9
37
(K9a
18
)
A knot diagram
1
Linearized knot diagam
6 8 1 9 2 5 3 7 4
Solving Sequence
2,8 3,6
1 5 7 9 4
c
2
c
1
c
5
c
7
c
8
c
4
c
3
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, −u
7
− 2u
3
− u
2
+ 2a + u − 1, u
8
− u
7
+ 2u
6
− 2u
5
+ 4u
4
− 3u
3
+ 2u
2
+ 1i
I
u
2
= h−u
3
+ b − 1, u
5
− u
3
+ 2u
2
+ 2a + u − 1, u
6
+ u
4
+ 2u
3
+ u
2
+ u + 2i
I
u
3
= h−u
3
+ b − u + 1, −u
2
+ a + u − 1, u
4
− u
3
+ 2u
2
− 2u + 1i
I
u
4
= hb − a − u − 1, a
2
+ 3au + 2a − 1, u
2
+ u + 1i
I
u
5
= hb − u, a − u − 2, u
2
+ u + 1i
I
u
6
= hb + u, a + 2u − 1, u
2
+ 1i
* 6 irreducible components of dim
C
= 0, with total 26 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
= hb−u, −u
7
−2u
3
−u
2
+2a+u−1, u
8
−u
7
+2u
6
−2u
5
+4u
4
−3u
3
+2u
2
+1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
−u
2
a
6
=
1
2
u
7
+ u
3
+
1
2
u
2
−
1
2
u +
1
2
u
a
1
=
1
2
u
7
− u
6
+ ··· +
1
2
u +
1
2
u
2
a
5
=
1
2
u
7
+ u
3
+
1
2
u
2
−
3
2
u +
1
2
u
a
7
=
−u
u
3
+ u
a
9
=
−u
3
u
5
+ u
3
+ u
a
4
=
u
2
− u + 1
1
2
u
7
+ u
5
+ ··· +
1
2
u −
1
2
a
4
=
u
2
− u + 1
1
2
u
7
+ u
5
+ ··· +
1
2
u −
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
6
+ 2u
5
− 4u
4
+ 6u
3
− 12u
2
+ 6u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
u
8
+ u
7
+ 2u
6
+ 2u
5
+ 4u
4
+ 3u
3
+ 2u
2
+ 1
c
3
, c
4
, c
9
u
8
− 2u
7
+ 6u
6
− 8u
5
+ 10u
4
− 9u
3
+ 5u
2
− 3u + 2
c
6
, c
8
u
8
+ 3u
7
+ 8u
6
+ 10u
5
+ 14u
4
+ 11u
3
+ 12u
2
+ 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
y
8
+ 3y
7
+ 8y
6
+ 10y
5
+ 14y
4
+ 11y
3
+ 12y
2
+ 4y + 1
c
3
, c
4
, c
9
y
8
+ 8y
7
+ 24y
6
+ 30y
5
+ 8y
4
− 5y
3
+ 11y
2
+ 11y + 4
c
6
, c
8
y
8
+ 7y
7
+ 32y
6
+ 82y
5
+ 146y
4
+ 151y
3
+ 84y
2
+ 8y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.862697 + 0.615401I
a = −0.361509 + 0.665983I
b = 0.862697 + 0.615401I
7.44069 + 0.66722I 4.81639 − 2.10627I
u = 0.862697 − 0.615401I
a = −0.361509 − 0.665983I
b = 0.862697 − 0.615401I
7.44069 − 0.66722I 4.81639 + 2.10627I
u = 0.578102 + 1.055330I
a = −1.26281 + 1.67027I
b = 0.578102 + 1.055330I
−1.73404 + 6.79402I −3.11839 − 7.09473I
u = 0.578102 − 1.055330I
a = −1.26281 − 1.67027I
b = 0.578102 − 1.055330I
−1.73404 − 6.79402I −3.11839 + 7.09473I
u = −0.666851 + 1.155530I
a = 0.88635 + 1.91065I
b = −0.666851 + 1.155530I
3.94193 − 10.98940I 0.47099 + 7.14773I
u = −0.666851 − 1.155530I
a = 0.88635 − 1.91065I
b = −0.666851 − 1.155530I
3.94193 + 10.98940I 0.47099 − 7.14773I
u = −0.273948 + 0.520074I
a = 0.737965 − 0.414347I
b = −0.273948 + 0.520074I
0.221012 − 1.276800I 1.83102 + 5.88514I
u = −0.273948 − 0.520074I
a = 0.737965 + 0.414347I
b = −0.273948 − 0.520074I
0.221012 + 1.276800I 1.83102 − 5.88514I
5
II. I
u
2
= h−u
3
+ b − 1, u
5
− u
3
+ 2u
2
+ 2a + u − 1, u
6
+ u
4
+ 2u
3
+ u
2
+ u + 2i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
−u
2
a
6
=
−
1
2
u
5
+
1
2
u
3
− u
2
−
1
2
u +
1
2
u
3
+ 1
a
1
=
−
1
2
u
5
− u
4
+ ··· −
3
2
u −
1
2
−u
4
− u
2
− u − 1
a
5
=
−
1
2
u
5
−
1
2
u
3
− u
2
−
1
2
u −
1
2
u
3
+ 1
a
7
=
−u
u
3
+ u
a
9
=
−u
3
u
5
+ u
3
+ u
a
4
=
−
1
2
u
5
+
1
2
u
3
+
1
2
u +
3
2
−u
5
− u
3
− 2u
2
− 1
a
4
=
−
1
2
u
5
+
1
2
u
3
+
1
2
u +
3
2
−u
5
− u
3
− 2u
2
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
4
− 4u
3
− 8u − 6
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
u
6
+ u
4
− 2u
3
+ u
2
− u + 2
c
3
, c
4
, c
9
(u
3
+ 2u − 1)
2
c
6
, c
8
u
6
+ 2u
5
+ 3u
4
+ 2u
3
+ u
2
+ 3u + 4
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
y
6
+ 2y
5
+ 3y
4
+ 2y
3
+ y
2
+ 3y + 4
c
3
, c
4
, c
9
(y
3
+ 4y
2
+ 4y −1)
2
c
6
, c
8
y
6
+ 2y
5
+ 3y
4
− 2y
3
+ 13y
2
− y + 16
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.931903 + 0.428993I
a = −0.180233 + 0.631115I
b = 0.705204 + 1.038720I
6.15087 + 5.13794I 3.31793 − 3.20902I
u = −0.931903 − 0.428993I
a = −0.180233 − 0.631115I
b = 0.705204 − 1.038720I
6.15087 − 5.13794I 3.31793 + 3.20902I
u = 0.226699 + 1.074330I
a = 0.41474 − 1.96546I
b = 0.226699 − 1.074330I
−4.07707 −8.63587 + 0.I
u = 0.226699 − 1.074330I
a = 0.41474 + 1.96546I
b = 0.226699 + 1.074330I
−4.07707 −8.63587 + 0.I
u = 0.705204 + 1.038720I
a = −0.484509 − 0.229988I
b = −0.931903 + 0.428993I
6.15087 + 5.13794I 3.31793 − 3.20902I
u = 0.705204 − 1.038720I
a = −0.484509 + 0.229988I
b = −0.931903 − 0.428993I
6.15087 − 5.13794I 3.31793 + 3.20902I
9
III. I
u
3
= h−u
3
+ b − u + 1, −u
2
+ a + u − 1, u
4
− u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
−u
2
a
6
=
u
2
− u + 1
u
3
+ u − 1
a
1
=
u
−u
3
− u
a
5
=
−u
3
+ u
2
− 2u + 2
u
3
+ u − 1
a
7
=
−u
u
3
+ u
a
9
=
−u
3
2u − 1
a
4
=
−u
2
+ 1
u
3
− 2u
2
+ 2u − 1
a
4
=
−u
2
+ 1
u
3
− 2u
2
+ 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
+ 4u − 2
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
2
− u + 1)
2
c
2
, c
3
, c
4
c
7
, c
9
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
6
(u
2
+ u + 1)
2
c
8
u
4
+ 3u
3
+ 2u
2
+ 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
(y
2
+ y + 1)
2
c
2
, c
3
, c
4
c
7
, c
9
y
4
+ 3y
3
+ 2y
2
+ 1
c
8
y
4
− 5y
3
+ 6y
2
+ 4y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.621744 + 0.440597I
a = 0.570696 + 0.107280I
b = −0.500000 + 0.866025I
− 2.02988I 0. + 3.46410I
u = 0.621744 − 0.440597I
a = 0.570696 − 0.107280I
b = −0.500000 − 0.866025I
2.02988I 0. − 3.46410I
u = −0.121744 + 1.306620I
a = −0.57070 − 1.62477I
b = −0.500000 − 0.866025I
2.02988I 0. − 3.46410I
u = −0.121744 − 1.306620I
a = −0.57070 + 1.62477I
b = −0.500000 + 0.866025I
− 2.02988I 0. + 3.46410I
13
IV. I
u
4
= hb − a − u − 1, a
2
+ 3au + 2a − 1, u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u + 1
a
6
=
a
a + u + 1
a
1
=
−2au − a + 2
−au + u + 1
a
5
=
−u − 1
a + u + 1
a
7
=
−u
u + 1
a
9
=
−1
0
a
4
=
a
a + u + 1
a
4
=
a
a + u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u + 2
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
9
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
2
, c
7
(u
2
− u + 1)
2
c
6
u
4
+ 3u
3
+ 2u
2
+ 1
c
8
(u
2
+ u + 1)
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
9
y
4
+ 3y
3
+ 2y
2
+ 1
c
2
, c
7
, c
8
(y
2
+ y + 1)
2
c
6
y
4
− 5y
3
+ 6y
2
+ 4y + 1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.121744 − 0.425428I
b = 0.621744 + 0.440597I
− 2.02988I 0. + 3.46410I
u = −0.500000 + 0.866025I
a = −0.62174 − 2.17265I
b = −0.121744 − 1.306620I
− 2.02988I 0. + 3.46410I
u = −0.500000 − 0.866025I
a = 0.121744 + 0.425428I
b = 0.621744 − 0.440597I
2.02988I 0. − 3.46410I
u = −0.500000 − 0.866025I
a = −0.62174 + 2.17265I
b = −0.121744 + 1.306620I
2.02988I 0. − 3.46410I
17
V. I
u
5
= hb − u, a − u − 2, u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
u + 1
a
6
=
u + 2
u
a
1
=
u
−u − 1
a
5
=
2
u
a
7
=
−u
u + 1
a
9
=
−1
0
a
4
=
u + 2
u
a
4
=
u + 2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u + 2
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
9
u
2
− u + 1
c
6
, c
8
u
2
+ u + 1
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
y
2
+ y + 1
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 1.50000 + 0.86603I
b = −0.500000 + 0.866025I
− 2.02988I 0. + 3.46410I
u = −0.500000 − 0.866025I
a = 1.50000 − 0.86603I
b = −0.500000 − 0.866025I
2.02988I 0. − 3.46410I
21
VI. I
u
6
= hb + u, a + 2u − 1, u
2
+ 1i
(i) Arc colorings
a
2
=
1
0
a
8
=
0
u
a
3
=
1
1
a
6
=
−2u + 1
−u
a
1
=
−u − 1
−1
a
5
=
−u + 1
−u
a
7
=
−u
0
a
9
=
u
u
a
4
=
−u + 2
−u + 1
a
4
=
−u + 2
−u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
22
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
9
u
2
+ 1
c
6
, c
8
(u + 1)
2
23
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
9
(y + 1)
2
c
6
, c
8
(y − 1)
2
24
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 1.00000 − 2.00000I
b = − 1.000000I
−1.64493 −4.00000
u = − 1.000000I
a = 1.00000 + 2.00000I
b = 1.000000I
−1.64493 −4.00000
25
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
7
(u
2
+ 1)(u
2
− u + 1)
3
(u
4
+ u
3
+ ··· + 2u + 1)(u
6
+ u
4
+ ··· − u + 2)
· (u
8
+ u
7
+ 2u
6
+ 2u
5
+ 4u
4
+ 3u
3
+ 2u
2
+ 1)
c
3
, c
4
, c
9
(u
2
+ 1)(u
2
− u + 1)(u
3
+ 2u − 1)
2
(u
4
+ u
3
+ 2u
2
+ 2u + 1)
2
· (u
8
− 2u
7
+ 6u
6
− 8u
5
+ 10u
4
− 9u
3
+ 5u
2
− 3u + 2)
c
6
, c
8
(u + 1)
2
(u
2
+ u + 1)
3
(u
4
+ 3u
3
+ 2u
2
+ 1)
· (u
6
+ 2u
5
+ 3u
4
+ 2u
3
+ u
2
+ 3u + 4)
· (u
8
+ 3u
7
+ 8u
6
+ 10u
5
+ 14u
4
+ 11u
3
+ 12u
2
+ 4u + 1)
26
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
7
(y + 1)
2
(y
2
+ y + 1)
3
(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
6
+ 2y
5
+ 3y
4
+ 2y
3
+ y
2
+ 3y + 4)
· (y
8
+ 3y
7
+ 8y
6
+ 10y
5
+ 14y
4
+ 11y
3
+ 12y
2
+ 4y + 1)
c
3
, c
4
, c
9
(y + 1)
2
(y
2
+ y + 1)(y
3
+ 4y
2
+ 4y − 1)
2
(y
4
+ 3y
3
+ 2y
2
+ 1)
2
· (y
8
+ 8y
7
+ 24y
6
+ 30y
5
+ 8y
4
− 5y
3
+ 11y
2
+ 11y + 4)
c
6
, c
8
(y − 1)
2
(y
2
+ y + 1)
3
(y
4
− 5y
3
+ 6y
2
+ 4y + 1)
· (y
6
+ 2y
5
+ 3y
4
− 2y
3
+ 13y
2
− y + 16)
· (y
8
+ 7y
7
+ 32y
6
+ 82y
5
+ 146y
4
+ 151y
3
+ 84y
2
+ 8y + 1)
27
