def g_quad(graph, node_list, gui=True):
'''
Return quadratic form (numpy array) of networkx graph 'graph', ordered
according to the node names in 'node_list'.
Q(v,v) = weight(v)
Q(v,w) = 1 if v, w are connected by an edge; 0 otherwise
Thus Q is the adjacency matrix everywhere except the diagonal, and just
the weights down the diagonal.
'''
assert len(node_list) == len(graph.nodes(data=False))
assert is_forest(graph)
num_bad = num_bad_vertices(graph, node_list)
if num_bad > 2:
if gui:
tkMessageBox.showwarning('Bad vertices',
'More than two bad vertices. (There are %i.)'%num_bad)
raise ValueError('More than two bad vertices. (There are %i.)'%num_bad)
# create adjacency matrix, ordered according to node_list
adj = nx.to_numpy_matrix(graph, nodelist=node_list, dtype=numpy.int)
for index, node in enumerate(node_list): # change diagonal
adj[index, index] = graph.node[node]['weight']
if not is_negative_definite(adj):
if gui:
tkMessageBox.showwarning('Quadratic form',
'Quadratic form is not negative definite')
print adj
raise ValueError('quadratric form is not negative definite')
return adj
def s_quad_form(listdata, gui=True):
'''
Returns the tuple (numpy.array quadratic_form, bool minus) of the weighted
'star-like' tree associated with the plumbed 3-manifold given by Seifert
data 'listdata'.
minus = True if orientation of the manifold described by listdata has been
reversed, False otherwise
Input: listdata [e, (p1, q1),...,(pr, qr)]
Note: You should run correct_form on listdata; s_quad_form does not do this
automatically
The 'star-like' tree is formed as follows:
Center node is a vertex with weight -e.
There are r branches coming off of the center vertex.
Each branch has the vertices [-a0, -a1, ... , -an], where a0,a1,...,an
are the numbers in the continued fraction expansion of pi/qi, for each i.
(Note: the continued fraction expansion has minus signs; see function
'cont_fraction'.)
The quadratic form of the weighted graph is defined by
Q(v, v) = m(v), where m is the weight of the vertex
Q(v, w) = 1 if v, w are connected by an edge, 0 otherwise
'''
new_data, minus = alter_data(listdata, gui) # make e > 0, -pi < qi < 0
tree = [new_data[0]] # list of weights in star tree [e, a0, a1,...]
branch_lengths = [] # list of lengths of branches in star tree, not
# including start node
for pair in new_data[1:]:
branch = cont_fraction(pair[0], -pair[1])
branch_lengths.append(len(branch))
tree.extend(branch)
size = sum(branch_lengths) + 1 # add start node
row = 0
column = 0
# make the quadratic form matrix
quad = numpy.zeros(shape=(size, size), dtype=numpy.int)
# fill in the diagonal
for index in range(size):
quad[index,index] = -tree[index]
# fill in the others (upper right triangle)
cur_position = 1
for length in branch_lengths:
quad[0,cur_position] = 1 # star node
for index in range(cur_position, cur_position+length-1):
quad[index,index+1] = 1 # adjacent
cur_position += length
symmetric(quad)
if not is_negative_definite(quad):
if gui:
tkMessageBox.showwarning('Quadratic form',
'Quadratic form is not negative definite')
print quad
raise ValueError('quadratric form is not negative definite')
# check get the same thing from weighted_graph.py; also checks negative def.
assert numpy.array_equal(quad, g_quad(make_graph(new_data), \
['N%i' %num for num in range(size)]))
return quad, minus
