def polygon(n):
"""
This constructs a polygon with n sides.
INPUT:
A positive integer n, at least 1.
OUTPUT:
A closed ribbon graph.
EXAMPLE:
>>> RibbonGraph.polygon(4) # doctest:+ELLIPSIS
<__main__.RibbonGraph object at 0x...>
"""
if n<1: raise ValueError
a = [ ribbon.halfedge() for i in xrange(n) ]
b1 = [ ribbon.halfedge() for i in xrange(n) ]
b2 = [ ribbon.halfedge() for i in xrange(n) ]
for i in xrange(n-1):
b1[i].e = b2[i+1]
b2[i+1].e = b1[i]
b1[n-1].e = b2[0]
b2[0].e = b1[n-1]
for i in xrange(n):
a[i].c = b1[i]
b1[i].c = b2[i]
b2[i].c = a[i]
h = ribbon.justgraph(a+b1+b2)
return RibbonGraph(h,a)
def Artin_generator(n,k):
"""Constructs the Artin generator s_k of n string braif group.
INPUT: n a positive integer, k an integer with 0 < |k| < n
OUTPUT: A Morphism
>>> s1 = Artin_generator(3,1)
>>> s2 = Artin_generator(3,2)
"""
if not 0 < abs(k) < n:
raise ValueError
# These are also defined in knots.py
in_over = ribbon.Features('head','blue',True)
in_under = ribbon.Features('head','blue',False)
out_over = ribbon.Features('tail','blue',True)
out_under = ribbon.Features('tail','blue',False)
r = range(n)
do = [ ribbon.halfedge() for i in r ]
co = [ ribbon.halfedge() for i in r ]
for a in do:
a.decorations = in_over
a.IsI = True
for a in co:
a.decorations = out_over
a.IsI = True
for i in r:
do[i].c = co[i]
co[i].c = do[i]
p =abs(k)
do[p-1].IsI = False
do[p].IsI = False
co[p-1].IsI = False
co[p].IsI = False
do[p-1].c = do[p]
do[p].c = co[p]
co[p].c = co[p-1]
co[p-1].c = do[p-1]
if k > 0:
do[k-1].decorations = in_over
do[k].decorations = in_under
co[k-1].decorations = out_under
co[k].decorations = out_over
elif k < 0:
do[p-1].decorations = in_under
do[p].decorations = in_over
co[p-1].decorations = out_over
co[p].decorations = out_under
else:
raise RuntimeError
g = ribbon.justgraph( set(do+co) )
return Morphism(g, do, co)
def vogel_move(f,u,v):
# This goes into an infinite loop
phi = f.graph.copy()
new = phi.codomain
a = phi.map[u]
b = a.e
c = phi.map[v]
d = c.e
x = [ ribbon.halfedge() for i in range(4) ]
for i in range(4):
x[i-1].c = x[i]
y = [ ribbon.halfedge() for i in range(4) ]
for i in range(4):
y[i-1].c = y[i]
a.e = x[0]; x[0].e = a
d.e = x[3]; x[3].e = d
b.e = y[1]; y[1].e = b
c.e = y[2]; y[2].e = c
x[1].e = y[0]; y[0].e = x[1]
x[2].e = y[3]; y[3].e = x[2]
switch = {'head':'tail','neither':'neither','tail':'head'}
ut = a.decorations.directed
ot = switch[ut]
if False:
x[0].decorations._replace(a.decorations.colour)
x[1].decorations._replace(c.decorations.colour)
x[2].decorations._replace(a.decorations.colour)
x[3].decorations._replace(c.decorations.colour)
y[0].decorations._replace(d.decorations.colour)
y[1].decorations._replace(b.decorations.colour)
y[2].decorations._replace(d.decorations.colour)
y[3].decorations._replace(b.decorations.colour)
for a in x:
a.decorations._replace(colour='red')
for a in y:
a.decorations._replace(colour='red')
g = ribbon.justgraph( new.he.union( set(x+y) ) )
u = phi.map[f.outside[0]]
outside = [u]
s = u.e.c
while s != u:
outside.append(s)
s = s.e.c
return LinkDiagram(g,outside)
def vertex(x):
"""
Constructs a single vertex of valency n.
INPUT:
A positive integer n, at least 1 or a list of features.
OUTPUT:
A closed ribbon graph.
EXAMPLES:
>>> RibbonGraph.vertex(3) # doctest:+ELLIPSIS
<__main__.RibbonGraph object at 0x...>
"""
if isinstance(x, Iterable):
n = len(x)
else:
n = x
if n<1: raise ValueError("Not enough points.")
a = [ ribbon.halfedge() for i in xrange(n) ]
for i in xrange(n-1):
a[i].c = a[i+1]
a[n-1].c = a[0]
if isinstance(x, Iterable):
for i, r in zip(xrange(n),x):
if not isinstance(r, ribbon.features):
raise ValueError
a[i].decorations = r
h = ribbon.justgraph(a)
return RibbonGraph(h,a)
def from_PlanarDiagram(PD):
"""Produces a LinkDiagram from the planar diagram notation.
INPUT: A list or set of 4-tuples
OUTPUT: A closed justgraph
EXAMPLES:
>>> LinkDiagram.from_PlanarDiagram([[1,2,3,4],[4,3,6,5],[2,1,5,6]]) #doctest: +ELLIPSIS
<__main__.LinkDiagram object at 0x...>
>>> g = LinkDiagram.from_PlanarDiagram([[1,2,3,4],[4,3,6,5],[2,1,5,6]])
>>> [ len(v) for v in g.vertices ]
[4, 4, 4]
>>> [ len(v) for v in g.edges ]
[2, 2, 2, 2, 2, 2]
>>> x = [ len(v) for v in g.faces ]
>>> x.sort()
>>> x
[2, 2, 2, 3, 3]
>>> x = [ len(v) for v in g.graph.get_orbits( lambda a: a.e.c ) ]
>>> x.sort()
>>> x
[2, 2, 2, 3, 3]
"""
if not all([ len(x) == 4 for x in PD ]):
raise ValueError("Not a valid planar diagram code")
for i in range(2*len(PD)):
p = [ x for x in PD if i+1 in x ]
if len(p) != 2:
raise ValueError("Not a valid planar diagram code")
D = dict()
he = set()
for x in PD:
a = [ ribbon.halfedge() for i in range(4) ]
for i in range(4):
a[i-1].c = a[i]
D[tuple(x)] = a
he = he.union(set(a))
for i in range(2*len(PD)):
p = [ x for x in PD if i+1 in x ]
a0 = D[tuple(p[0])][p[0].index(i+1)]
a1 = D[tuple(p[1])][p[1].index(i+1)]
a0.e = a1
a1.e = a0
g = ribbon.justgraph(he)
outside = g.get_orbits( lambda a: a.e.c )[0]
return LinkDiagram(g,outside)
def map(self):
labels = ('VE','VF','EV','EF','FV','FE',)
flags = dict([])
for l in labels:
for i in range(self.degree):
flags[(i,l,)] = ribbon.halfedge()
for i in range(self.degree):
flags[(i,'VE',)].e = flags[(i,'EV',)]
flags[(i,'EV',)].e = flags[(i,'VE',)]
flags[(i,'VF',)].e = flags[(i,'FV',)]
flags[(i,'FV',)].e = flags[(i,'VF',)]
flags[(i,'FE',)].e = flags[(i,'EF',)]
flags[(i,'EF',)].e = flags[(i,'FE',)]
#flags[(i,'VE',)]).c = flags[(self.sigma[i],'VF',)]
flags[(self.sigma[i],'VE',)].c = flags[(i,'VF',)]
flags[(i,'EV',)].c = flags[(i,'EF',)]
flags[(i,'VF',)].c = flags[(i,'VE',)]
flags[(self.phi[i],'FV',)].c = flags[(i,'FE',)]
flags[(i,'FE',)].c = flags[(i,'FV',)]
#flags[(i,'EF',)].c = flags[(self.alpha[i],'EV',)]
flags[(self.alpha[i],'EF',)].c = flags[(i,'EV',)]
for i in range(self.degree):
flags[(i,'EV',)].decorations = ribbon.Features('neither','red',True)
flags[(i,'VE',)].decorations = ribbon.Features('neither','red',True)
flags[(i,'FV',)].decorations = ribbon.Features('neither','blue',True)
flags[(i,'VF',)].decorations = ribbon.Features('neither','blue',True)
flags[(i,'FE',)].decorations = ribbon.Features('neither','green',True)
flags[(i,'EF',)].decorations = ribbon.Features('neither','green',True)
g = ribbon.justgraph(set(flags.values()))
a = self.faces[0]
x = flags[(a[0],'FE',)]
out = [x]
y = x.c
while y != x:
out.append(y)
y = y.c
out.reverse()
sut = [ flags[(i,'FE',)] for i in a ] + [ flags[(i,'FV',)] for i in a ]
if set(sut) != set(out):
raise RuntimeError
return g, out
def closure(self, bv=None):
"""Construct the closure of a web.
INPUT: A web and a boundary vector.
OUTPUT: A closed web.
EXAMPLES:
>>> RibbonGraph.polygon(5).closure() #doctest: +ELLIPSIS
<closedgraph.ClosedGraph object at 0x...>
>>> RibbonGraph.polygon(5).closure().graph.count_vertices()
[0, 0, 5, 10]
"""
if bv == None:
bv = [1] * len(self.bd)
if sum(bv) != len(self.bd):
raise ValueError("Boundary vector is not consistent with web.")
if len(bv) < 3:
raise ValueError("Not enough corners.")
C = len(bv)
B = len(self.bd)
phi = self.jg.copy()
he = phi.codomain.he
rim = ribbon.Features('neither','black',True)
switch = {'head':'tail','neither':'neither','tail':'head'}
ci = [ ribbon.halfedge() for i in xrange(C) ]
for a in ci:
a.decorations = rim
co = [ ribbon.halfedge() for i in xrange(C) ]
for a in co:
a.decorations = rim
he = he.union(ci+co)
bi = [ ribbon.halfedge() for i in xrange(B) ]
for a in bi:
a.decorations = rim
bo = [ ribbon.halfedge() for i in xrange(B) ]
for a in bo:
a.decorations = rim
bc = [ ribbon.halfedge() for i in xrange(B) ]
he = he.union(bi+bo+bc)
for i in xrange(C):
ci[i].c = co[i]
co[i].c = ci[i]
nb = [ phi.map[a] for a in self.bd ]
for i in xrange(B):
bi[i].c = bo[i]
bo[i].c = bc[i]
bc[i].c = bi[i]
bc[i].e = nb[i]
nb[i].e = bc[i]
for a in bc:
f = a.e.decorations
a.decorations = ribbon.Features(switch[f.directed],f.colour,True)
p = 0
for i, a in enumerate(bv):
r = co[i-1]
for j in xrange(a):
bi[p].e = r
r.e = bi[p]
r = bo[p]
p += 1
r.e = ci[i]
ci[i].e = r
ng = ribbon.justgraph(he)
u = co[0]
outside = [u]
s = u.e.c
while s != u:
outside.append(s)
s = s.e.c
return closedgraph.ClosedGraph(ng, outside)
def from_DT(DT):
"""Construct a knot diagram from the Dowker-Thistlewaite code.
The theory is given in the original paper:
Classification of knot projections
C. H. Dowker & Morwen B. Thistlewaite
Topology and its Applications 16 (1983), 19--31
INPUT:
A list of positive even integers
OUTPUT:
A closed justgraph
EXAMPLES:
>>> LinkDiagram.from_DT(DT([4,6,2])) #doctest: +ELLIPSIS
<__main__.LinkDiagram object at 0x...>
>>> LinkDiagram.from_DT(DT([4,6,8,2])) #doctest: +ELLIPSIS
<__main__.LinkDiagram object at 0x...>
>>> LinkDiagram.from_DT(DT([4,8,10,2,6])) #doctest: +ELLIPSIS
<__main__.LinkDiagram object at 0x...>
>>> LinkDiagram.from_DT(DT([6,8,10,2,4])) #doctest: +ELLIPSIS
<__main__.LinkDiagram object at 0x...>
>>> g = LinkDiagram.from_DT(DT([4,6,2]))
>>> [ len(v) for v in g.vertices ]
[4, 4, 4]
>>> [ len(v) for v in g.edges ]
[2, 2, 2, 2, 2, 2]
>>> x = [ len(v) for v in g.faces ]
>>> x.sort()
>>> x
[2, 2, 2, 3, 3]
>>> g = LinkDiagram.from_DT(DT([4,6,8,2]))
>>> [ len(v) for v in g.vertices ]
[4, 4, 4, 4]
>>> [ len(v) for v in g.edges ]
[2, 2, 2, 2, 2, 2, 2, 2]
>>> x = [ len(v) for v in g.faces ]
>>> x.sort()
>>> x
[2, 2, 3, 3, 3, 3]
>>> x = [ len(v) for v in g.graph.get_orbits( lambda a: a.c.e ) ]
>>> x.sort()
>>> x
[2, 2, 3, 3, 3, 3]
>>> g = LinkDiagram.from_DT(DT([4,8,10,2,6]))
>>> [ len(v) for v in g.vertices ]
[4, 4, 4, 4, 4]
>>> [ len(v) for v in g.edges ]
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
>>> x = [ len(v) for v in g.faces ]
>>> x.sort()
>>> x
[2, 2, 2, 3, 3, 4, 4]
>>> x = [ len(v) for v in g.graph.get_orbits( lambda a: a.c.e ) ]
>>> x.sort()
>>> x
[2, 2, 2, 3, 3, 4, 4]
>>> g = LinkDiagram.from_DT(DT([8,10,2,12,4,6]))
>>> x = [ len(v) for v in g.vertices ]
>>> x.sort()
>>> x
[4, 4, 4, 4, 4, 4]
>>> [ len(v) for v in g.edges ]
[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
>>> x = [ len(v) for v in g.faces ]
>>> x.sort()
>>> x
[2, 2, 2, 3, 3, 3, 4, 5]
>>> x = [ len(v) for v in g.graph.get_orbits( lambda a: a.c.e ) ]
>>> x.sort()
>>> x
[2, 2, 2, 3, 3, 3, 4, 5]
REFS:
http://katlas.org/wiki/DT_%28Dowker-Thistlethwaite%29_Codes
"""
emb = DT.orientation
m = 2 * len(emb)
left = [ ribbon.halfedge() for i in range(m) ]
right = [ ribbon.halfedge() for i in range(m) ]
for i in range(m):
left[i].e = right[i-1]
right[i-1].e = left[i]
for i, k in enumerate(DT.code):
a = abs(k)-1
x = [ left[2*i], left[a], right[2*i], right[a] ]
if k > 0:
x[0].decorations = in_over
x[1].decorations = in_under
x[2].decorations = out_over
x[3].decorations = out_under
elif k < 0:
x[0].decorations = in_under
x[1].decorations = in_over
x[2].decorations = out_under
x[3].decorations = out_over
else:
raise ValueError("Not a valid Dowker-Thistlewaite code")
if emb[i] == -1:
x.reverse()
elif emb[i] != 1:
raise RuntimeError
for i in xrange(4):
x[i-1].c = x[i]
g = ribbon.justgraph( set(left).union(set(right)) )
outside = g.get_orbits( lambda a: a.e.c )[0]
return LinkDiagram(g,outside)