Python reidemeister_presentationの例

コード例 #1

ファイル: test_fp_groups.py プロジェクト: chiranthsiddappa/sympy

def test_subgroup_presentations():
    F, x, y = free_group("x, y")
    f = FpGroup(F, [x**3, y**5, (x*y)**2])
    H = [x*y, x**-1*y**-1*x*y*x]
    p1 = reidemeister_presentation(f, H)
    assert str(p1) == "((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))"

    H = f.subgroup(H)
    assert (H.generators, H.relators) == p1

    f = FpGroup(F, [x**3, y**3, (x*y)**3])
    H = [x*y, x*y**-1]
    p2 = reidemeister_presentation(f, H)
    assert str(p2) == "((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))"

    f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
    H = [x]
    p3 = reidemeister_presentation(f, H)
    assert str(p3) == "((x_0,), (x_0**4,))"

    f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
    H = [x]
    p4 = reidemeister_presentation(f, H)
    assert str(p4) == "((x_0,), (x_0**6,))"

    # this presentation can be improved, the most simplified form
    # of presentation is <a, b | a^11, b^2, (a*b)^3, (a^4*b*a^-5*b)^2>
    # See [2] Pg 474 group PSL_2(11)
    # This is the group PSL_2(11)
    F, a, b, c = free_group("a, b, c")
    f = FpGroup(F, [a**11, b**5, c**4, (b*c**2)**2, (a*b*c)**3, (a**4*c**2)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b])
    H = [a, b, c**2]
    gens, rels = reidemeister_presentation(f, H)
    assert str(gens) == "(b_1, c_3)"
    assert len(rels) == 18

コード例 #2

def test_subgroup_presentations():
    F, x, y = free_group("x, y")
    f = FpGroup(F, [x**3, y**5, (x*y)**2])
    H = [x*y, x**-1*y**-1*x*y*x]
    p1 = reidemeister_presentation(f, H)
    assert str(p1) == "((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))"

    H = f.subgroup(H)
    assert (H.generators, H.relators) == p1

    f = FpGroup(F, [x**3, y**3, (x*y)**3])
    H = [x*y, x*y**-1]
    p2 = reidemeister_presentation(f, H)
    assert str(p2) == "((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))"

    f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
    H = [x]
    p3 = reidemeister_presentation(f, H)
    assert str(p3) == "((x_0,), (x_0**4,))"

    f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
    H = [x]
    p4 = reidemeister_presentation(f, H)
    assert str(p4) == "((x_0,), (x_0**6,))"

    # this presentation can be improved, the most simplified form
    # of presentation is <a, b | a^11, b^2, (a*b)^3, (a^4*b*a^-5*b)^2>
    # See [2] Pg 474 group PSL_2(11)
    # This is the group PSL_2(11)
    F, a, b, c = free_group("a, b, c")
    f = FpGroup(F, [a**11, b**5, c**4, (b*c**2)**2, (a*b*c)**3, (a**4*c**2)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b])
    H = [a, b, c**2]
    gens, rels = reidemeister_presentation(f, H)
    assert str(gens) == "(b_1, c_3)"
    assert len(rels) == 18

コード例 #3

ファイル: test_fp_groups.py プロジェクト: Kogorushi/sympy

def test_subgroup_presentations():
    F, x, y = free_group("x, y")
    f = FpGroup(F, [x**3, y**5, (x*y)**2])
    H = [x*y, x**-1*y**-1*x*y*x]
    p1 = reidemeister_presentation(f, H)
    assert str(p1) == "((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))"

    f = FpGroup(F, [x**3, y**3, (x*y)**3])
    H = [x*y, x*y**-1]
    p2 = reidemeister_presentation(f, H)
    assert str(p2) == "((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))"

    f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
    H = [x]
    p3 = reidemeister_presentation(f, H)
    assert str(p3) == "((x_0,), (x_0**4,))"

    f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
    H = [x]
    p4 = reidemeister_presentation(f, H)
    assert str(p4) == "((x_0,), (x_0**6,))"

    # this presentation can be improved, the most simplified form
    # of presentation is <a, b | a^11, b^2, (a*b)^3, (a^4*b*a^-5*b)^2>
    # See [2] Pg 474 group PSL_2(11)
    # This is the group PSL_2(11)
    F, a, b, c = free_group("a, b, c")
    f = FpGroup(F, [a**11, b**5, c**4, (b*c**2)**2, (a*b*c)**3, (a**4*c**2)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b])
    H = [a, b, c**2]
    k = ("((a_0, b_1, c_3), "
        "(c_3**3, b_1**5, a_0**4*b_1**-2*a_0**-1*b_1**2, "
        "c_3*b_1**-2*c_3*b_1**-2*c_3*b_1**-2, b_1**-1*a_0*b_1**2*c_3**-1*b_1**-1*a_0*b_1**2*c_3**-1, "
        "a_0**11, a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**2*c_3, "
        "a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1, "
        "a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1, "
        "a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
        "b_1**-2*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**2*c_3**-1*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2, "
        "b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**2*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
        "b_1**2*c_3*b_1**-2*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2*c_3**-1*b_1*c_3, "
        "b_1**2*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**-2*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
        "c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
        "c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2, "
        "b_1**2*a_0*b_1**2*c_3**-1*b_1**-1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2*c_3**-1*b_1**2*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1, "
        "c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0, "
        "a_0**5*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0**5*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0**5*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1))"
        )
    assert str(reidemeister_presentation(f, H)) == k

コード例 #4

def test_subgroup_presentations():
    F, x, y = free_group("x, y")
    f = FpGroup(F, [x**3, y**5, (x*y)**2])
    H = [x*y, x**-1*y**-1*x*y*x]
    p1 = reidemeister_presentation(f, H)
    assert str(p1) == "((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))"

    f = FpGroup(F, [x**3, y**3, (x*y)**3])
    H = [x*y, x*y**-1]
    p2 = reidemeister_presentation(f, H)
    assert str(p2) == "((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))"

    f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
    H = [x]
    p3 = reidemeister_presentation(f, H)
    assert str(p3) == "((x_0,), (x_0**4,))"

    f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
    H = [x]
    p4 = reidemeister_presentation(f, H)
    assert str(p4) == "((x_0,), (x_0**6,))"

    # this presentation can be improved, the most simplified form
    # of presentation is <a, b | a^11, b^2, (a*b)^3, (a^4*b*a^-5*b)^2>
    # See [2] Pg 474 group PSL_2(11)
    # This is the group PSL_2(11)
    F, a, b, c = free_group("a, b, c")
    f = FpGroup(F, [a**11, b**5, c**4, (b*c**2)**2, (a*b*c)**3, (a**4*c**2)**3, b**2*c**-1*b**-1*c, a**4*b**-1*a**-1*b])
    H = [a, b, c**2]
    k = ("((a_0, b_1, c_3), "
        "(c_3**3, b_1**5, a_0**4*b_1**-2*a_0**-1*b_1**2, "
        "c_3*b_1**-2*c_3*b_1**-2*c_3*b_1**-2, b_1**-1*a_0*b_1**2*c_3**-1*b_1**-1*a_0*b_1**2*c_3**-1, "
        "a_0**11, a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**2*c_3, "
        "a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1, "
        "a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1, "
        "a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
        "b_1**-2*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*a_0*b_1**2*c_3**-1*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2, "
        "b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**2*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
        "b_1**2*c_3*b_1**-2*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2*c_3**-1*b_1*c_3, "
        "b_1**2*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1**-2*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
        "c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1, "
        "c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2, "
        "b_1**2*a_0*b_1**2*c_3**-1*b_1**-1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*b_1**2*c_3**-1*b_1**2*c_3*b_1*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1, "
        "c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0*c_3**-1*b_1*a_0*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0, "
        "a_0**5*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0**5*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1*a_0**5*b_1**-2*a_0*b_1**2*c_3**-1*b_1**-2*c_3**-1*b_1))"
        )
    assert str(reidemeister_presentation(f, H)) == k