9
29
(K9a
31
)
A knot diagram
1
Linearized knot diagam
5 9 6 2 8 3 1 4 7
Solving Sequence
1,5 2,8
6 4 3 7 9
c
1
c
5
c
4
c
3
c
7
c
9
c
2
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
− 2u
8
+ 6u
7
− 9u
6
+ 13u
5
− 19u
4
+ 14u
3
− 12u
2
+ 4b + 7u − 1,
3u
9
− 6u
8
+ 18u
7
− 31u
6
+ 47u
5
− 65u
4
+ 58u
3
− 56u
2
+ 8a + 29u − 11,
u
10
− u
9
+ 4u
8
− 7u
7
+ 8u
6
− 14u
5
+ 11u
4
− 10u
3
+ 7u
2
− 2u − 1i
I
u
2
= h3488u
15
+ 8516u
14
+ ··· + 887b + 5098, 5348u
15
+ 12394u
14
+ ··· + 887a + 7607,
u
16
+ 3u
15
+ ··· + 2u + 1i
I
u
3
= hb − 1, 2a − 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 27 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I.
I
u
1
= hu
9
− 2u
8
+ · · · + 4b − 1, 3u
9
− 6u
8
+ · · · + 8a − 11, u
10
− u
9
+ · · · − 2u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
8
=
−
3
8
u
9
+
3
4
u
8
+ ··· −
29
8
u +
11
8
−
1
4
u
9
+
1
2
u
8
+ ··· −
7
4
u +
1
4
a
6
=
1
16
u
9
−
1
8
u
8
+ ··· +
31
16
u −
17
16
1
8
u
9
−
1
4
u
8
+ ··· +
15
8
u −
1
8
a
4
=
u
u
3
+ u
a
3
=
1
16
u
9
−
1
8
u
8
+ ··· +
31
16
u −
1
16
1
8
u
9
−
1
4
u
8
+ ··· +
7
8
u −
1
8
a
7
=
−
1
8
u
9
+
1
4
u
8
+ ··· −
15
8
u +
9
8
−
1
4
u
9
+
1
2
u
8
+ ··· −
7
4
u +
1
4
a
9
=
−
7
8
u
9
+
3
4
u
8
+ ··· −
17
8
u +
15
8
3
4
u
9
−
1
2
u
8
+ ··· −
7
4
u +
1
4
a
9
=
−
7
8
u
9
+
3
4
u
8
+ ··· −
17
8
u +
15
8
3
4
u
9
−
1
2
u
8
+ ··· −
7
4
u +
1
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
37
16
u
9
−
3
8
u
8
−
55
8
u
7
+
145
16
u
6
−
41
16
u
5
+
351
16
u
4
−
35
8
u
3
+
19
2
u
2
−
219
16
u +
85
16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
6
u
10
+ u
9
+ 4u
8
+ 7u
7
+ 8u
6
+ 14u
5
+ 11u
4
+ 10u
3
+ 7u
2
+ 2u − 1
c
2
, c
5
2(2u
10
+ 3u
9
− 4u
8
− 8u
7
+ 9u
6
+ 7u
5
− 5u
4
− 2u
3
− u + 1)
c
7
, c
9
u
10
− 4u
8
+ u
7
+ 5u
6
− 3u
5
+ 12u
4
+ 18u
3
− 7u
2
− 11u − 4
c
8
u
10
− 3u
9
+ 3u
8
+ 8u
7
− 7u
6
− 30u
5
+ 80u
4
− 60u
3
+ 41u
2
− 30u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
y
10
+ 7y
9
+ ··· − 18y + 1
c
2
, c
5
4(4y
10
− 25y
9
+ ··· − y + 1)
c
7
, c
9
y
10
− 8y
9
+ ··· − 65y + 16
c
8
y
10
− 3y
9
+ ··· − 244y + 64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.642531 + 0.377867I
a = 0.425417 + 0.618053I
b = −0.388235 + 0.305929I
−1.18006 − 1.03831I −4.73685 + 3.71172I
u = 0.642531 − 0.377867I
a = 0.425417 − 0.618053I
b = −0.388235 − 0.305929I
−1.18006 + 1.03831I −4.73685 − 3.71172I
u = −0.296868 + 1.222110I
a = −0.265428 + 0.874553I
b = 0.310628 + 1.327070I
4.73127 + 5.96240I 6.55763 − 6.45237I
u = −0.296868 − 1.222110I
a = −0.265428 − 0.874553I
b = 0.310628 − 1.327070I
4.73127 − 5.96240I 6.55763 + 6.45237I
u = 0.090479 + 1.266340I
a = −0.180352 − 0.660546I
b = 1.72873 − 0.67558I
8.92450 − 2.36890I 11.53570 + 2.96432I
u = 0.090479 − 1.266340I
a = −0.180352 + 0.660546I
b = 1.72873 + 0.67558I
8.92450 + 2.36890I 11.53570 − 2.96432I
u = 1.36651
a = −0.570064
b = −1.16409
0.587104 12.3230
u = −0.50395 + 1.40837I
a = 0.204381 − 1.196050I
b = −1.50564 − 0.50027I
10.4508 + 12.2059I 7.05765 − 6.58910I
u = −0.50395 − 1.40837I
a = 0.204381 + 1.196050I
b = −1.50564 + 0.50027I
10.4508 − 12.2059I 7.05765 + 6.58910I
u = −0.230893
a = 2.70203
b = 0.873110
1.26306 9.09880
5
II. I
u
2
= h3488u
15
+ 8516u
14
+ · · · + 887b + 5098, 5348u
15
+ 12394u
14
+ · · · +
887a + 7607, u
16
+ 3u
15
+ · · · + 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
8
=
−6.02931u
15
− 13.9729u
14
+ ··· − 2.86809u − 8.57610
−3.93236u
15
− 9.60090u
14
+ ··· − 2.30440u − 5.74746
a
6
=
7.33709u
15
+ 15.6888u
14
+ ··· + 5.98309u + 15.1251
3.02593u
15
+ 7.05299u
14
+ ··· + 3.38331u + 5.16347
a
4
=
u
u
3
+ u
a
3
=
−4.12176u
15
− 9.11838u
14
+ ··· + 4.54791u − 3.85457
−0.676437u
15
− 1.99098u
14
+ ··· + 2.04397u − 0.525366
a
7
=
−2.09696u
15
− 4.37204u
14
+ ··· − 0.563698u − 2.82864
−3.93236u
15
− 9.60090u
14
+ ··· − 2.30440u − 5.74746
a
9
=
−4.00451u
15
− 9.22661u
14
+ ··· − 1.97971u − 5.55017
−1.15896u
15
− 3.23788u
14
+ ··· − 0.784667u − 1.39346
a
9
=
−4.00451u
15
− 9.22661u
14
+ ··· − 1.97971u − 5.55017
−1.15896u
15
− 3.23788u
14
+ ··· − 0.784667u − 1.39346
(ii) Obstruction class = −1
(iii) Cusp Shapes =
14664
887
u
15
+
36136
887
u
14
+ ··· +
12068
887
u +
27426
887
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
6
u
16
− 3u
15
+ ··· − 2u + 1
c
2
, c
5
u
16
− u
15
+ ··· + 136u + 47
c
7
, c
9
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
2
c
8
(u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
y
16
+ 11y
15
+ ··· + 20y
2
+ 1
c
2
, c
5
y
16
− 9y
15
+ ··· − 13044y + 2209
c
7
, c
9
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
2
c
8
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.181988 + 1.048500I
a = 1.25894 + 1.17937I
b = 0.463640
4.13490 7.89446 + 0.I
u = −0.181988 − 1.048500I
a = 1.25894 − 1.17937I
b = 0.463640
4.13490 7.89446 + 0.I
u = −1.142130 + 0.104845I
a = −0.895766 − 0.516597I
b = −1.334530 − 0.318930I
5.66955 + 6.44354I 5.42845 − 5.29417I
u = −1.142130 − 0.104845I
a = −0.895766 + 0.516597I
b = −1.334530 + 0.318930I
5.66955 − 6.44354I 5.42845 + 5.29417I
u = 0.309237 + 1.112330I
a = 0.034672 − 0.683601I
b = 0.108090 − 0.747508I
1.13045 − 2.57849I 0.27708 + 3.56796I
u = 0.309237 − 1.112330I
a = 0.034672 + 0.683601I
b = 0.108090 + 0.747508I
1.13045 + 2.57849I 0.27708 − 3.56796I
u = −0.072810 + 1.153150I
a = −1.02661 + 1.10040I
b = 1.180120 + 0.268597I
4.33052 + 1.13123I 3.41522 − 0.51079I
u = −0.072810 − 1.153150I
a = −1.02661 − 1.10040I
b = 1.180120 − 0.268597I
4.33052 − 1.13123I 3.41522 + 0.51079I
u = −0.597255 + 0.026660I
a = 1.20070 − 1.29659I
b = 0.108090 − 0.747508I
1.13045 − 2.57849I 0.27708 + 3.56796I
u = −0.597255 − 0.026660I
a = 1.20070 + 1.29659I
b = 0.108090 + 0.747508I
1.13045 + 2.57849I 0.27708 − 3.56796I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.50715 + 1.45748I
a = 0.219942 + 0.896459I
b = −1.334530 + 0.318930I
5.66955 − 6.44354I 5.42845 + 5.29417I
u = 0.50715 − 1.45748I
a = 0.219942 − 0.896459I
b = −1.334530 − 0.318930I
5.66955 + 6.44354I 5.42845 − 5.29417I
u = −0.60300 + 1.44597I
a = −0.091711 − 0.669730I
b = −1.37100
9.79260 9.86404 + 0.I
u = −0.60300 − 1.44597I
a = −0.091711 + 0.669730I
b = −1.37100
9.79260 9.86404 + 0.I
u = 0.280801 + 0.318917I
a = 3.29984 − 0.74872I
b = 1.180120 − 0.268597I
4.33052 − 1.13123I 3.41522 + 0.51079I
u = 0.280801 − 0.318917I
a = 3.29984 + 0.74872I
b = 1.180120 + 0.268597I
4.33052 + 1.13123I 3.41522 − 0.51079I
10
III. I
u
3
= hb − 1, 2a − 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
8
=
0.5
1
a
6
=
0.25
1.5
a
4
=
1
2
a
3
=
0.75
0.5
a
7
=
−0.5
1
a
9
=
0.5
1
a
9
=
0.5
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2.25
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
9
u − 1
c
2
2(2u − 1)
c
4
, c
6
, c
7
u + 1
c
5
2(2u + 1)
c
8
u
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
, c
7
, c
9
y − 1
c
2
, c
5
4(4y − 1)
c
8
y
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.500000
b = 1.00000
0 −2.25000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
(u − 1)(u
10
+ u
9
+ ··· + 2u − 1)
· (u
16
− 3u
15
+ ··· − 2u + 1)
c
2
4(2u − 1)(2u
10
+ 3u
9
− 4u
8
− 8u
7
+ 9u
6
+ 7u
5
− 5u
4
− 2u
3
− u + 1)
· (u
16
− u
15
+ ··· + 136u + 47)
c
4
, c
6
(u + 1)(u
10
+ u
9
+ ··· + 2u − 1)
· (u
16
− 3u
15
+ ··· − 2u + 1)
c
5
4(2u + 1)(2u
10
+ 3u
9
− 4u
8
− 8u
7
+ 9u
6
+ 7u
5
− 5u
4
− 2u
3
− u + 1)
· (u
16
− u
15
+ ··· + 136u + 47)
c
7
(u + 1)(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
2
· (u
10
− 4u
8
+ u
7
+ 5u
6
− 3u
5
+ 12u
4
+ 18u
3
− 7u
2
− 11u − 4)
c
8
u(u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1)
2
· (u
10
− 3u
9
+ 3u
8
+ 8u
7
− 7u
6
− 30u
5
+ 80u
4
− 60u
3
+ 41u
2
− 30u + 8)
c
9
(u − 1)(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
2
· (u
10
− 4u
8
+ u
7
+ 5u
6
− 3u
5
+ 12u
4
+ 18u
3
− 7u
2
− 11u − 4)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
(y − 1)(y
10
+ 7y
9
+ ··· − 18y + 1)(y
16
+ 11y
15
+ ··· + 20y
2
+ 1)
c
2
, c
5
16(4y − 1)(4y
10
− 25y
9
+ ··· − y + 1)
· (y
16
− 9y
15
+ ··· − 13044y + 2209)
c
7
, c
9
(y − 1)(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
2
· (y
10
− 8y
9
+ ··· − 65y + 16)
c
8
y(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
2
· (y
10
− 3y
9
+ ··· − 244y + 64)
16
