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pportilla committed Jan 16, 2019
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r"""
Mixed t\^ete-\`a-t\^ete graphs.
- Pablo Portilla (2019)
"""

#*****************************************************************************
# Copyright (C) 2016 Pablo Portilla <p.portilla89@gmail.com>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
# http://www.gnu.org/licenses/
#*****************************************************************************

from sage.structure.sage_object import SageObject
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.unique_representation import CachedRepresentation
from sage.groups.perm_gps.permgroup_element import PermutationGroupElement
from sage.rings.integer_ring import ZZ
from sage.misc.cachefunc import cached_method
from sage.misc.flatten import flatten
from copy import copy
from sage.geometry.ribbon_graph import *
from sage.functions.other import floor
from sage.rings.rational import Rational
from sage.matrix.matrix_space import MatrixSpace
from sage.geometry.tat_graph import *

def _find(l, k):
r"""
Return the two coordinates of the element ``k`` in the list of
lists ``l``.
INPUT:
- ``l`` -- a list of lists
- ``k`` -- a candidate to be in a list in ``l``
OUTPUT:
A list with two integers describing the position of the first
instance of `k`` in ``l``.
TESTS::
sage: from sage.geometry.ribbon_graph import _find
sage: A = [[2,3,4],[4,5,2],[8,7]]
sage: _find(A,2)
[0, 0]
sage: _find(A,7)
[2, 1]
sage: _find(A,5)
[1, 1]
sage: _find(A,-1)
Traceback (most recent call last):
...
ValueError: element -1 not found
"""
for i,lst in enumerate(l):
if k in lst:
return [i, lst.index(k)]
raise ValueError("element {} not found".format(k))


class MixedTatGraph(SageObject):
r"""
INPUT:
- ``list_ribbon`` -- a list of nested ribbon graphs.
- ``metric`` -- a list of as many rational numbers as darts has the first
element of``ribbon``.
- ``relative_boundary=[]`` -- a subset of ``ribbon.boundary()`` that
constitutes the relative boundary components of ``ribbon``. It is, by
default, initialized to an empty list (for defining pure t\^ete-\`a-t\^ete graphs)
Alternatively, one can pass in 2 integers and this will construct
a bipartite t\^ete-\`a-t\^ete graph which realizes the corresponding
Brieskorn-Pham singularity.
EXAMPLES:
sage: T33 = bipartite_tat_graph(3,3); T33
Tete-a-tete graph of order 3 on a ribbon graph of genus 1 and 3 boundary components.
sage: T33.sigma()
(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)
sage: T33.rho()
(1,12)(2,15)(3,18)(4,11)(5,14)(6,17)(7,10)(8,13)(9,16)
sage: print(T33.metric())
[1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4]
sage: T33._relative_boundary
[]
sage: B33 = blow_up(T33, 1,1/8); B33 ; B33._relative_boundary
Relative t\^ete-\`a-t\^ete graph of order 3 on a ribbon graph of genus 1 and 6 boundary components; where 3 boundary components are part of the relative boundary and might be permuted by the automorphism induced.
[[19, 24, 23, 22, 21, 20], [25, 30, 29, 28, 27, 26], [31, 36, 35, 34, 33, 32]]
sage: T517=bipartite_tat_graph(5,17);T517
Tete-a-tete graph of order 85 on a ribbon graph of genus 32 and 1 boundary components.
"""
def __init__(self, list_ribbon, metric, relative_boundary=[]):
r"""
Initialize ``self``.
"""
for i in range(len(relative_boundary)):
assert relative_boundary[i] in ribbon.boundary()

assert check_tat_property(ribbon, metric, relative_boundary) == True
self._ribbon = ribbon
self._metric = metric
self._sigma = ribbon._sigma
self._rho = ribbon._rho
self._boundary = self._ribbon.boundary()
self._relative_boundary = relative_boundary
self._basis = self._ribbon.homology_basis()
self._mu = 2*self._ribbon.genus() + self._ribbon.number_boundaries()-1


def _repr_(self):
r"""
Return basic information about the mixed t\^ete-\`a-t\^ete graph in string
format.
EXAMPLES:
Example of a relative t\^ete-\`a-t\^ete graph::
sage: s0 = PermutationGroupElement('(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)')
sage: r0 = PermutationGroupElement('(1,9)(2,11)(3,4)(5,14)(6,7)(8,17)(10,18)(12,13)(15,16)')
sage: R0 = RibbonGraph(s0,r0)
sage: m0 = 6*[1/8,3/8,1/8];
sage: perm_bound = [[1, 9, 7, 6, 4, 3], [10, 18, 16, 15, 13, 12]]
sage: T0 = TatGraph(R0,m0,relative_boundary = perm_bound); T0
Relative t\^ete-\`a-t\^ete graph of order 6 on a ribbon graph of genus 1 and 3 boundary components; where 2 boundary components are part of the relative boundary and might be permuted by the automorphism induced.
Example of a pure t\^ete-\`a-t\^ete graph::
sage: T23 = bipartite_tat_graph(2,3); T23
Tete-a-tete graph of order 6 on a ribbon graph of genus 1 and 1 boundary components.
"""
if not self._relative_boundary:
return "Tete-a-tete graph of order {} on a ribbon graph of genus {} and {} boundary components.".format(self.order(), self._ribbon.genus(), self._ribbon.number_boundaries())
else:
return "Relative t\^ete-\`a-t\^ete graph of order {} on a ribbon graph of genus {} and {} boundary components; where {} boundary components are part of the relative boundary and might be permuted by the automorphism induced.".format(self.order(), self._ribbon.genus(), self._ribbon.number_boundaries(), len(self._relative_boundary))

def sigma(self):
r"""
Return the permutation `\sigma` of ``self._ribbon``.
EXAMPLES::
sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
sage: R = RibbonGraph(s1, r1)
sage: m = 6*[1/2]
sage: T = TatGraph(R,m)
sage: T.sigma()
(1,3,5)(2,4,6)
"""
return self._ribbon.sigma()

def rho(self):
r"""
Return the permutation `\rho` of ``self._ribbon``.
EXAMPLES::
sage: s1 = PermutationGroupElement('(1,3,5)(2,4,6)')
sage: r1 = PermutationGroupElement('(1,2)(3,4)(5,6)')
sage: R = RibbonGraph(s1, r1)
sage: m = 6*[1/2]
sage: T = TatGraph(R,m)
sage: T.rho()
(1,2)(3,4)(5,6)
"""
return self._ribbon.rho()

def metric(self):
r"""
Return a vector containing the metric of the graph where the `i`th
(starting at `0`) value of the vector corresponds to the dart `i+1`.
EXAMPLES::
sage: T23 = bipartite_tat_graph(2,3); T23.metric()
[1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4, 1/4]
"""
return self._metric

def ribbon(self):
r"""
Return the underlying ribbon graph of the t\^ete-\`a-t\^ete graph ``self``.
EXAMPLES::
sage: T23 = bipartite_tat_graph(2,3); T23.ribbon(); T23.ribbon().sigma(); T23.ribbon().rho()
Ribbon graph of genus 1 and 1 boundary components
(1,2,3)(4,5,6)(7,8)(9,10)(11,12)
(1,8)(2,10)(3,12)(4,7)(5,9)(6,11)
"""
return self._ribbon


def action_homology(self):
r"""
Return matrix representing the action of the t\^ete-\`a-t\^ete automorphism
on the first homology group.
OUTPUT:
- Return a `self._mu() \times self._mu()` matrix that represents
the action of the t\^ete-\`a-t\^ete automorphism on the first homology group
with respect to the basis self._ribbon.homology_basis(). This
matrix has only `0`, `1` and `-1` as entries.
EXAMPLES::
sage: T34 = bipartite_tat_graph(3,4); T34
Tete-a-tete graph of order 12 on a ribbon graph of genus 3 and 1 boundary components.
sage: T34._ribbon.homology_basis()
[[[6, 17], [18, 2], [1, 15], [14, 5]],
[[7, 20], [21, 3], [1, 15], [14, 5]],
[[8, 23], [24, 4], [1, 15], [14, 5]],
[[10, 16], [18, 2], [1, 15], [13, 9]],
[[11, 19], [21, 3], [1, 15], [13, 9]],
[[12, 22], [24, 4], [1, 15], [13, 9]]]
sage: T34.action_homology(); T34.order(); T34.action_homology()**12
[ 0 0 0 1 -1 0]
[ 0 0 0 1 0 -1]
[ 0 0 0 1 0 0]
[-1 1 0 1 -1 0]
[-1 0 1 1 0 -1]
[-1 0 0 1 0 0]
12
[1 0 0 0 0 0]
[0 1 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1]
sage: T33 = bipartite_tat_graph(3,3); T33
Tete-a-tete graph of order 3 on a ribbon graph of genus 1 and 3 boundary components.
sage: T33.action_homology(); T33.order(); T33.action_homology()**3
[ 0 0 1 -1]
[ 0 0 1 0]
[-1 1 1 -1]
[-1 0 1 0]
3
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
sage: BT33 = blow_up(T33, 0, 1/8)
sage: BT33.action_homology(); BT33.action_homology()**3
[ 0 -1 1 1 0 0 0]
[ 0 0 1 0 0 -1 0]
[ 0 0 0 -1 0 0 1]
[ 0 0 1 1 -1 0 0]
[ 0 0 1 0 0 0 0]
[ 1 0 0 -1 0 0 0]
[ 0 0 1 1 0 0 0]
[1 0 0 0 0 0 0]
[0 1 0 0 0 0 0]
[0 0 1 0 0 0 0]
[0 0 0 1 0 0 0]
[0 0 0 0 1 0 0]
[0 0 0 0 0 1 0]
[0 0 0 0 0 0 1]
"""
#Set the space of matrices that we will be working on and
#a copy of the zero matrix that will be modified to get
#the action of the monodromy.
M = MatrixSpace(ZZ, self._mu, self._mu)
T = copy(M.zero_matrix())

#we run along all the darts of each of the elements of the
#basis of the homology. We check which is their image; if they
#intersect with positive orientation (with respect to the fixed
#orientation of homology_basis) then we set a +1 in the corresponding
#column, if they intersect with negative orientation, a -1.
for i in range (self._mu):
for j in range (self._mu):
for k in range (len(self._basis[i])):
if (safewalk(self._ribbon,
self._metric,
self._basis[i][k][0],
self._relative_boundary)
== self._basis[j][0][0]):
T[i,j] = 1
elif (safewalk(self._ribbon,
self._metric,
self._basis[i][k][0],
self._relative_boundary)
== self._basis[j][0][1]):

T[i,j] = -1
return T


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