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README.md

-# volume_rigidity
-
-
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+Volume rigidity matrix:
+volume_rigidity_matroid.py:
+	get_volume_rigidity_matrix(V,H,p,d)
+	is_volume_rigid(V,H,p,d)
+	
+example_1.py verifies if one specific triangulation is volume rigid.
+example_2.py verifies if all irreducible triangulations of the torus are volume rigid.
+example_3.py verifies if all irreducible triangulations of the projective plane are volume rigid.
+example_4.py verifies if all irreducible triangulations of the Klein bottle are volume rigid.
\ No newline at end of file

example_1.py

+import volume_rigidity_matroid
+import numpy as np
+
+#triangulation
+triangulation = [[1,4,6],[1,2,6],[2,6,7],[2,3,7],[3,5,7],[1,3,5],[4,5,6],[5,6,8],[6,7,8],[4,7,8],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[1,3,8],[1,4,8]]
+n = 0
+for F in triangulation:
+    d = len(F) #the vertices are embedded in R^{d-1}, in our case in R^2
+    n = max(max(F),n)    
+H = set([tuple([s[0]-1,s[1]-1,s[2]-1]) for s in triangulation]) #we need to shift the vertices by 1
+V = range(n) #the set of vertices is 0,...,7
+
+#build the vertex embedding
+num_rows = d-1
+num_columns = n
+low = 1
+high = 10000
+p = np.random.randint(low, high, (num_rows, num_columns))
+p = np.vstack([p, np.ones((1,n), dtype=int)])
+
+#vertify if the triangulation is volume rigid
+print(volume_rigidity_matroid.is_volume_rigid(V,H,p,d))

example_2.py

+#process if all the minimal triangulations of the torus are volume rigid
+
+import volume_rigidity_matroid
+import numpy as np
+
+def is_triangulation_volume_rigid(triangulation):
+    n=0
+    for F in triangulation:
+        d = len(F) #the vertices are embedded in R^{d-1}, in our case in R^2
+        n = max(max(F),n)    
+    H = set([tuple([s[0]-1,s[1]-1,s[2]-1]) for s in triangulation]) #we need to shift the vertices by 1
+    V = range(n) #the set of vertices is 0,...,7
+
+    #build the vertex embedding
+    num_rows = d-1
+    num_columns = n
+    low = 1
+    high = 10000
+    p = np.random.randint(low, high, (num_rows, num_columns))
+    p = np.vstack([p, np.ones((1,n), dtype=int)])
+
+    #vertify if the triangulation is volume rigid
+    return volume_rigidity_matroid.is_volume_rigid(V,H,p,d)
+
+def main():
+    triangulations=[
+        [[1,2,4],[2,3,5],[2,4,5],[1,3,4],[3,4,6],[3,5,6],[4,6,7],[4,5,7],[1,5,7],[1,3,7],[2,3,7],[2,6,7],[1,2,6],[1,5,6]],
+        [[1,2,4],[2,5,6],[2,4,5],[2,3,6],[1,3,4],[3,4,6],[4,6,8],[4,5,8],[6,7,8],[5,6,7],[1,5,8],[3,7,8],[1,3,8],[2,3,7],[1,2,7],[1,5,7]],
+        [[1,2,4],[2,4,6],[2,3,6],[1,3,4],[3,6,8],[3,4,8],[4,5,8],[4,6,7],[4,5,7],[6,7,8],[1,5,7],[1,2,7],[2,7,8],[2,5,8],[2,3,5],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[1,3,4],[3,4,8],[3,6,7],[3,7,8],[4,5,6],[5,6,7],[4,5,8],[1,5,7],[1,2,7],[2,7,8],[2,5,8],[1,3,5],[2,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[1,3,4],[3,6,7],[3,4,7],[4,5,6],[5,6,7],[4,7,8],[4,5,8],[1,3,5],[3,5,8],[2,3,8],[2,7,8],[2,5,7],[1,2,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[1,3,7],[1,4,7],[4,5,6],[4,6,8],[6,8,9],[6,7,9],[5,7,9],[4,5,7],[1,5,8],[1,2,8],[2,8,9],[2,3,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[1,3,7],[1,4,7],[4,5,6],[5,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,9],[1,5,8],[1,2,8],[2,8,9],[2,3,9],[3,5,9],[1,3,5]],
+        [[1,2,3],[2,4,6],[2,3,6],[3,6,7],[1,3,4],[3,4,7],[4,6,8],[4,5,8],[6,7,8],[7,8,9],[4,7,9],[4,5,9],[1,5,8],[1,2,8],[2,3,8],[3,8,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,6,7],[2,3,7],[3,4,7],[1,3,4],[4,5,8],[4,6,8],[6,8,9],[6,7,9],[4,7,9],[4,5,9],[1,5,8],[1,2,8],[2,8,9],[2,3,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,4,7],[1,3,4],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,9],[1,5,8],[1,2,8],[2,8,9],[2,3,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,4,7],[1,3,4],[4,5,8],[4,6,8],[6,8,9],[6,7,9],[5,7,9],[4,5,7],[1,5,8],[1,2,8],[2,8,9],[2,3,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,4,7],[1,3,4],[4,5,6],[5,6,8],[6,8,9],[6,7,9],[4,7,9],[4,5,9],[1,2,5],[2,5,8],[2,8,9],[2,3,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,6,7],[2,3,7],[3,4,7],[1,3,4],[4,5,6],[5,6,8],[6,8,9],[6,7,9],[4,7,9],[4,5,9],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,4,7],[1,3,4],[4,5,6],[5,6,8],[6,8,9],[6,7,9],[4,7,9],[4,5,9],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,4,7],[1,3,4],[4,5,6],[5,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,9],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,4,7],[1,3,4],[4,5,6],[5,6,8],[6,7,8],[7,8,9],[5,7,9],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,4,7],[1,3,4],[4,5,6],[5,6,8],[6,8,9],[6,7,9],[5,7,9],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,6,7],[2,3,7],[3,4,7],[1,3,4],[4,5,6],[5,6,8],[6,7,8],[4,7,8],[4,8,9],[4,5,9],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,6,7],[2,3,7],[3,4,7],[1,3,4],[4,5,6],[5,6,7],[5,7,8],[4,7,8],[4,8,9],[4,5,9],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,4,7],[1,3,4],[4,5,6],[5,6,7],[5,7,8],[4,7,8],[4,8,9],[4,5,9],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[3,5,9],[1,3,5]],
+        [[1,2,4],[2,4,6],[2,6,10],[2,7,10],[2,3,7],[3,4,7],[1,3,4],[4,5,6],[5,6,10],[4,7,10],[5,9,10],[4,8,10],[4,5,8],[3,8,10],[3,9,10],[3,5,8],[1,3,5],[2,3,9],[2,5,9],[1,2,5]]
+    ]
+    for triangulation in triangulations:
+        print(is_triangulation_volume_rigid(triangulation))
+
+
+if __name__ == "__main__":
+    main()

example_3.py

+#process if all the minimal triangulations of the projective plane are volume rigid
+
+import volume_rigidity_matroid
+import numpy as np
+
+def is_triangulation_volume_rigid(triangulation):
+    n=0
+    for F in triangulation:
+        d = len(F) #the vertices are embedded in R^{d-1}, in our case in R^2
+        n = max(max(F),n)    
+    H = set([tuple([s[0]-1,s[1]-1,s[2]-1]) for s in triangulation]) #we need to shift the vertices by 1
+    V = range(n) #the set of vertices is 0,...,7
+
+    #build the vertex embedding
+    num_rows = d-1
+    num_columns = n
+    low = 1
+    high = 10000
+    p = np.random.randint(low, high, (num_rows, num_columns))
+    p = np.vstack([p, np.ones((1,n), dtype=int)])
+
+    #vertify if the triangulation is volume rigid
+    return volume_rigidity_matroid.is_volume_rigid(V,H,p,d)
+
+def main():
+    triangulations=[
+        [[1,2,4],[1,3,4],[2,4,5],[2,3,5],[1,3,5],[1,5,6],[1,2,6],[2,3,6],[3,4,6],[4,5,6]],
+        [[1,2,4],[1,3,4],[2,4,5],[2,3,5],[1,3,6],[1,2,6],[2,6,7],[2,3,7],[3,4,7],[4,6,7],[4,5,6],[3,5,6]]
+    ]
+    for triangulation in triangulations:
+        print(is_triangulation_volume_rigid(triangulation))
+
+
+if __name__ == "__main__":
+    main()

example_4.py

+#process if all the minimal triangulations of the Klein bottle are volume rigid
+
+import volume_rigidity_matroid
+import numpy as np
+
+def is_triangulation_volume_rigid(triangulation):
+    n=0
+    for F in triangulation:
+        d = len(F) #the vertices are embedded in R^{d-1}, in our case in R^2
+        n = max(max(F),n)    
+    H = set([tuple([s[0]-1,s[1]-1,s[2]-1]) for s in triangulation]) #we need to shift the vertices by 1
+    V = range(n) #the set of vertices is 0,...,7
+
+    #build the vertex embedding
+    num_rows = d-1
+    num_columns = n
+    low = 1
+    high = 10000
+    p = np.random.randint(low, high, (num_rows, num_columns))
+    p = np.vstack([p, np.ones((1,n), dtype=int)])
+
+    #vertify if the triangulation is volume rigid
+    return volume_rigidity_matroid.is_volume_rigid(V,H,p,d)
+
+def main():
+    triangulations=[
+        [[1,4,6],[1,2,6],[2,6,7],[2,3,7],[3,5,7],[1,3,5],[4,5,6],[5,6,8],[6,7,8],[4,7,8],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[1,3,8],[1,4,8]],
+        [[1,4,6],[1,2,6],[2,3,6],[3,6,7],[3,5,7],[1,3,5],[4,5,6],[5,6,8],[6,7,8],[4,7,8],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[1,3,8],[1,4,8]],
+        [[1,2,4],[2,3,4],[3,4,6],[3,6,7],[3,5,7],[1,3,5],[4,5,6],[5,6,8],[6,7,8],[4,7,8],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[1,3,8],[1,4,8]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[1,3,7],[1,5,7],[4,5,6],[5,6,8],[6,7,8],[4,7,8],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[3,4,8],[1,3,4]],
+        [[1,4,6],[1,2,6],[2,3,6],[3,6,7],[3,5,7],[1,3,5],[4,5,6],[5,6,8],[6,7,8],[4,7,8],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[3,4,8],[1,3,4]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,5,7],[1,3,5],[4,5,6],[5,6,8],[6,7,8],[4,7,8],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[3,4,8],[1,3,4]],
+        [[1,4,6],[1,2,6],[2,3,6],[3,5,6],[1,3,5],[4,5,7],[4,6,7],[6,7,9],[5,7,9],[6,8,9],[5,6,8],[4,5,8],[4,8,9],[1,2,5],[2,5,9],[2,3,9],[1,3,9],[1,4,9]],
+        [[1,4,6],[1,2,6],[2,3,6],[3,6,7],[3,5,7],[1,3,5],[4,5,6],[5,6,8],[6,7,8],[4,7,8],[4,5,7],[1,2,5],[2,5,9],[2,3,9],[3,4,9],[1,3,4]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,5,6],[1,3,5],[4,6,7],[4,5,7],[5,7,9],[6,7,9],[6,8,9],[5,6,8],[4,8,9],[4,5,8],[1,2,5],[2,5,9],[2,3,9],[3,4,9],[1,3,4]],
+        [[1,4,6],[1,2,6],[2,3,6],[3,5,6],[1,3,5],[4,6,7],[6,7,8],[5,6,8],[4,5,7],[5,7,9],[7,8,9],[4,5,8],[4,8,9],[1,2,5],[2,5,9],[2,3,9],[1,3,9],[1,4,9]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,5,6],[1,3,5],[4,5,7],[4,6,7],[5,7,9],[6,7,8],[7,8,9],[5,6,8],[4,8,9],[4,5,8],[1,2,5],[2,5,9],[2,3,9],[3,4,9],[1,3,4]],
+        [[1,2,4],[2,3,4],[3,4,6],[3,6,7],[3,5,7],[1,3,5],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,5,8],[1,2,8],[2,3,8],[3,8,9],[1,3,9],[1,4,9]],
+        [[1,2,4],[2,3,4],[3,4,6],[3,6,7],[1,3,7],[1,5,7],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,5,8],[1,2,8],[2,3,8],[3,8,9],[1,3,9],[1,4,9]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[1,3,7],[1,5,7],[4,5,6],[5,6,8],[6,7,8],[7,8,9],[5,7,9],[4,5,9],[1,5,8],[1,2,8],[2,8,9],[2,3,9],[3,4,9],[1,3,4]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,5,7],[1,3,5],[4,5,6],[5,6,8],[6,7,8],[7,8,9],[5,7,9],[4,5,9],[1,2,5],[2,5,8],[2,8,9],[2,3,9],[3,4,9],[1,3,4]],
+        [[1,2,4],[2,4,6],[2,6,7],[2,3,7],[1,3,7],[1,5,7],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,5,8],[1,2,8],[2,3,8],[3,8,9],[3,4,9],[1,3,4]],
+        [[1,2,4],[2,4,6],[2,6,7],[2,3,7],[3,5,7],[1,3,5],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[3,4,9],[1,3,4]],
+        [[1,2,4],[2,4,6],[2,6,7],[2,3,7],[3,5,7],[1,3,5],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,5,8],[1,2,8],[2,3,8],[3,8,9],[3,4,9],[1,3,4]],
+        [[1,2,4],[2,4,6],[2,6,7],[2,3,7],[1,3,7],[1,5,7],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,2,5],[2,5,8],[2,8,9],[2,3,9],[1,3,9],[1,4,9]],
+        [[1,2,4],[2,4,6],[2,6,7],[2,3,7],[1,3,7],[1,5,7],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,2,5],[2,5,8],[2,8,9],[2,3,9],[3,4,9],[1,3,4]],
+        [[1,2,4],[2,4,6],[2,6,7],[2,3,7],[1,3,7],[1,5,7],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[1,3,9],[1,4,9]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[3,5,7],[1,3,5],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,5,8],[1,2,8],[2,8,9],[2,3,9],[3,4,9],[1,3,4]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[1,3,7],[1,5,7],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,2,5],[2,5,8],[2,3,8],[3,8,9],[1,3,9],[1,4,9]],
+        [[1,2,4],[2,4,6],[2,3,6],[3,6,7],[1,3,7],[1,5,7],[4,5,8],[4,6,8],[6,7,8],[7,8,9],[4,7,9],[4,5,7],[1,5,8],[1,2,8],[2,8,9],[2,3,9],[3,4,9],[1,3,4]],
+        [[1,2,6],[1,4,6],[2,6,7],[4,6,7],[2,3,7],[1,3,7],[1,5,7],[4,7,8],[4,5,8],[5,8,9],[7,8,9],[7,9,10],[5,7,10],[4,5,10],[4,9,10],[1,5,9],[1,2,9],[2,4,9],[2,3,4],[1,3,4]]
+    ]
+    for triangulation in triangulations:
+        print(is_triangulation_volume_rigid(triangulation))
+
+
+if __name__ == "__main__":
+    main()

volume_rigidity_matroid.py

+import numpy as np
+from itertools import combinations
+
+def get_matrix_minor(M,i,j):
+    minor = M[:]
+    minor = np.delete(minor, (i),axis=0) # deletes i-th row
+    minor = np.delete(minor, (j),axis=1) #deletes j-th column
+    return minor
+    
+
+
+#The function volume_rigidity_matrix computes the volume-rigidity matrix as defined in the article arXiv:2211.00574
+
+#input:
+# d: cardinality of members of H
+# V: set of vertices
+# p: vertex embedding of the vertices in R^{d-1} and a final row of 1's
+# H: the hypergraph as set of tuples of size d
+def get_volume_rigidity_matrix(V,H,p,d):
+    volume_rigidity_matrix = np.empty(((d-1)*len(V), 0),dtype=int) #create an empty matrix with d*len(V) rows and 0 columns.
+    for F in H: # we build column vectors
+        selected_rows = range(d)
+        selected_columns = F
+        M_F = p[np.ix_(selected_rows, selected_columns)] #select columns of p corresponding to vertices of F
+        column_vector_F = np.empty(0, dtype=int) 
+        for v in V: # we build 
+            for i in range(d-1):
+                if v not in F:
+                    column_vector_F = np.append(column_vector_F,[0]) #add 0 d times for each vector not in F
+                if v in F:
+                    index_i = i
+                    index_v = F.index(v) #position of v in F
+                    minor_F_i_v = ((-1)**(index_i+index_v))*np.linalg.det(get_matrix_minor(M_F,index_i,index_v))
+                    column_vector_F = np.append(column_vector_F, minor_F_i_v)
+        volume_rigidity_matrix = np.c_[volume_rigidity_matrix, column_vector_F]
+    return volume_rigidity_matrix
+
+def is_volume_rigid(V,H,p,d):
+    volume_rigidity_matrix = get_volume_rigidity_matrix(V,H,p,d)
+    return np.linalg.matrix_rank(volume_rigidity_matrix) == (d-1)*len(V)-(d*d-d-1)