combinatorial) orientation of each of the cells of M, and a (combinatorial) triangulation of M without adding new vertices. So each dface now is replaced by a collection of (combinatorial) simplices. The realization (M, p) yields a realization \((\widehat{M},p)\) of the new CWcomplex.
Definition 5.7
Assume that a generic realization (M, p) is fixed. Choose a triangulation \((\widehat{M},p)\) as is described above.
For a dedge \(e\in \widehat{M}\) and a dface \(f \in \widehat{M}\) containing g, choose \(\mathbf {n}(e,f)\) to be the unit normal to the oriented cell f at its simplicial face g whose orientation is induced by the orientation of f.
A realvalued function s on the set of dcells of M is called a selfstress if for every dedge e of \(\widehat{M}\), condition \((*)\) is fulfilled (cf. [15]).
This definition is proven to be independent of the choice of the combinatorial triangulation and also of the choice of the orientations of the faces.
The notion of stressed realizations has the following physical meaning. One imagines that the dfaces are realized by planar soap film. The faces are made of different types of soap, that is, with different physical property. Each of the faces creates a tension, which should be in equilibrium at the dedges. The tension is always orthogonal to the boundary of a face and lies in the affine hull of the face. A selfintersecting face produces both compression and tension as is depicted in Fig. 12.
We say that an Rframework (M, p) is selfstressable whenever there exists a nonzero stress.
Proposition 5.8
The following two statements hold:

(i)
Each Rframework yields a dframework \((E,I,F,\mathbf {n})\) which agrees with Definition 2.1. The incidences are dictated by the combinatorics of M. Its selfstressability agrees with the Definition 2.2.

(ii)
For \(d=1\), an Rframework is a planar realization of some graph. Its selfstressability agrees with the classical notion of selfstresses of graphs in the plane.
Assume now that M is the dskeleton of some \((d\,{+}\,1)\)dimensional manifold \(\overline{M}\), that is, the chambers are welldefined.
Definition 5.9
A lift of \((\overline{M}, p)\) is an assignment of a linear function \(h_C:\mathbb {R}^{d+1}\!\rightarrow \mathbb {R}\) to each chamber C. By definition, a lift satisfies the following: whenever two chambers C and \(C'\) share a dface f, the restrictions of \(h_C\) and \(h_{C'}\) on the affine span \(\langle p(f)\rangle \) coincide. A lift is nontrivial if (at least some of) the functions \(h_C\) are different for different chambers.
Let us fix some chamber C. Lifts that are identically zero on C form a linear space \(\text{ Lift }(\overline{M},p)\).
Theorem 5.10
Let M be the dskeleton of some \((d\,{+}\,1)\)dimensional manifold \(\overline{M}\) (cf. [15]).

(i)
If the first homology group of \(\overline{M}\) vanishes, that is,
$$\begin{aligned} H_1(\overline{M},\mathbb {Z}_2)=0, \end{aligned}$$then the linear space \(\mathrm {Lift}(\overline{M},p)\) and the space of selfstresses \(\mathrm {Stress}(M,p)\) are canonically isomorphic.

(ii)
Liftability of \((\overline{M},p)\) implies selfstressability of (M, p).
Remark 5.11
The theorem gives another proof of Corollary 4.9. Indeed, each realization of \(K_5\) can be viewed as a projection of a fourdimensional simplex. In other words, it is liftable, and hence stressable.
Each dvertex v of an Rframework yields in a natural way a spherical framework via the following algorithm:

1.
We may assume that each face is a simplex, otherwise triangulate the faces.

2.
Take an affine plane h which is orthogonal to the affine span \(\langle p(v)\rangle \). Clearly, we have \(\dim h=3\).

3.
Take a small sphere \(S^2\) lying in the plane h and centered at the intersection point \(O= h\cap \langle p(v)\rangle \).

4.
For each dface f incident to v, take the projection \(\mathrm{Pr}_h (f)\) to the plane h. Since f is a simplex, the intersection \(\mathrm{Pr}_h (f)\cup S^2\) is a geodesic arc.
This yields a framework \(\mathscr {S}_v\) placed in the sphere \(S^2\). Selfstressability of spherical graphs is well understood since it reduces to selfstressability of planar graphs (see [4, 10, 18]). A face loop is called local with respect a dvertex v if all the dfaces and dedges participating in the path are incident to v.
Lemma 5.12
The two statements are equivalent:

(i)
The stress monodromy of each local (with respect to some dvertex v) face loop is trivial.

(ii)
The spherical framework \(\mathscr {S}_v\) is selfstressable.
Proof
Triviality of any local stress monodromy implies that stresses can be assigned to the faces incident to v in such a way that locally the equilibrium condition holds.
The same stress assignment gives a selfstress of \(\mathscr {S}_v\), and vice versa. \(\square \)
Example 5.13
If a dvertex v has exactly four incident dedges, then \(\mathscr {S}_v\) is stressable. Indeed, in this case \(\mathscr {S}_v\) is a \(K_4\) placed on the sphere, which is always stressable.
Theorem 5.14
Assume that \(\mathscr {R}\) is a trivalent Rframework.

(i)
\(\mathscr {R}\) is selfstressable if and only if for each face loop the stress monodromy is trivial.

(ii)
\(\mathscr {R}\) is selfstressable if and only if the following two conditions hold:

(a)
For each vertex v, the induced spherical framework \(\mathscr {S}_v\) is selfstressable.

(b)
For some generators \(g_1,\ldots ,g_k\) of the first homology group \(H_1(\mathscr {R})\), and some collection of representatives \(\gamma _1,\ldots ,\gamma _k\) that are facecycles, all the stress monodromies are trivial (one representative for one generator).

(a)

(iii)
In particular, if \(\mathscr {R}\) is oneconnected, its selfstressability is equivalent to selfstressability of \(\mathscr {S}_v\) for all the dvertices.
Proof
(i) Take any face, assign to it any stress, and extend it to other faces. Triviality of the monodromy guarantees that no contradiction will arise.
(ii) Keeping in mind Lemma 3.13, observe that any face loop is a linear combination of \(\gamma _1,\ldots ,\gamma _k\) and some local face loops. By Lemma 5.12, the monodromies of local loops are trivial. It remains to apply (i). Now (iii) follows. \(\square \)
6.3 Some examples
Let us start by two elementary examples.

(1)
Take the Schlegel diagram of a 3dimensional cube (that is the projection of the edges of a cube). It is a trivalent graph in the plane. It is selfstressable since it is liftable by construction. However, one easily can redraw the graph keeping the combinatorics in such a way that the realization is no longer selfstressable. For this, it is sufficient to generically perturb the positions of the vertices.

(2)
Now let us work out an analogous example in \(\mathbb {R}^3\). Take the Schlegel diagram of a 4dimensional cube (that is, the projections of all the 2faces). It is a (3, 4)valent Rframework. It is selfstressable since it is liftable by construction. But, unlike (1), by Theorem 5.14, any other realization is selfstressable. Indeed, we have a oneconnected Rframework. Each vertex induces a spherical realization of \(K_4\) which is always stressable.
The above examples suggest general questions:
Are all trivalent Rframeworks selfstressable? Are all (3, 4)valent Rframeworks selfstressable?
The answer is negative, which is demonstrated in the following construction, which is interesting for its own sake.
Example 5.15
Take a triangular prism P in \(\mathbb {R}^3\), a pyramid over P, and the projection of the 2skeleton of the pyramid back to \(\mathbb {R}^3\). We obtain an Rframework \(\mathscr {R}_0\) which is defined with some freedom: firstly, one may alter the position of the vertex of the pyramid, and secondly, one may apply a projective transform to P.
Lemma 5.16
The Rframework \(\mathscr {R}_0\) is selfstressable, and the space of stresses is onedimensional.
Proof
The Rframework \(\mathscr {R}_0\) is a projection of the 2skeleton of a 4dimensional tetrahedron. In other words, \(\mathscr {R}_0\) is liftable, and therefore, selfstressable. Since it is trivalent, the space of stresses is at most onedimensional. \(\square \)
The prism P has two disjoint triangular faces. They are also faces of \(\mathscr {R}_0\); let us call them green. The edges of these faces are also called green. The faces of \(\mathscr {R}_0\) that are not green are called white. The edges that are not green are called white. Fix also one of the white faces, let us call it the test face for \(\mathscr {R}_0\).
It is easy to check that any two of the white faces of \(\mathscr {R}_0\) are connected by a face path which uses white edges only.
6.3.1 Main construction, first step
Take two copies of \(\mathscr {R}_0\), say, \(\mathscr {R}_1\) and \(\mathscr {R}_2\) such that one of the green faces of \(\mathscr {R}_1\) coincides with a green face of for \(\mathscr {R}_2\). Patch \(\mathscr {R}_1\) and \(\mathscr {R}_2\) along these faces, and eliminate these green faces. Define the result by \(\mathscr {R}_{12}\).
Lemma 5.17
\(\mathscr {R}_{12}\) is selfstressable, and the space of the selfstresses has dimension 1.
Proof
Selfstressability: Take a selfstress \(s_1\) of \(\mathscr {R}_1\) and a stress \(s_2\) of \(\mathscr {R}_2\) such that \(s_1s_2\) on the patched green faces vanishes. Then \(s_1s_2\) represents a nontrivial stress of \(\mathscr {R}_{12}\).
Dimension one: Consider a stress s on \(\mathscr {R}_{12}\). Take a stress \(s_1\) of \(\mathscr {R}_1\) which agrees with s on the test face of \(F_1\) and a stress \(s_2\) of \(\mathscr {R}_2\) which agrees with s on the test face of \(\mathscr {R}_2\). Take \(ss_1s_2\). It is a stress on \(\mathscr {R}_{12}\) plus the green face which is zero everywhere on \(\mathscr {R}_{12}\), except, maybe, the green face, which possible only if \(ss_1s_2\) vanishes everywhere. Therefore, each stress of \(\mathscr {R}_{12}\) is a linear combination of two stresses of \(\mathscr {R}_1\) and \(\mathscr {R}_2\) that cancel each other on the green face. \(\square \)
6.3.2 Main construction, second step
We proceed in the same manner: we take one more copy of \(\mathscr {R}_0\), which is called \(\mathscr {R}_3\) and patch it to \(\mathscr {R}_{12}\) along green faces, and eliminate the green face which was used for the patch. We get \(\mathscr {R}_{123}\). Analogously, we have:
Lemma 5.18
\(R_{123}\) is selfstressable, and the space of the stresses has dimension 1.
6.3.3 Main construction, next steps
Now we have a chain of three copies of \(\mathscr {R}_0\) patched together. Only two green faces survive. By adjusting the shapes of the components, we may assume that these green faces coincide. Patch the last two green faces and remove them. After that, we get an Rframework \(\widetilde{\mathscr {R}}\). Generically we have:
Lemma 5.19
\(\widetilde{\mathscr {R}}\) is not selfstressable, but it is locally selfstressable, that is \(\mathscr {R}_v\) is selfstressable for each vertex v.
Proof
Nonselfstressability Before the last step, the space of stresses was onedimensional. After the last step (\(=\) after patching two last green faces), the dimension can only drop. Let us patch back the two last green faces and get a framework \(\widetilde{\mathscr {R}}'\).
Assume \({\widetilde{\mathscr {R}}}\) is selfstressable. This means that \(\widetilde{\mathscr {R}}'\) has a stress which sums up to zero on the last two green faces. By the above lemmata, the value of the stress on the test face of \(\mathscr {R}_1\) uniquely defines the stress on the first green face and the stress on the test face of \(\mathscr {R}_2\) (and all the faces of \(\mathscr {R}_1\) and \(\mathscr {R}_2\)), this uniquely defines the stress on the test face of \(\mathscr {R}_3\), and so on. We conclude that the stress on the second green face is also uniquely defined, and generically, the green stresses do not cancel each other.
Local selfstressability follows from the fact that putting back any of the green faces creates a selfstressable Rframework. \(\square \)
Let us observe that \(\widetilde{\mathscr {R}}\) is trivalent, but not (3, 4)valent. Each of the green edges has four incident faces, and there are vertices of valency higher that 4. However one can prove that a number of local surgeries turns it to a (3, 4)valent Rframework.
Eventually we arrive at a (3, 4) Rframework which is locally selfstressable, but not globally selfstressable.
Notes
The support of a CWcomplex is the topological space represented by the complex. That is, one forgets the combinatorics and leaves the topology only.
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Karpenkov, O., Müller, C., Panina, G. et al. Equilibrium stressability of multidimensional frameworks. European Journal of Mathematics 8, 33–61 (2022). https://doi.org/10.1007/s40879021005233
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DOI: https://doi.org/10.1007/s40879021005233
Keywords
 Framework
 Tensegrity
 Equilibrium stress
 Selfstress
 Discrete multiplicative 1form
 Cayley algebra
 Maxwell–Cremona correspondence
 Lifting
Mathematics Subject Classification
 52C25
 52C35
 52B70