123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the...

25
s(i,j) s(i,j+1) The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the point. The YY move (associativity) and its signed direction. The broken surface representation of the resulting foam. The seams and vertex in the projection. A s(i+1,j) s(i,j) A 123 1 2 3 (23) (12) 123 1 2 3 (12)3 1(23) 1 1 2 2 3 3 (23) (12) s(i,j) The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the point. The twist move. The broken surface representation. The double line in the projection. d(i,j) Tw f(i,j) The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the point. The label f indicates a fold. The I move. The broken surface representation. The double line in the projection. d(i,j) I d(i,j) The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the point. The II move. The broken surface representation. The double lines in the projection. d(i,j) d(i,j) d(i+1,j) d(i,j) The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the point. The III move and its signed direction. The broken surface representation. The double lines in the projection. III d(i+1,j) d(i+1,j) d(i,j) =(1*3)*(2*3) 1 1*3 2 3 3 1*2 2*3 (1*2)*3 1 2 3 2*3 1*2 1*3 (1*2)*3 =(1*3)*(2*3) 1 1*3 2*3 1 2 2 3 3 1*2 2*3 (1*2)*3 (1*3)*(2*3) III 1

Transcript of 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the...

Page 1: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

s(i,j)

s(i,j+1)

The chart depiction.Here i indicates thenumber of sheets in front of the pointand j indicates thenumber of sheets behind the point.

The YY move (associativity) and itssigned direction.

The broken surface representation of the resulting foam.

The seams and vertexin the projection.

A

s(i+1,j)

s(i,j)

A

123

1

2

3

(23)

(12)

123

1

2

3

(12)3 1(23)

1 12 23 3

(23)(12)

s(i,j)The chart depiction.Here i indicates thenumber of sheets in front of the pointand j indicates thenumber of sheets behind the point.

The twist move.

The broken surface representation.

The double linein the projection.

d(i,j)

Tw

f(i,j)The chart depiction.Here i indicates thenumber of sheets in front of the pointand j indicates thenumber of sheets behind the point. The label f indicatesa fold.

The I move.

The broken surface representation.

The double linein the projection.

d(i,j)

I

d(i,j)The chart depiction.Here i indicates thenumber of sheets in front of the pointand j indicates thenumber of sheets behind the point.

The II move.

The broken surface representation.

The double linesin the projection.

d(i,j)

d(i,j) d(i+1,j)

d(i,j)

The chart depiction.Here i indicates thenumber of sheets in front of the pointand j indicates thenumber of sheets behind the point.

The III move and itssigned direction.

The broken surface representation.

The double lines in the projection.

III

d(i+1,j)

d(i+1,j) d(i,j)

=(1*3)*(2*3)

11*32

3

3

1*2 2*3

(1*2)*3

1

2

3 2*3

1*2

1*3

(1*2)*3=(1*3)*(2*3)

1

1*3

2*3

12 23 31*2

2*3 (1*2)*3 (1*3)*(2*3)

III

1

Page 2: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

d(i,j) s(i+1,j)

d(i+1,j)s(i,j+1) d(i,j)

YI

The chart depiction.Here i indicates thenumber of sheets in front of the pointand j indicates thenumber of sheets behind the point.

The YI move and itssigned direction.

The broken surface representation of the resulting foam.

The double lines andseams in the projection.

12

12

3

1*32*3

(12)*3=(1*3)(2*3)

11 22

3

3

2*3

12 1*3

(12)*3 3

3

(1*3)(2*3)

YI

1

23

12

31*3

2*3

(12)*3

d(i,j)s(i+1,j) d(i+1,j)

s(i,j+1)d(i,j)

The chart depiction.Here i indicates thenumber of sheets in front of the pointand j indicates thenumber of sheets behind the point.

The YI move and itssigned direction.

The broken surface representation of the resulting foam.

The double lines andseams in the projection.

YI

1

2

32

3

23

23

1*2

(1*2)*3=1*(23)

(1*2)*3=1*(23)

1

2

31*2

23

1 12 23 3

23 23

1*2

1*(23)(1*2)*3

IY

(12)3 1(23)

1 12 23 3

(23)(12)<1,2>

<12,3><2,3>

<1,23>

1 12 23 3

23 23

1*2

1*(23)(1*2)*3

IY<1><2>

<1*2><3>

<2,3>

<2,3>

<1><23>

11 22

3

3

2*3

12 1*3

(12)*3 3

3

(1*3)(2*3)

YI<1,2>

<12><3>

<2><3>

<1><3>

<1*3,2*3>

1

1*3

2*3

12 23 31*2

2*3 (1*2)*3 (1*3)*(2*3)

III<1><2>

<1*2><3><2><3>

<2><3>

<1><3>

<1*3><2*3>

YY

2

Page 3: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

123 1|23

12|31|2|3

1234

123|4

1|2|3|4

1|2|34

1|23|4 12|3|4

1|234 12|34

3

Page 4: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

1234

123|4

1|2|3|4

1|2|34

1|23|4 12|3|4

1|234 12|34

<1,2,3>

<1,23,4>

<2,3,4>

<1,2,4>

<12,3,4>

<1,2,3,4>

<1,2,3><4>

<1,2,3><1,23><4>

<2,3><4><12,3><4>

<1,2><4><1*4,2*4,3*4>

<1><2,3><4>

<1><2,3>

<1><23><4><2,3><4>

<1*4><2*4,3*4>

<1><2><4>

<1*2><3><4>

<2,3><4>

<1,2><34>

<2><3,4>

<1><3,4>

<1*3,2*3><4>

<1,2><3>

<12><3,4>

<1,2><3,4>

<1><2><3,4>

<1><2><3>

<1*3><2*3><4>

<1><2><34>

<2><3,4>

<1*2><3,4>

<2><3,4>

<1><3,4>

<1,2><3><4>

<1,2><3>

<1*3,2*3><4>

<1><3><4>

<2><3><4>

<12><3><4>

<1,2><4>

<1*4,2*4><3*4>

<1><2,3,4>

<2,3,4>

<1,><2,34>

<1*2><3,4>

<1><23,4>

<1><2,3>

<2,3,4>

<1><2><3><4>

<1><2><3>

<1*3><2*3><4>

<1><3><4>

<2><3><4>

<2><3><4><1*2><3><4>

<1><2><4>

<1*4><2*4><3*4>

4

Page 5: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

0

1

2 3

4

5

1

2

3

4

5

12

2334

45

123

234

345

1234

2345

12345

0

1

2 3

4

1

2

3

4

12

2334

123

234

1234

0

1

2 3

5

1

2

3

12

23

45

123

345

2345

12345

0

1

2

4

5

1

2

5

12

2334

45

123

234

345

1234

2345

123450

1

3

4

5

1

4

5

23

45

123

234

1234

2345

12345

0

2 3

4

5

3

4

5

12

34

45

123

345

1234

12345

1

2 3

4

5

2

3

4

5

2334

45234

345

2345

<1,2,3,4>

<12,3,4,5>

<1,23,4,5><1,2,34,5>

<1,2,3,45>

<12,3,4,5>

5

Page 6: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

<1,2,3>

<1,23,4>

<2,3,4>

<12,3,4>

<2,3,4>

<1,2,3,4><1,2,3><4>

<1,2,3><1,23><4>

<2,3><4><12,3><4>

<1,2><4><1*4,2*4,3*4>

<1,2><34>

<2><3,4>

<1><3,4>

<1*3,2*3><4>

<1,2><3>

<12><3,4>

<1,2><3,4>

<1,2><3><4>

<1,2><3>

<1*3,2*3><4>

<1><3><4>

<2><3><4>

<12><3><4>

<1,2><4>

<1*4,2*4><3*4>

6

Page 7: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

<1><2,3><4>

<1><2,3>

<1><23><4><2,3><4>

<1*4><2*4,3*4>

<1><2><4>

<1*2><3><4>

<2,3><4>

<1><2><3,4>

<1><2><3>

<1*3><2*3><4>

<1><2><34>

<2><3,4>

<1*2><3,4>

<2><3,4>

<1><3,4>

<1><2,3,4>

<2,3,4>

<1,><2,34>

<1*2><3,4>

<1><23,4>

<1><2,3>

<2,3,4>

<1><2><3><4>

<1><2><3>

<1*3><2*3><4>

<1><3><4>

<2><3><4>

<2><3><4><1*2><3><4>

<1><2><4>

<1*4><2*4><3*4>

7

Page 8: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

123

124

134

234

134

124

123

234

123

124

134

234

134

124

123

234

8

Page 9: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

9

Page 10: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

<2,34,5><123,4,5>

<1,2,3>

<12,3,4>

<1,2,3>

<123,4,5>

<1,23,4>

<2,3,4>

<12,34,5>

<3,4,5>

<12,3,45>

<1,2,345>

<1,234,5>

<3,4,5>

<1,2,345> <2,3,45>

<1,23,45>

<2,3,4>

<1,234,5>

<23,4,5>

-<1,2,3,4>

-<12,3,4,5>

<1,2,3,45>

<1,23,4,5>

-<1,2,34,5>

<2,3,4,5>

<1,2,34>

10

Page 11: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

(3,4)|5

(3,4)|5

(3,4)|5

(1,2,34)*5 (1,2,34)*5

(1,2)|5 (1,2)|5

(1,2)|5

(2,3,4)*5

(2,3)|5

(2,3)|5

(23,4)|5

(2,3,4)

(2,34)|5

(2,3,4)

(12,3,4)*5

(12,3,4)*5

(1,23,4)*5

(1,23,4)*5

(1,2,3)*5

(1,234)|5

(1,234)|5

(1,23)|5

(1,2,3)(1,2,3)

(123,4)|5

(123,4)|5

(12,3)|5

(12,34)|5

(1,2,34)

(1,23,4)

(12,3,4)

(1,2,3,4)*5

(1,2,3)|5

(1,23,4)|5

-(1,2,3,4)

-(1,2,34)|5

-(12,3,4)|5

(2,3,4)|5

11

Page 12: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

(12,3)*4|5

(1,2)|4

-(12,3)|(45)

-(1,2)|(4,5)

-(1,2,3)*4|5

-(1,2,3)|4

(1,23)|(45)

(2,3)|(4,5)

(1,2,3)|(45)

(12,3)|4(1,2,3)*4

(1,2)*4|5 (12)|(4,5)

(1,2)|(45)

(1,2)|(45)

3|(4,5)3|(4,5)

3|(4,5)

(1,2,3)*(45)(1,2,3)*(45)

(1,2,3)*(45)

(1,2,3)*(45)

1|(4,5)

1|(4,5)

1|(4,5)1|(4,5)

(2,3)*4|5

(2,3)*4|5

2|(4,5)

2|(4,5)

(1,23)*4|5

(1,23)*4|5

(2,3)|4

(2,3)|4

(2,3)|4

(123)|(4,5)(123)|(4,5)

(1,2,3)

(1,2,3)

(1,23)|4

(1,23)|(45)

(23)|(4,5)

(2,3)|(45) (12,3)|(45)

(1,2)|4

(12,3)*4|5

12

Page 13: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

(2,3)|5

(2,3)|5

((2,3)|4)*5((1,23)|4)*5

((1,23)|4)*5

3|4|53|4|5

3|4|5

3|4|5

(1,2,3)*(45)

(1,2,3)*(45)

2|4|5

2|4|5

(1,2,3)*(45)

(1,2,3)*(45)

(2,3)*4|5

(2,3)*4|5(2,3)|4

(2,3)|4

(2,3)|4

(23)|4|5

(2,3)|4|5

(1,23)|5

(123)|4|5(123)|4|5

(1,2,3)

(1,2,3)

(12,3)|5

(1,2)|5

(1,2)|5

(1,2)|5

(1,2,3)*5

((12,3)|4)*5

((12,3)|4)*5((1,2)|4)*5

((1,2)|4)*5

((1,2,3)|4)*5

(1,2,3)|5

(12,3)|4 (12)|4|5

(12,3)*4|5

(12,3)*4|5

-(12,3)|4|5

(1,2)|4(1,2)|4

(1,2)*4|5

1|4|5

-(1,2)|4|5

1|4|51|4|5

1|4|5

(1,2,3)*4

(1,23)*4|5

(1,23)*4|5

-(1,2,3)*4|5

(1,23)|4

(1,23)|4|5

-(1,2,3)|4

13

Page 14: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

2*3|(4,5)

(1,2)|(3,45)

(1,2)*3|(4,5)

1|(3,4,5)

2|(3,4,5)

-(1,2)|(34,5)

-(1,2)|(3,4)

-(12)|(3,4,5)

2|(3,45)(3,4,5)

1*3|(4,5) (1,2)*3|(45)

(12)*3|(4,5)

(1,2)|3(3,4,5)

(1,2)*(34)|5

1|(3,4)

(1,2)*3|4

1|(34,5)

2|(3,4)

(1,2)|(345)

(3,4,5)

2|(34,5)

12|(34,5)

(1,2)|(34)

(12)|(3,4) (12)|(3,45)

1|(3,45)

2*3|(4,5)

1|(3,45)

(1,2)*(34)|5

(3,4,5)

(1,2)*(34)|5

1|(3,4)

(1,2)*(34)|5

2|(3,4)

2|(3,4)

1|(34,5)

(3,4,5)

(1,2)*3|4 (12)*3|(4,5)

(3,4,5)

(1,2)|3

(1,2)|3

(1,2)|(345)

(3,4,5)

14

Page 15: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

-(12)|(3,4)|5

(1,2)|3|5

(1,2)*3|4|5 ((1,2)|(3,4))*5

-(1,2)|(34)|5-(1,2)|(3,4)

2|(3,4)|51|(3,4)|5

((1,2)*3|4)*5

((1,2)*3|4)*5 ((1,2)*3|4)*5

1|3|5 2|3|51|3|5

1|3|5

2|3|5

2|3|5

2*3|4|5

2*3|4|5

2*3|4|5

(1|(3,4))*5

(1|(3,4))*5

(1|(3,4))*5

(1,2)*(34)|5

(1,2)*(34)|5

(1,2)*(34)|5 (1,2)*(34)|5

(3,4)|5

(3,4)|5

(3,4)|5

(3,4)|5

(3,4)|5

(3,4)|5(3,4)|5

1*3|4|5

(3,4)|5

(2|(3,4))*5

(1,2)*3|5((1,2)|3)*5

(1,2)|3

(1,2)|3

(1,2)|3

(1,2)|5

(1,2)|5

(1,2)|5

2|(3,4)

2|(3,4)

2|(3,4)

1|(34)|5

1|(34)|5

1|(3,4)

1|(3,4)

(1,2)*3|4

(1,2)*3|4((1,2)|(34))*5

((1,2)|(34))*5

(12)*3|4|5

(12)*3|4|5

(12)|3|5

((12)|(3,4))*5

((12)|(3,4))*5

(12)|(34)|5

(12)|(3,4)

(1,2)|(34)

2|(34)|5

15

Page 16: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

-(1,2)|3|4

-(1,2)|3)*4|5

-(1,2)|(4,5)

-(12)|3|(4,5)

(1,2)*3|(4,5)

1|3|(4,5)

2|3|(4,5)

(1,2)|3|(45)

((12)|3)*4|5

(1,2)|4 (1,2)|4

((12)|3)*4|5

3|(4,5)

(1,2)|3

(1,2)|3

3|(4,5)

1|(4,5)

2|3|4

2|3|4

2|3|4

2|3|4

(((1,2)*3)*4)|5

((1,2)|*3)|(45)

2*3|(4,5)

2*3|(4,5)

1|3|(45)

1|3|(45)

(12)*3|(4,5)

(1,2)*3)|4

(12)|3|(45)

(1,2)|(45)

2|(4,5)

2|(4,5)

(2|3)*4|5(2|3)*4|5

(1|3)*4|5

(1|3)*4|5

3|(4,5)

2|3|(45)

(12)|3|4

(12)|(4,5)

((1,2)|3)*4(1,2)*4|5

1*3|(4,5)

1|3|4

1|3|4

(12)*3|(4,5)

(1,2)|3

3|(4,5)

3|(4,5)

3|(4,5)(((1,2)*3)*4)|5

(1,2)*3)|4

(((1,2)*3)*4)|5 (((1,2)*3)*4)|5

1|(4,5)

1|(4,5)

1|(4,5)

((1,2)|3)*(45)

3|(4,5)

3|(4,5)

(1,2)|(45)

((1,2)|3)*(45)

((1,2)|3)*(45)((1,2)|3)*(45)

16

Page 17: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

-(1,2)|3|4

-((1,2)|3)*4|5

-(1,2)|4|5

-12|3|4|5

(1,2)*3|4|5

1|3|4|5 2|3|4|5

((1,2)|3|4)*5(1,2)|3|5

((12)|3)*4|5

(1,2)|4 (1,2)|4

((12)|3)*4|5

3|4|53|4|5

3|4|5

3|4|5

(1,2)|3

(1,2)|3

(1,2)|3

(1,2)|5

(1,2)|5

(1,2)|5

3|4|5 3|4|5

3|4|5

3|4|5

1|4|5

1|4|5

1|4|5

1|4|5

(((1,2)|3)*4)*5

(((1,2)|3)*4)*5

(((1,2)|3)*4)*5(((1,2)|3)*4)*5

2|3|4

2|3|4

2|3|4

2|3|4

((1,2)|3)*4|5

((1,2)|3)*4|5

((1,2)|3)*4|5((1,2)|3)*4|5

((1,2)|*3)*5|4

((1,2)|*3)*5|4 ((1,2)|*3)*5|4

1|3|51|3|5

1|3|5

2*3|4|5

2*3|4|5

2*3|4|5

2|3|5

2|3|5

2|3|5

(1|3|4)*5

(1|3|4)*5

(1|3|4)*5

(12)*3|4|5

(12)*3|4|5

(1,2)*3)|4

(1,2)*3)|4

(12)*5|3|4

(12)*5|3|4

((1,2)|3)*4)*5

((1,2)|3)*4)*5

2|4|5

2|4|5

(2|3)*4|5

(2|3)*4|5

(1|3)*4|5

(1|3)*4|5

3|4|5

(2|3|4)*5

((1,2)|3)*5

(12)|3|5

(1,2)*3|5

(12)|3|4 (12)4|5

((1,2)|3)*4 (1,2)*4|5

1*3|4|5

1|3|4

1|3|4

17

Page 18: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

1|(2,3)

(23,4,5)

1*(23)|(4,5)

(2,3,4)

1*2|(3,4)

1|(2,34)

(2,3,4)

1|(23,4)

(2,3,4)

(2,34,5)

(3,4,5)

1|(234,5)

1*2|(34,5)

(3,4,5)

1|(2,345)

1*2|(3,45)

1|(23,45)

(23,4,5)

(2,34,5)

(2,3,45)

-(2,3,4,5)

-1*2|(3,4,5)

-1|(2,34,5)

(2,3,4,5)

1|(23,4,5)

1|(2,3,45)

-1|(2,3,4)

1*2|(3,4)

1|(2,3)

1|(234,5)(2,34,5)

(3,4,5)

1|(2,345)

1|(2,3)

1*(23)|(4,5)1*(23)|(4,5)

(23,4,5)

(2,3,45)

(2,3,45)

18

Page 19: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

1|(2,3)|5

(1|(2,3,4))*5

-1|(2,34)|5

-1*2|(3,4)|5

-(2,3,4)|5

1|(23,4)|5

-1|(2,3,4)

(2,3,4)

(1*2)|(34)|5

(2,3,4)|5

(3,4)|5

(1*2)|(3,4)(1*2)|(3,4)

(2,34)|5

(2,34)|5

1|(2,34)

(2,3,4)*5((23),4)|5

(2,3)|5

1|(2,3)

1|(23,4)

1|(234)|5

(2,3,4)

((23),4)|5

1|(2,3)

1|(2,3)

(1*(23)|4|5

(1*(23)|4|5

(1*(23)|4|5

(2,3)|5

(1*(23)|4|5

(2,3,4)*5

(1*2)|3|5

(1*2)|3|5

(2,3,4)*5

1|(234)|5

(2,3,4)(2,34)|5

(1|(2,34))*5

1|2|5

(3,4)|5

1|2|5

1|2|5

1|2|5

(3,4)|5

(2,3,4)*5

(1*2)|(3,4))*5

(1*2)|(3,4))*5

(1|(2,3))*5

(2,3)|5

1|(23)|5

(1|(23,4))*5

(2,3)|5

(23,4)|5

(2,3,4)*5

(1|(23,4))*5

1|(2,34)

(1|(2,34))*5

19

Page 20: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

-(12)|(3,4)|5

(1,2)|3|5

(1,2)*3|4|5 ((1,2)|(3,4))*5

-(1,2)|(34)|5-(1,2)|(3,4)

2|(3,4)|51|(3,4)|5

2|(4,5)

((1*2)|3)*4|5 (1*2)|3|4

2*3|4|5

2*3|4|5

2*3|4|5

(1|(3,4))*5

(1|(3,4))*5

(1|(3,4))*5

(1,2)*(34)|5

(1,2)*(34)|5

(1,2)*(34)|5 (1,2)*(34)|5

(3,4)|5

(3,4)|5

(3,4)|5

(3,4)|5

(3,4)|5

(3,4)|5(3,4)|5

1*3|4|5

(3,4)|5

(2|(3,4))*5

(1,2)*3|5((1,2)|3)*5

(1,2)|3

(1,2)|3

(1,2)|3

(1,2)|5

(1,2)|5

(1,2)|5

2|(3,4)

2|(3,4)

2|(3,4)

1|(34)|5

1|(34)|5

1|(3,4)

1|(3,4)

(1,2)*3|4

(1,2)*3|4((1,2)|(34))*5

((1,2)|(34))*5

(12)*3|4|5

(12)*3|4|5

(12)|3|5

((12)|(3,4))*5

((12)|(3,4))*5

(12)|(34)|5

(12)|(3,4)

(1,2)|(34)

2|(34)|5

2|(4,5) 2|(4,5)

((1*2)|3)*4|5 (1*2)|3|4

(1*2)|3|4

((1*2)|3)*4|5

20

Page 21: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

(2,3)*4|5(23)|4|5

(23)|4|5

(23)|4|5

(2,3)*4|5

1|(2,3)|5

(1|(2,3)|4)*5

1|(23)|4|5

-(2,3)|4|5

-(1*2)|3|4|5 -1|2|4|5

-(1|(2,3))*4|5-1|(2,3)|4

(2,3)|4|5(1|(23)|4)*5

(2,3)|5(2,3)|5

(1|(23)|4)*5

(1|(2,3))*5

((2,3)|4)*5

(1*2|3|4)*5 (1|2|4)*5

(1|2|4)*5

((2,3)|4)*5

(1|(2,3))*(45)

3|4|5

(1|(2,3))*(45)

2|4|5

2|4|5

(2,3)|4

1|2|5

1|2|5

(1*2)|3|5

(2,3)|5

1|(2,3)

(2,3)|4

3|4|5

3|4|5

(1|(2,3))*(45)

(1|(2,3))*(45)

(1*2|3|4)*5

((2,3)|4)*5

((2,3)|4)*5

((2,3)|4)*5

(1*2)|3|5

1|(2,3)

1|(2,3)

1|(23)|5

(1*(23)|4|5(1*(23)|4|5

(1*(23)|4|5

(1*(23)|4|5

1|(23)|5

1|(23)|4

1|4|5 1|4|5

(2,3)|4

1|4|5

1|(23)|5

(1|2)*4|5

1|4|5

((1*2)|3)*4|5

1|2|4

(1|(2,3))*4

(2,3)*4|5

1|2|4

2|4|5

(1*2)|3|4

(2,3)*4|5

(2,3)*4|5

(1*2)|3|4

(2,3)|4

(1*2)|3|4

3|4|5

(2,3)|5

1|2|5

1|2|5

3|4|5

1*2|4|5

2|4|5 ((1*2)|3)*4|5

((1*2)|3)*4|5

2|4|5

21

Page 22: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

(1*2)|(3,4)2|(3,4)

2|(34,5)

1|2|3

(3,4,5)

(1|2)*3|4

1|(3,4)

2|(3,4)

1|2|34

-1|2|(3,4)

(1*2)|(34,5)

1|2|(345)

(3,4,5)

(1*2)|(3,45)

(3,4,5)

2|(3,45)

(2*3)|(4,5)

-2|(3,4,5)

-(1*2)|(3,4,5)

2|(3,45) (1|2)*3|(45)

1|(3,45)

1|2|(3,45)

(2*3)|(4,5)

(1*3)|(4,5)(1|2)*(34)|5

(1|2)*3|(4,5)

(3,4,5)

1|(34,5) 1|(3,4,5)

2|(34,5)

-(1|2)|(34,5)

2|(3,4,5)

2|(34,5)

1|2|3

1|2|31|2|3

(3,4,5)

(3,4,5)

(3,4,5)

(1|2)*3|4

(1|2)*(34)|5(1|2)*(34)|5

(1|2)*(34)|5

1|(3,4)

2|(3,4)

2|(3,4)

1|(34,5)

2|(3,45)

(1*(23))|(4,5)

1|(3,45)

(3,4,5)

1|2|(345)

(1*(23))|(4,5)

(1*(23))|(4,5)

(2*3)|(4,5)

(1*2)|(3,4)

(2*3)|(4,5)

22

Page 23: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

1|3|(4,5)

(1|2)*3|(4,5)

1|2|3|(45)

-1|2|(4,5)

-(1|2|3)*4|5 -1|2|3|4

-2|3|(4,5)-(1*2)|3|(4,5)

1|3|(45)

2|(4,5)

(1|2|3)*(45)

3|(4,5) (1*(23)|(4,5)

1|2|3

(1|2)*(34)|5

3|(4,5)

2|3|4

2|3|4

2|3|4

2|3|4

1|(4,5)

2|(4,5)

((1*2)|3)*4|5

1|2|4

(1*2)|3|4

(2|3)*4|5

1|2|(45)

(12)*3|4|5

2|(4,5)

2|3|(45)

(1|2)*3|4

1|3|4(1|3)*4|5

(1|3)*4|5

(1|2|3)*4

2|3|4

3|(4,5)

(1*2)|3|45

(1*2)|(4,5)

2|3|(45) (1|2)*3|(45)

3|(4,5) (1*2)|(4,5)

(1*2)|3|4

(1*2)|3|4

(2|3)*4|5

(2|3)*4|5

1|2|4

1|2|4

((1*2)|3)*4|5

((1*2)|3)*4|5

2|(4,5)2|(4,5)

(1|2)*4|5

1|(4,5)

1|(4,5) 1|(4,5)

(1|2|3)*(45)

(1|2|3)*(45) (1|2|3)*(45)

(2|3)*4|5

(2|3)*4|5

(1|2)*(34)|5

(1|2)*(34)|5

(1|2)*(34)|5

1|3|4

3|(4,5)

3|(4,5)

3|(4,5)

(1|2)*3|4

1|2|3

1|2|3 1|2|3

(2*3)|(4,5)

(1*(23)|(4,5)

(1*(23)|(4,5)

2|3|(45)

(2*3)|(4,5)2|(4,5)

3|(4,5)

3|(4,5)

1|2|(45)

2|(4,5)

1|3|(45)

2|3|(4,5)

23

Page 24: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

1|(3,4)1|2|(34)

(1|2|(34))*5

2|(34)|5

1*2|(3,4)

(3,4)|5

2|3|5

(2*3)|4|5

2|(3,4)

(3,4)|5

(1|2)*3|4(1*2)*3|4|5

(1*2)|3|5

(1*2|(3,4))*5

(2|(3,4))*5

(1*2)|3|5

(3,4)|5

(1*2)|(34)|5

1|2|5

1|(34)|5

2|(34)|5

(1|2)*(34)|5

(1*3)|4|5

1|3|5

2*3|4|5

(3,4)|5

(1|(3,4))*5

2|3|5

(2|(3,4))*5

(1|2|3)*5

1|2|3

(1|2)*3|5

((1|2)*3|4)*5

(1|(3,4))*5

2*3|4|5

2*3|4|5

1|3|5

1|3|5

((1|2)*3|4)*5

((1|2)*3|4)*5

2|3|5

2|3|5

(1|(3,4))*5

(3,4)|5

(3,4)|5

(1|2|(34))*5

1|2|5

1|2|5

1|2|5

(1*2|(3,4))*5

(2|(3,4))*5

(2|(3,4))*5(2|(3,4))*5

(1*2)*3|4|5

(1*2)*3|4|5

(1*2)*3|4|5

2|3|51|2|3

1|2|3

1|2|3

1*2|(3,4)

2|(34)|5

2|(3,4)

2|(3,4)

2|(3,4)

(1|2)*(34)|5

(1|2)*(34)|5

(1|2)*(34)|5

1|(34)|5

1|(3,4)

(1|2)*3|4

(3,4)|5

(3,4)|5

(3,4)|5

(2*3)|4|5

-1|2|(3,4)

1|2|3|5 (1|2)*3|4|5

1|(3,4)|52|(3,4)|5(1|2|(3,4))*5

-1|2|(34)|5

-(1*2)|(3,4)|5

-2|(3,4)|5

24

Page 25: 123 1|23...d(i,j) s(i+1,j) s(i,j+1)d(i+1,j)d(i,j) YI The chart depiction. Here i indicates the number of sheets in front of the point and j indicates the number of sheets behind the

(1*2|3)*4|5

(1*2|3)*4|51*2|3|4

1*2|3|4

2|4|5

2|4|5

(2|3)*4|5

(2|3)*4|5

1|2|4 1|2|4

(1|2|3)*4

2|3|4

((1|2)*3)*4|5

1|3|4

1|3|4

2|3|4

(1|2)*3|4

(1|2)*3|41|2|3

1|2|3

3|4|5

3|4|5

3|4|5

3|4|5

((1|2)*3)*4|5

((1|2)*3)*4|5

1|2|3

1|2|3

3|4|5

(2|3|4)*5

(2|3|4)*5 (2|3|4)*5

(2|3|4)*51*2|3|5

1*2|3|5

1|2|5

1|2|5

1|2|5

1|2|5

(1|2)*3|5

1|3|5

1|3|5

1|3|5

2|3|5

2|3|5

(1|2|3)*5

2|3|5

((1|2)*3|4)*5

((1|2)*3|4)*5

((1|2)*3|4)*5

2*3|4|5

2*3|4|5

2*3|4|5

(1*2|3|4)*5

(1*2|3|4)*5

3|4|5

3|4|5

3|4|5

3|4|5

1*2|4|5

2|3|4

(1|2|4)*5

(1|2|4)*5

2*3|4|52*3|4|5

(1*2)*3|4|5

(1*2)*3|4|5 (1*2)*3|4|5

(1*2)*3|4|5

2|3|5

2|3|5

((1|2|3)*4)*5

((1|2|3)*4)*5

((1|2|3)*4)*5((1|2|3)*4)*5

(1|2)*3|4|5

1|2|3|5

(1|2|3|4)*5

2|3|4|51|3|4|5

-2|3|4|5 -1*2|3|4|5

-1|2|4|5

-(1|2|3)*4|5

-1|2|3|4(1|2)*4|5

1|4|5

1|4|5

1|4|5

1|4|5

2|4|5

2|4|5

2|3|4

2|3|4

(1|3|4)*5

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(1|3|4)*5

3|4|5

(2|3|4)*51*3|4|5

(1|3)*4|5

(1|3)*4|5

(2|3)*4|5

(2|3)*4|5

25


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