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Sasakian Geometry$

Charles Boyer and Krzysztof Galicki

Print publication date: 2007

Print ISBN-13: 9780198564959

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780198564959.001.0001

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(p.559) Appendix B

(p.559) Appendix B

Source:
Sasakian Geometry
Publisher:
Oxford University Press

B.1. Reid's List of K3 Surfaces as hypersurfaces in CP 3(w)

b2(X)

X

22

X 4 ∨ P(1,1,1,1), X 6 ∨ P(1,1,1,3)

21

X 5 ∨ P(1,1,1,2), X 12 ∨ P(1,1,4,6)

20

X 8 ∨ P(1,1,2,4), X 10 ∨ P(1,1,3,5)

19

X 6 ∨ P(1,1,2,2), X 7 ∨ P(1,1,2,3), X 9 ∨ P(1,1,3,4)

17

X 10 ∨ P(1,2,2,5), X 18 ∨ P(1,2,6,9)

16

X 8 ∨ P(1,2,2,3), X 12 ∨ P(1,2,3,6), X 14 ∨ P(1,2,4,7), X 16 ∨ P(1,2,5,8)

15

X 9 ∨ P(1,2,3,3), X 10 ∨ P(1,2,3,4), X 11 ∨ P(1,2,3,5), X 12 ∨ P(1,2,4,5), X 15 ∨ P(1,2,5,7), X 24 ∨ P(1,3,8,12),

14

X 16 ∨ P(1,3,4,8), X 18 ∨ P(1,3,5,9), X 22 ∨ P(1,3,7,11), X 30 ∨ P(1,4,10,15)

13

X 12 ∨ P(1,3,4,4), X 13 ∨ P(1,3,4,5), X 14 ∨ P(2,2,3,7), X 15 ∨ P(1,3,4,7), X 15 ∨ P(1,3,5,6), X 20 ∨ P(1,4,5,10), X 21 ∨ P(1,3,7,10), X 22 ∨ P(1,4,6,11), X 28 ∨ P(1,4,9,14), X 36 ∨ P(1,5,12,18), X 42 ∨ P(1,6,14,21)

12

X 12 ∨ P(2,2,3,5), X 16 ∨ P(1,4,5,6), X 18 ∨ P(1,4,6,7), X 26 ∨ P(1,5,7,13), X 30 ∨ P(1,6,8,15),

11

X 12 ∨ P(2,3,3,4), X 18 ∨ P(2,3,4,9), X 21 ∨ P(1,5,7,8), X 24 ∨ P(1,6,8,9), X 30 ∨ P(2,3,10,15),

10

X 14 ∨ P(2,3,4,5), X 16 ∨ P(2,3,4,7), X 20 ∨ P(2,3,5,10), X 22 ∨ P(2,4,5,11), X 24 ∨ P(2,3,7,12), X 26 ∨ P(2,3,8,13),

9

X 15 ∨ P(2,3,5,5), X 15 ∨ P(3,3,4,5), X 17 ∨ P(2,3,5,7), X 18 ∨ P(2,3,5,8), X 20 ∨ P(2,4,5,9), X 20 ∨ P(2,4,5,9), X 21 ∨ P(2,3,7,9), X 24 ∨ P(2,3,8,11), X 26 ∨ P(2,5,6,13)

8

X 18 ∨ P(3,4,5,6), X 20 ∨ P(2,5,6,7), X 24 ∨ P(3,4,5,12), X 32 ∨ P(2,5,9,16), X 42 ∨ P(3,4,14,21),

7

X 19 ∨ P(3,4,5,7), X 20 ∨ P(3,4,5,8), X 21 ∨ P(3,5,6,7), X 27 ∨ P(2,5,9,11), X 28 ∨ P(3,4,7,14), X 30 ∨ P(4,5,6,15), X 34 ∨ P(3,4,10,17), X 36 ∨ P(3,4,11,18), X 48 ∨ P(3,5,16,24),

6

X 24 ∨ P(3,4,7,10), X 24 ∨ P(4,5,6,9), X 30 ∨ P(3,4,10,13), X 32 ∨ P(4,5,7,16), X 34 ∨ P(4,6,7,15,17), X 38 ∨ P(3,5,11,19), X 54 ∨ P(4,5,18,27)

5

X 25 ∨ P(4,5,7,9), X 27 ∨ P(5,6,7,9), X 28 ∨ P(4,6,7,11), X 33 ∨ P(3,5,11,14), X 38 ∨ P(5,6,8,19), X 40 ∨ P(5,7,8,20), X 42 ∨ P(2,5,14,21), X 44 ∨ P(4,5,13,22), X 66 ∨ P(5,6,22,33)

4

X 24 ∨ P(3,6,7,8), X 30 ∨ P(5,6,8,11), X 36 ∨ P(7,8,9,12), X 50 ∨ P(7,8,10,25),

(p.560) B.2. Differential topology of 2k(S 3 × S 4) and 2k(S 5 × S 6)

Table B.2.1 2k#(S 3 × S 4)

k

τ‎k

D 2(k)

D 2 ( k ) | b P 8 |

1

1

28

1

2

3

28

1

3

6

14

½

4

10

14

½

5

15

28

1

6

21

4

1 7

7

28

1

1 28

8

36

7

¼

9

45

28

1

10

55

28

1

20

210

2

1 14

48

1176

1

1 28

50

1275

28

1

100

5050

14

½

496

123256

1

1 28

500

125250

14

½

Table B.2.2 2k#(S5 × S6)

k

τ‎,k

D 3(k)

D 3 ( k ) | b P 12 |

1

1

992

1

2

3

992

1

3

6

496

½

4

10

496

½

5

15

992

1

6

21

992

1

7

28

248

¼

8

36

248

¼

9

45

992

1

10

55

992

1

31

496

2

1 496

48

1176

124

50

1275

992

1

62

1953

32

1 31

124

7750

16

1 62

248

30876

8

11 24

496

123256

4

12 48

500

125250

496

½

992

492528

2

14 96

(p.561) B.3. Tables of Kähler–Einstein metrics on hypersurfaces in CP(w)

Table B.3.1 lists the infinite series solutions appearing in Theorem 5.4.16, while Table B.3.2 lists the sporadic ones. The last column of the tables indicate whether the klt condition (which implies that Z w admits a Kähler–Einstein metric) holds (Y) or is unknown (?). The computer search gives the following complete list:

Table B.3.1 Infinite Series Examples of Z w of Index 1 ≤ I ≤ 10

I

w

d

b 2

K-E

1

(2,2k+1,2k+1,4k+1)

8k+4

8

Y

2

(4,2k+1,2k+1,4k)

8k+4

7

Y

2

(3,3k+1,6k+1,9k+3)

18k+6

6

Y

2

(3,3k+1,6k+1,9k)

18k+3

5

?

2

(3,3k,3k+1,3k+1)

9k+3

7

Y

2

(3,3k+1,3k+2,3k+2)

9k+6

5

Y

2

(4,2k+1,4k+2,6k+1)

12k+6

6

?

4

(6,6k+5,12k+8,18k+15)

18k+15

5

Y

4

(6,6k+5,12k+8,18k+9)

36k+24

3

?

4

(6,6k+5,12k+8,18k+15)

36k+30

4

Y

6

(8,4k+1,4k+3,4k+5)

12k+11

3

?

6

(9,3k+2,3k+5,6k+1)

12k+11

3

?

(p.562)

Table B.3.2 Sporadic Examples of Z w of Index 1 ≤ I ≤ 10

I

w

Monomials of f w

d

b 2

K-E

1

(1,2,3,5)

z 0 10 , z 1 5 , z 2 3 z 1 , z 3 2 , ( 17 ) *

10

9

Y

1

(1,3,5,7)

z 0 15 , z 1 5 , z 2 3 , z 3 2 z 0 , ( 19 )

15

9

Y

1

(1,3,5,8)

z 0 16 , z 1 5 z 0 , z 2 3 z 0 , z 3 2 , ( 20 )

16

10

Y

1

(2,3,5,9)

z 0 9 , z 1 6 , z 2 3 z 1 , z 3 2 , ( 13 )

18

7

Y

1

(3,3,5,5)

g 5 ( z 0 , z 1 ) , f 3 ( z 2 , z 3 )

15

5

Y

1

(3,5,7,11)

z 0 6 z 2 , z 1 5 , z 2 2 z 3 , z 3 2 z 0 , ( 8 )

25

5

Y

1

(3,5,7,14)

z 0 7 z 2 , z 1 5 z 0 , g 2 ( z 2 2 , z 3 ) , ( 9 )

28

6

Y

1

(3,5,11,18)

g 2 ( z 0 6 z 3 ) , z 1 5 z 2 , z 2 3 z 0 , ( 10 )

36

6

Y

1

(5,14,17,21)

z 1 7 z 3 , z 1 4 , z 2 3 z 0 , z 3 2 z 1 , z 0 5 z 1 z 2

56

4

Y

1

(5,19,27,31)

z 0 10 z 3 , z 1 4 z 0 , z 2 3 , z 3 2 z 1 , z 0 7 z 1 z 2

81

3

Y

1

(5,19,27,50)

g 2 ( z 0 10 , z 3 ) , z 1 5 z 0 , z 2 3 z 1 , z 0 7 z 1 2 z 3

100

4

Y

1

(7,11,27,37)

z 0 10 z 1 , z 1 4 z 3 , z 2 3 , z 3 3 z 0

81

3

Y

1

(7,11,27,44)

g 2 ( z 1 4 , z 3 ) , z 0 11 z 1 , z 2 3 z 0 , z 0 4 , z 1 3 z 2

88

4

Y

1

(9,15,17,20)

z 0 5 z 1 , z 1 4 , z 2 3 z 0 , z 3 3

60

3

Y

1

(9,15,23,23)

z 0 6 z 1 , z 1 4 z 0 , z 2 3 , z 2 2 z 3 , z 2 z 3 2 , z 3 3

69

5

Y

1

(11,29,39,49)

z 0 8 z 2 , z 1 4 z 0 , z 2 3 , z 2 3 z 1

127

3

Y

1

(11, 49, 69, 128)

z 0 17 z 2 , z 1 5 z 0 , z 2 4 , z 2 2 z 3 , z 3 2

256

2

Y

1

(13,23,35,57)

z 0 8 z 1 , z 1 4 z 2 , z 2 2 z 3 , z 3 2 z 0

127

3

Y

1

(13, 35, 81, 128)

z 0 17 z 1 , z 1 5 z 2 , z 2 3 z 0 , z 3 2

256

2

Y

2

(2,3,4,5)

z 0 6 , z 1 4 , z 2 3 , z 3 2 z 0 , ( 10 )

12

5

?

2

(2,3,4,7)

z 0 7 , z 1 4 z 0 , z 3 2 z 0 , z 2 3 , ( 11 )

14

6

?

2

(3,4,5,10)

z 0 5 z 2 , z 1 5 , z 4 2 , z 2 3 , ( 9 )

20

5

Y

2

(3,4,6,7)

g 3 ( z 0 2 , z 0 ) z 1 3 z 2 , z 3 2 z 1 , ( 8 )

18

6

?

2

(3,4,10,15)

z 0 10 , z 1 5 z 3 , z 2 3 , z 3 2 , ( 10 )

30

7

Y

2

(3,7,8,13)

z 0 7 z 2 , z 1 3 z 2 , z 2 2 z 3 , z 3 2 z 0 , ( 7 )

29

5

?

2

(3,10,11,19)

z 0 10 z 3 , z 1 3 z 2 , z 2 2 z 3 , z 3 2 z 0 , ( 7 )

41

5

?

2

(5,13,19,22)

z 0 7 z 3 , z 1 4 z 0 , z 2 3 , z 3 2 z 1 , z 0 5 z 1 z 2

57

3

Y

2

(5,13,19,35)

g 2 ( z 0 7 , z 3 ) , z 1 5 z 0 , z 2 3 z 1 , z 0 5 z 1 2 z 2

70

3

Y

2

(6,9,10,13)

z 0 6 , z 1 4 , z 2 3 z 0 , z 3 2 z 2 , z 0 3 z 1 2

36

4

Y

2

(7,8,19,25)

z 0 7 z 1 , z 1 4 z 3 , z 2 3 , z 3 2 z 0 , z 0 2 z 1 3 z 2

57

3

Y

2

(7,8,19,32)

z 0 4 z 1 , z 1 4 , z 2 3 z 0 , z 3 3

64

4

Y

2

(9,12,13,16)

z 0 8 z 1 , g 2 ( z 1 4 , z 3 ) z 2 3 z 0 , ( 7 )

48

3

Y

2

(9,12,19,19)

z 0 4 z 1 , z 1 4 , z 2 3 z 0 , z 3 3

57

5

Y

2

(9,19,24,31)

z 0 5 z 1 , z 1 4 z 0 , z 2 3 , z 2 2 z 3 , z 2 z 3 3 , z 3 3

81

3

Y

2

(10,19,35,43)

z 0 9 , z 1 3 z 2 , z 2 3 z 0 , z 3 2 z 1

105

3

Y

2

(11,21,28,47)

z 0 7 z 2 , z 1 5 z 0 , z 2 3 , z 3 2 z 0

105

3

Y

2

(11,25,32,41)

z 0 6 z 3 , z 1 3 z 2 , z 3 2 z 0 , z 3 2 z 1

107

3

Y

2

(11,25,34,43)

z 0 10 , z 1 4 z 0 , z 2 2 z 3 , z 3 2 z 1

111

3

Y

2

(11, 43, 61, 113)

z 0 15 z 2 , z 1 5 z 0 , z 2 3 z 1 , z 3 2

226

2

Y

2

(13,18,45,61)

z 0 9 z 1 , z 1 5 z 2 , z 2 3 , z 3 2 z 0

135

3

Y

2

(13,20,29,47)

z 0 6 z 3 , z 1 3 z 3 , z 2 3 z 1 , z 3 2 z 0

107

3

Y

2

(13,20,31,49)

z 0 7 z 1 , z 1 4 z 2 , z 2 3 z 3 , z 3 2 z 0

111

3

Y

2

(13,31,71,113)

z 0 15 z 1 , z 1 5 z 2 , z 2 3 z 0 , z 3 2

226

2

Y

2

(14,17,29,41)

z 0 5 z 3 , z 1 5 z 2 , z 2 3 z 0 , z 3 2 z 1

99

3

Y

3

(5,7,11,13)

z 0 4 z 3 , z 1 4 z 0 , z 2 3 , z 2 3 z 1 , z 0 3 z 1 z 2

33

3

?

3

(5,7,11,20)

g 2 ( z 0 4 z 3 ) , z 1 5 z 0 , z 2 3 z 1 , z 0 3 z 1 z 2

40

4

Y

3

(11,21,29,37)

z 0 6 z 2 , z 1 4 z 0 , z 2 2 z 3 , z 3 2 z 1

95

3

Y

3

(11,37,53,98)

z 0 13 z 2 , z 1 5 z 0 , z 2 3 z 1 , z 3 2

196

2

Y

3

(13,17,27,41)

z 0 6 z 1 , z 1 4 z 2 , z 2 2 z 3 , z 3 2 z 0

95

3

Y

3

(13,27,61,98)

z 0 13 z 1 , z 1 5 z 2 , z 2 3 z 0 , z 3 2

196

2

Y

3

(15,19,43,74)

z 0 7 z 2 , z 1 7 z 0 , z 2 3 z 1 , z 3 2

148

2

Y

4

(5,6,8,9)

z 0 3 z 3 , z 1 4 , z 2 3 , z 2 3 z 1 , z 0 2 z 1 z 2

24

3

?

4

(5,6,8,15)

g 2 ( z 0 3 z 3 ) , z 1 5 , z 2 3 z 1 , z 0 2 z 1 2 z 2

30

4

?

4

(9,11,12,17)

z 0 5 , z 1 3 z 2 , z 2 3 z 0 , z 3 2 z 1

45

3

?

4

(10,13,25,31)

z 0 5 z 2 , z 1 5 z 0 , z 2 3 , z 3 2 z 1

75

3

Y

4

(11,17,20,27)

z 0 4 z 3 , z 1 3 z 2 , z 2 3 z 0 , z 3 2 z 1

71

3

?

4

(11,17,24,31)

z 0 5 z 2 , z 1 4 z 0 , z 2 2 z 3 , z 3 2 z 1

79

3

Y

4

(11,31,45,83)

z 0 11 z 2 , z 1 5 z 0 , z 2 3 z 1 , z 3 2

166

2

Y

4

(13,14,19,29)

z 0 4 z 2 , z 1 3 z 3 , z 2 3 z 1 , z 3 2 z 0

71

2

?

4

(13,14,23,33)

z 0 5 z 1 , z 1 4 z 2 , z 2 2 z 3 , z 3 2 z 0

79

3

Y

4

(13,23,51,83)

z 0 11 z 1 , z 1 5 z 2 , z 2 3 z 0 , z 3 2

166

2

Y

5

(11,13,19,25)

z 0 4 z 2 , z 1 4 z 0 , z 2 2 z 3 , z 3 2 z 1

63

3

?

5

(11,25,37,68)

z 0 9 z 2 , z 1 5 z 0 , z 2 3 z 1 , z 3 2

136

2

Y

5

(13,19,41,68)

z 0 9 z 1 , z 1 5 z 2 , z 2 3 z 0 , z 3 2

136

2

Y

6

(7,10,15,19)

z 0 5 z 1 , z 1 3 z 2 , z 2 3 , z 3 2 z 0

45

3

?

6

(11,19,29,53)

z 0 7 z 2 , z 1 5 z 0 , z 2 4 , z 3 2

106

2

Y

6

(13,15,31,53)

z 0 7 z 1 , z 1 5 z 2 , z 2 3 z 0 , z 3 2

106

2

Y

7

(11,13,21,38)

z 0 5 z 2 , z 1 5 z 0 , z 2 3 z 1 , z 3 2

76

2

Y

8

(7,11,13,23)

z 0 5 z 1 , z 1 3 z 2 , z 2 3 z 0 , z 3 2

46

2

?

8

(7,18,27,37)

z 0 9 z 1 , z 1 3 z 2 , z 2 3 , z 3 2 z 0

81

3

?

9

(7,15,19,32)

z 0 7 z 1 , z 1 3 z 2 , z 2 3 z 0 , z 3 2

64

2

?

10

(7,19,25,41)

z 0 9 z 1 , z 1 3 z 2 , z 2 3 z 0 , z 3 2

82

2

?

10

(7,26,39,55)

z 0 13 z 1 , z 1 3 z 2 , z 2 3 , z 3 2 z 0

117

3

?

(*) (for lack of space only the total number of monomial terms in f w is indicated)

(p.563) The computer program indicates that there are neither series solutions nor sporadic solutions satisfying the hypothesis of Theorem 5.4.16 for I > 10. In fact, an easy argument shows that there are no such solutions for su.ciently large I.1

(p.564) B.4. Positive Breiskorn–Pham Links in Dimension 5

Table B.4.1 Positive BP Links with b 2(M) = 0, H 2(L(a),Z ≠ 0

M

L(a)

N 𝒮ℱ

M 2

L(2, 3, 9, 15), L(2, 3, 3, k), k = 3(6)

M 3

L(2, 3, 8, 20), L(2, 3, 4, k), k = 4, 8(12)

M 4

L(3, 3, 3, 4), L(3, 3, 4, 9)

2

M 5

L(3, 3, 5, 6), L(2, 3, 5, k), k = 6, 12, 18, 24(30), L(2, 4, 5, 8), L(2, 4, 5, 12), L(2, 4, 5, 16)

M 7

L(2, 4, 7, 8), L(2, 4, 6, 7), L(2, 3, 7, 36), L(2, 3, 7, 30), L(2, 3, 7, 24), L(2, 3, 7, 18), L(2, 3, 7, 12), L(3, 3, 3, 7), L(2, 4, 4, 7), L(2, 3, 6, 7)

10

M 8

L(3, 3, 3, 8)

1

M 9

L(2, 4, 4, 9)

1

M 10

L(3, 3, 3, 10)

1

M 11

L(2, 3, 11, 12), L(3, 3, 3, 11), L(2, 4, 4, 11), L(2, 3, 6, 11), L(2, 4, 6, 11)

5

M k,k >12,k= 1,5(6)

L(2, 3, 6, k), L(3, 3, 3, k), L(2, 4, 4, k)

3

M k,k >12,k= 2,4(6)

L(3, 3, 3, k)

1

M k,k >12,k= 3(6)

L(2, 4, 4, k)

1

2 M 3

L(2, 3, 5, k), k = 10, 20(30)

3 M 3

L(2, 3, 8, 8), L(2, 3, 8, 16), L(3, 4, 4, 4), L(2, 3, 7, 14), L(2, 3, 7, 28)

5

4 M 3

L(2, 3, 10, 10)

1

2 M 5

L(2, 5, 6, 6)

1

4 M 2

L(2, 3, 5, k), k = 15(30)

6 M 2

L(2, 5, 5, 5), L(2, 3, 7, 21)

2

7 M 2

L(2, 3, 9, 9)

1

(p.565)

Table B.4.2 Positive BP Links with b 2(M) > 0, H 2(L(a),Z tor ≠ 0

M

L(a)

N 𝒮ℱ

2M #M 2

L(2, 3, 4, k), k = 6(12) L(2, 4, 6, 9), L(2, 3, 6, 8)

2M #M∞4

L(2, 3, 8, 18), L(2, 3, 6, 8)

2

2M #M 5

L(2, 3, 10, 12), L(2, 3, 6, 10)

2

2M #M k, (3, k) = 1,k >6

L(2, 3, 6, 2k)

1

3M #M 3

L(2, 4, 6, 8), L(2, 4, 4, 6)

2

3M #M k, (2, k) = 1,k >4

L(2, 4, 4, 2k)

1

4M #M 2

L(3, 3, 4, 6)

1

4M #M 3

L(2, 3, 9, 12), L(2, 3, 6, 9)

2

4M #M k, (2, k) = 1,k >4

L(2, 3, 6, 3k)

1

4M #2M 2

L(2, 4, 5, 10)

1

5M #2M 2

L(2, 4, 6, 6)

1

6M #M 2

L(3, 3, 3, 6), L(2, 3, 8, 12)

2

6M #M k,k >2

L(3, 3, 3, 3k)

1

7M #M k,k >1

L(2, 4, 4, 4k)

1

8M #M k,k >1

L(2, 3, 6, 6k)

1

(p.566)

Table B.4.3 Positive BP Links diffeomorphic to S 5

L(2, 2, p, q)

(p, q) = 1

L(2, 3, 3, k)

k = 1, 5(6)

L(2, 3, 4, k)

k = 1, 2, 5, 7, 10, 11(12)

L(2, 3, 5, k)

k ≠ 0, 6, 10, 12, 15, 18, 20, 24(30)

L(2, 3, 7, k)

6 < k < 42, k ≠ 12, 14, 18, 21, 24, 28, 30, 36

L(2, 3, 8, k)

k = 10, 11, 13, 14, 17, 19, 22, 23

L(2, 3, 9, k)

k = 11, 13, 17

L(2, 3, 10, k)

k = 11, 13, 14

L(2, 3, 11, k)

k = 11, 13

L(2, 4, 5, k)

k = 6, 7, 9, 11, 13, 14, 17, 18, 19

L(2, 4, 6, k)

k = 7, 11

L(2, 4, 7, k)

k = 9

L(2, 5, 5, k)

k = 7, 9

L(2, 5, 6, k)

k = 7

L(3, 3, 4, k)

k = 5, 7, 11

L(3, 3, 5, k)

k = 7

L(3, 4, 4, k)

k = 5

Table B.4.4 Positive BP Links with b 2(M) > 0, H 2(L(a),ℤ)tor = 0

M

L(a)

(n - 1)M

L(2, 2, p, q), (p, q) = n > 1

2M

L(2, 3, 3, k), k = 2, 4(6), L(2, 3, 4, k), k = 3, 9(12)

4M

L(2, 3, 3, k), k = 0(6)

6M

L(2, 3, 4, k), k = 0(12)

8M

L(2, 3, 5, k), k = 0(30)

M

L(2, 4, 6, 10)

2M

L(2, 3, 8, k), k = 9, 15, 21, L(2, 3, 9, k), k = 10, 14, 16 L(3, 3, 4, 10)

4M

L(2, 4, 5, k), k = 5, 15, L(2, 5, 5, k), k = 6, 8

6M

L(2, 4, 7, 7), L(3, 3, 4, k), k = 4, 8

8M

L(3, 3, 5, 5)

(p.567) B.5. The Yau–Yu Links in Dimensions 5

Table B.5.1 The Yau–Yu Links in Dimensions 5

Type

f(z 0,z 1,z 2,z 3)

|W|/degree(f)

I (BP)

z 0 a + z 1 b + z 2 c + z 3 d

1 a + 1 b + 1 c + 1 d

II

z 0 a + z 1 b + z 2 c + z 3 z 3 d

1 a + 1 b + 1 c + c 1 c d

III

z 0 a + z 1 b + z 2 c z 3 + z 2 z 3 d

1 a + 1 b + d 1 c d 1 + c 1 c d 1

IV

z 0 a + z 0 z 1 b + z 2 c + z 2 z 3 d

1 a + a 1 a b + 1 c + c 1 c d

V

z 0 a z 1 + z 0 z 1 b + z 2 c + z 2 z 3 d

b 1 a b 1 + a 1 a b 1 + 1 c + c 1 c d

VI

z 0 a z 1 + z 0 z 1 b + z 2 c z 3 + z 2 z 3 d

b 1 a b 1 + a 1 a b 1 + d 1 c d 1 + c 1 d c

VII

z 0 a + z 1 b + z 1 z 2 c + z 2 z 3 d

1 a + 1 b + b 1 b c + [ b ( c 1 ) + 1 ] b c d

VIII

z 0 a + z 1 b + z 1 z 2 c + z 1 z 3 d + z 2 p z 3 q p ( b 1 ) b c + q ( b 1 ) b d = 1

1 a + 1 b + b 1 b c + b 1 b d

IX

z 0 a + z 1 b z 3 + z 2 c z 3 + z 1 z 3 d + z 1 p z 2 q p ( d 1 ) b d 1 + q b ( d 1 ) c ( b d 1 ) = 1

1 a + ( d 1 ) b d 1 + b ( d 1 ) c ( b d 1 ) + b 1 b d 1

X

z 0 a + z 1 b z 2 + z 2 c z 3 + z 1 z 3 d

1 a + [ d ( c 1 ) + 1 ] b c d + 1 + [ b ( d 1 ) + 1 ] b c d + 1 + [ c ( b 1 ) + 1 ] b c d + 1

XI

z 0 a + z 0 z 1 b + z 1 z 2 c + z 2 z 3 d

1 a + a 1 a b + [ a ( b 1 ) + 1 ] a b c + [ a b ( c 1 ) + ( a 1 ) ] a b c d

XII

z 0 a + z 0 z 1 b + z 0 z 2 c + z 1 z 3 d + z 1 p z 2 q p ( a 1 ) a b + q ( a 1 ) a c = 1

1 a + a 1 a b + a 1 a c + [ a ( b 1 ) + 1 ] a b d

XIII

z 0 a + z 0 z 1 b + z 1 z 2 c + z 1 z 3 d + z 2 p z 3 q p [ a ( b 1 ) + 1 ] a b c + q [ a ( b 1 ) + 1 ] a b d = 1

1 a + a 1 a b + a 1 a c + [ a ( b 1 ) + 1 ] a b d

XIV

z 0 a + z 0 z 1 b + z 1 z 2 c + z 0 z 3 d + z 1 p z 3 q + z 2 r z 3 s p ( a 1 ) a b + q ( a 1 ) a c = 1 = r ( a 1 ) a c + s ( a 1 ) a d

1 a + a 1 a b + a 1 a c + a 1 a d

XV

z 0 a z 1 + z 0 z 1 b + z 0 z 2 c + z 2 z 3 d + z 1 p z 3 q p ( a 1 ) a b 1 + q b ( a 1 ) c ( a b 1 ) = 1

b 1 a b 1 + a 1 a b 1 + b ( a 1 ) c ( a b 1 ) + [ c ( a b 1 ) b ( a 1 ) ] c d ( a b 1 )

XVI

z 0 a z 1 + z 0 z 1 b + z 0 z 2 c + z 0 z 3 d + z 1 p z 2 q + z 2 r z 3 s p ( a 1 ) a b 1 + q b ( a 1 ) c ( a b 1 ) = 1 = r ( a 1 ) a c + s ( a 1 ) a d

( b 1 ) a b 1 + ( a 1 ) a b 1 + b ( a 1 ) c ( a b 1 ) + b ( a 1 ) d ( a b 1 )

XVII

z 0 a z 1 + z 0 z 1 b + z 1 z 2 c + z 0 z 3 d + z 1 p z 3 q + z 0 r z 2 s p ( a 1 ) a b 1 + q b ( a 1 ) d ( a b 1 ) = 1 = r ( b 1 ) a b 1 + s a ( b 1 ) a ( a b 1 )

b 1 a b 1 + a 1 a b 1 + a ( b 1 ) c ( a b 1 ) + b ( a 1 ) d ( a b 1 )

XVIII

z 0 a z 2 + z 0 z 1 b + z 1 z 2 c + z 1 z 3 d + z 2 p z 3 q p [ a ( b 1 ) + 1 ] a b c + 1 + q c [ a ( b 1 ) + 1 ] d ( a b c + 1 ) = 1

[ b ( c 1 ) + 1 ] a b c + 1 + [ c ( a 1 ) + 1 ] a b c + 1 + [ a ( b 1 ) + 1 ] c ( a b c + 1 ) + c [ a ( b 1 ) + 1 ] d ( a b c + 1 )

XIX

z 0 a z 3 + z 0 z 1 b + z 2 c z 1 + z 2 z 3 d

[ b ( d ( c 1 ) + 1 ) 1 ] a b c d 1 + [ d ( c ( a 1 ) + 1 ) 1 ] a b c d 1 + [ a ( b ( d 1 ) + 1 ) 1 ] a b c d 1 + [ c ( a ( b 1 ) + 1 ) 1 ] a b c d 1

(p.568)

Notes:

(1) The code for the C program used to generate the tables of the Theorem 5.4.16 are available at: http://www.math.unm.edu/~xgalicki/papers/publications.html.