def orbifold_euler_characteristics(g,n):
"""
Return the orbifold/virtual Euler characteristics of `M_{g,n}`,
computed according to Harer-Zagier.
"""
if g==0:
return factorial(n-3) * minus_one_exp(n-3)
elif g==1:
return Fraction(minus_one_exp(n), 12) * factorial(n-1)
else: # g > 1
return bernoulli(2*g) * factorial(2*g+n-3) / (factorial(2*g-2) * 2*g) * minus_one_exp(n)
def euler_characteristics(g, n):
"""
Return Euler characteristics of `M_{g,n}`.
The Euler characteristics is computed according to formulas and
tables found in:
* Bini-Gaiffi-Polito, arXiv:math/9806048, p.3
* Bini-Harer, arXiv:math/0506083, p. 10
"""
if g == 0:
# according to Bini-Gaiffi-Polito arXiv:math/9806048, p.3
return factorial(n - 3) * minus_one_exp(n - 3)
elif g == 1:
# according to Bini-Gaiffi-Polito, p. 15
if n > 4:
return factorial(n - 1) * Fraction(minus_one_exp(n - 1), 12)
else:
es = [1, 1, 0, 0]
return es[n - 1] # no n==0 computed in [BGP]
elif g == 2:
# according to Bini-Gaiffi-Polito, p. 14
if n > 6:
return factorial(n + 1) * Fraction(minus_one_exp(n + 1), 240)
else:
es = [1, 2, 2, 0, -4, 0, -24]
return es[n]
elif g > 2:
# according to Bini-Harer arXiv:math/0506083, p. 10
es = [ # n==1 n==2 n==3 n==4 n==5 n==6 n==7 n==8
[8, 6, 4, -10, 30, -660, 6540, 79200], # g==3
[-2, -10, -24, -24, -360, 2352, -37296, 501984], # g==4
[12, 26, 92, 182, 1674, -16716, 238980, -3961440], # g==5
[0, -46, -206, 188, -7512, 124296, -2068392, 37108656], # g==6
[38, 120, 676, -1862, 71866, -1058676, 21391644,
-422727360], # g==7
[
-166, -630, -5362, 16108, -680616, 12234600, -259464240,
5719946400
], # g==8
[
748, 2132, 29632, -323546, 7462326, -164522628, 3771668220,
-90553767840
], # g==9
[
-1994, 6078, -213066, 4673496, -106844744, 2559934440,
-64133209320, 1.664e+12
], # g==10
]
# g=0,1,2 already done above
return es[g - 3][n]
else:
raise ValueError("No Euler characteristics known for M_{g,n},"
" where g=%s and n=%s" % (g, n))
def orbifold_euler_characteristics(g, n):
"""
Return the orbifold/virtual Euler characteristics of `M_{g,n}`,
computed according to Harer-Zagier.
"""
if g == 0:
return factorial(n - 3) * minus_one_exp(n - 3)
elif g == 1:
return Fraction(minus_one_exp(n), 12) * factorial(n - 1)
else: # g > 1
return bernoulli(2 * g) * factorial(2 * g + n - 3) / (
factorial(2 * g - 2) * 2 * g) * minus_one_exp(n)
def add_markings(self, name, markings, per_row=8):
"""
Output a table showing how the different markings of the same
underlying fatgraph do number the boundary components.
"""
graph = markings.graph
n = graph.num_boundary_cycles
N = len(markings.numberings)
self._section_marked_graphs += N
if N == factorial(n):
self._current_graph_data.markings = (r"""
Fatgraph $%(name)s$ only has the identity automorphism, so the
marked fatgraphs $%(name)s^{(0)}$ to $%(name)s^{(%(num_markings)d)}$
are formed by decorating boundary cycles of $%(name)s$ with
all permutations of $(%(basetuple)s)$ in lexicographic order.
See Section ``Markings of fatgraphs with trivial automorphisms''
for a complete table.
""" % dict(name=name, num_markings=N,
basetuple=(str.join(',', [str(x) for x in xrange(n)]))))
if self._appendix_graph_markings is None:
# only need to do this once per document
self._appendix_graph_markings = self._fmt_numberings(
markings.numberings, 'G', n, per_row)
else:
self._current_graph_data.markings = self._fmt_numberings(
markings.numberings, name, n, per_row)
def euler_characteristics(g,n):
"""
Return Euler characteristics of `M_{g,n}`.
The Euler characteristics is computed according to formulas and
tables found in:
* Bini-Gaiffi-Polito, arXiv:math/9806048, p.3
* Bini-Harer, arXiv:math/0506083, p. 10
"""
if g==0:
# according to Bini-Gaiffi-Polito arXiv:math/9806048, p.3
return factorial(n-3)*minus_one_exp(n-3)
elif g==1:
# according to Bini-Gaiffi-Polito, p. 15
if n>4:
return factorial(n-1)*Fraction(minus_one_exp(n-1),12)
else:
es = [1,1,0,0]
return es[n-1] # no n==0 computed in [BGP]
elif g==2:
# according to Bini-Gaiffi-Polito, p. 14
if n>6:
return factorial(n+1)*Fraction(minus_one_exp(n+1),240)
else:
es = [1,2,2,0,-4,0,-24]
return es[n]
elif g>2:
# according to Bini-Harer arXiv:math/0506083, p. 10
es = [# n==1 n==2 n==3 n==4 n==5 n==6 n==7 n==8
[ 8, 6, 4, -10, 30, -660, 6540, 79200], # g==3
[ -2, -10, -24, -24, -360, 2352, -37296, 501984], # g==4
[ 12, 26, 92, 182, 1674, -16716, 238980, -3961440], # g==5
[ 0, -46, -206, 188, -7512, 124296, -2068392, 37108656], # g==6
[ 38, 120, 676, -1862, 71866, -1058676, 21391644, -422727360], # g==7
[ -166, -630, -5362, 16108, -680616, 12234600, -259464240, 5719946400], # g==8
[ 748, 2132, 29632, -323546, 7462326, -164522628, 3771668220, -90553767840], # g==9
[-1994, 6078, -213066, 4673496, -106844744, 2559934440, -64133209320, 1.664e+12], # g==10
]
# g=0,1,2 already done above
return es[g-3][n]
else:
raise ValueError("No Euler characteristics known for M_{g,n},"
" where g=%s and n=%s" % (g,n))
