def taut_polynomial_image(tri, angle, cycles = [], alpha = True, mode = "taut"):
"""
If cycles = [] then this is the taut polynomial. If cycles != []
then this is the image of the taut polynomial under the
epimorphism obtained by killing the given boundary cycles.
"""
taut_poly = taut_polynomial_via_tree(tri, angle, alpha = alpha, mode = mode)
ZG = group_ring(tri, angle, [], alpha = alpha)
ZH = group_ring(tri, angle, cycles, alpha = alpha)
P = ZH.polynomial_ring()
image_in_laurent = epimorphism_in_laurent(tri, angle, cycles, ZH)
epi = ZG.Hom(ZH)(image_in_laurent)
return normalise_poly( epi(taut_poly), ZH, P )
def veering_polynomial(tri,
angle,
alpha=True,
normalisation=True,
mode="lower"):
# set up
ZH = group_ring(tri, angle, [], alpha=alpha)
P = ZH.polynomial_ring()
if verbose > 0:
print(("angle", angle))
coorientations = is_transverse_taut(tri,
angle,
return_type="tet_vert_coorientations")
if mode == "upper":
coorientations = [[-x for x in coor] for coor in coorientations]
ET = edges_to_tetrahedra_matrix(tri, angle, coorientations, normalisation,
ZH, P)
#print(ET)
if normalisation == True:
return normalise_poly(ET.determinant(), ZH, P)
else:
perm = permutation_tet_top_diagonal(tri, angle, coorientations)
#print(perm, "sign", sign(perm))
return sign(perm) * ET.determinant()
def taut_polynomial(tri, angle, cycles = [], alpha = True, mode = "taut"):
# set up
ZH = group_ring(tri, angle, cycles, alpha = alpha)
P = ZH.polynomial_ring()
ET = edges_to_triangles_matrix(tri, angle, cycles, ZH, P, mode = mode)
# compute via minors
minors = ET.minors(tri.countTetrahedra())
return normalise_poly(gcd(minors), ZH, P)
def taut_polynomial_via_smith(tri, angle, cycles = [], alpha = True, mode = "taut"):
# set up
assert tri.homology().rank() == 1 # need the polynomial ring to be a PID
ZH = group_ring(tri, angle, cycles, alpha = alpha, ring = QQ) # ditto
P = ZH.polynomial_ring()
ET = edges_to_triangles_matrix(tri, angle, cycles, ZH, P, mode = mode)
# compute via smith normal form
ETs = ET.smith_form()[0]
a = tri.countEdges()
ETs_reduced = Matrix([row[:a] for row in ETs])
return normalise_poly(ETs_reduced.determinant(), ZH, P)
def taut_polynomial_via_tree_and_smith(tri, angle, cycles = [], alpha = True, mode = "taut"):
# set up
assert tri.homology().rank() == 1 # need the polynomial ring to be a PID
ZH = group_ring(tri, angle, cycles, alpha = alpha, ring = QQ) # ditto
P = ZH.polynomial_ring()
ET = edges_to_triangles_matrix(tri, angle, cycles, ZH, P, mode = mode)
_, non_tree_faces, _ = spanning_dual_tree(tri)
ET = ET.transpose()
ET = Matrix([row for i, row in enumerate(ET) if i in non_tree_faces]).transpose()
# compute via smith normal form
ETs = ET.smith_form()[0]
a = tri.countEdges()
ETs_reduced = Matrix([row[:a] for row in ETs])
return normalise_poly(ETs_reduced.determinant(), ZH, P)
def taut_polynomial_via_tree(tri, angle, cycles = [], alpha = True, mode = "taut"):
"""
If cycles = [] then this is the taut polynomial. If cycles != []
then this is the zero-Fitting invariant of the module obtained by
'extension of scalars' - that is, form the epimorphism obtained by
killing the given boundary cycles, apply it to the presentation
matrix, and only then compute the gcd of the (correct minors).
"""
# set up
ZH = group_ring(tri, angle, cycles, alpha = alpha)
P = ZH.polynomial_ring()
ET = edges_to_triangles_matrix(tri, angle, cycles, ZH, P, mode = mode)
_, non_tree_faces, _ = spanning_dual_tree(tri)
ET = ET.transpose()
ET = Matrix([row for i, row in enumerate(ET) if i in non_tree_faces]).transpose()
# compute via minors
minors = ET.minors(tri.countTetrahedra())
return normalise_poly(gcd(minors), ZH, P)
def edges_to_triangles_matrix_wrapper(tri, angle, cycles):
ZH = group_ring(tri, angle, cycles, alpha = True)
P = ZH.polynomial_ring()
return edges_to_triangles_matrix(tri, angle, cycles, ZH, P, mode = "taut")