9
16
(K9a
25
)
A knot diagram
1
Linearized knot diagam
5 6 8 3 2 9 1 4 7
Solving Sequence
1,5
2 6
3,7
8 4 9
c
1
c
5
c
2
c
7
c
4
c
9
c
3
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
+ a − u + 1, u
8
+ u
7
− 4u
6
− 3u
5
+ 5u
4
+ u
3
− u
2
+ 3u − 1i
I
u
2
= h−u
11
+ 4u
9
− u
8
− 5u
7
+ 3u
6
+ u
5
− 2u
4
+ u
3
+ b + u − 1,
− u
11
− u
10
+ 4u
9
+ 2u
8
− 7u
7
+ u
6
+ 5u
5
− 5u
4
+ u
3
+ 3u
2
+ a − 2u,
u
12
+ u
11
− 4u
10
− 2u
9
+ 7u
8
− u
7
− 5u
6
+ 5u
5
− u
4
− 3u
3
+ 2u
2
+ 1i
I
u
3
= hb + 1, a + 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 21 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
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
+ a − u + 1, u
8
+ u
7
− 4u
6
− 3u
5
+
5u
4
+ u
3
− u
2
+ 3u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
−u
−u
3
+ u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
7
=
−u
6
− u
5
+ 3u
4
+ 2u
3
− 2u
2
+ u − 1
u
a
8
=
−u
6
− u
5
+ 3u
4
+ 2u
3
− 2u
2
− 1
u
a
4
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
9
=
u
7
+ u
6
− 3u
5
− 2u
4
+ 2u
3
− u
2
+ u + 1
−u
2
a
9
=
u
7
+ u
6
− 3u
5
− 2u
4
+ 2u
3
− u
2
+ u + 1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
6
+ 2u
5
+ 10u
4
− 8u
3
− 12u
2
+ 10u − 16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
9
u
8
− u
7
− 4u
6
+ 3u
5
+ 5u
4
− u
3
− u
2
− 3u − 1
c
3
, c
8
u
8
− 3u
7
+ 3u
6
+ 2u
5
− 8u
4
+ 9u
3
− 3u
2
− 2u + 2
c
4
u
8
+ 3u
7
+ 5u
6
+ 4u
5
+ 2u
4
+ 13u
3
+ 13u
2
+ 16u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
9
y
8
− 9y
7
+ 32y
6
− 53y
5
+ 31y
4
+ 15y
3
− 15y
2
− 7y + 1
c
3
, c
8
y
8
− 3y
7
+ 5y
6
− 4y
5
+ 2y
4
− 13y
3
+ 13y
2
− 16y + 4
c
4
y
8
+ y
7
+ 5y
6
− 48y
5
− 58y
4
− 205y
3
− 231y
2
− 152y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.151337 + 0.673064I
a = −0.076017 − 0.952103I
b = 0.151337 + 0.673064I
1.48505 − 2.26376I −5.94128 + 4.53378I
u = 0.151337 − 0.673064I
a = −0.076017 + 0.952103I
b = 0.151337 − 0.673064I
1.48505 + 2.26376I −5.94128 − 4.53378I
u = 1.359440 + 0.207304I
a = 2.50827 − 1.24101I
b = 1.359440 + 0.207304I
−6.22518 − 3.55755I −14.5274 + 2.6249I
u = 1.359440 − 0.207304I
a = 2.50827 + 1.24101I
b = 1.359440 − 0.207304I
−6.22518 + 3.55755I −14.5274 − 2.6249I
u = −1.42757 + 0.33227I
a = −1.86256 − 1.18850I
b = −1.42757 + 0.33227I
−8.73978 + 9.88301I −15.2825 − 6.0696I
u = −1.42757 − 0.33227I
a = −1.86256 + 1.18850I
b = −1.42757 − 0.33227I
−8.73978 − 9.88301I −15.2825 + 6.0696I
u = −1.50912
a = −2.36273
b = −1.50912
−13.4445 −18.3370
u = 0.342714
a = −0.776649
b = 0.342714
−0.719034 −14.1600
5
II.
I
u
2
= h−u
11
+4u
9
+· · ·+b−1, −u
11
−u
10
+· · ·+a−2u, u
12
+u
11
+· · ·+2u
2
+1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
−u
−u
3
+ u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
7
=
u
11
+ u
10
− 4u
9
− 2u
8
+ 7u
7
− u
6
− 5u
5
+ 5u
4
− u
3
− 3u
2
+ 2u
u
11
− 4u
9
+ u
8
+ 5u
7
− 3u
6
− u
5
+ 2u
4
− u
3
− u + 1
a
8
=
u
10
− 3u
8
+ 2u
7
+ 2u
6
− 4u
5
+ 3u
4
− 3u
2
+ 3u − 1
u
11
− 4u
9
+ u
8
+ 5u
7
− 3u
6
− u
5
+ 2u
4
− u
3
− u + 1
a
4
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
9
=
−u
11
+ 4u
9
− 2u
8
− 6u
7
+ 6u
6
+ 2u
5
− 6u
4
+ 3u
3
+ 2u
2
− 2u
−u
11
+ 3u
9
− 2u
8
− 2u
7
+ 4u
6
− 3u
5
+ 3u
3
− 2u
2
+ u − 2
a
9
=
−u
11
+ 4u
9
− 2u
8
− 6u
7
+ 6u
6
+ 2u
5
− 6u
4
+ 3u
3
+ 2u
2
− 2u
−u
11
+ 3u
9
− 2u
8
− 2u
7
+ 4u
6
− 3u
5
+ 3u
3
− 2u
2
+ u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
8
+ 12u
6
− 4u
5
− 8u
4
+ 8u
3
− 4u
2
− 10
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
9
u
12
− u
11
− 4u
10
+ 2u
9
+ 7u
8
+ u
7
− 5u
6
− 5u
5
− u
4
+ 3u
3
+ 2u
2
+ 1
c
3
, c
8
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
c
4
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
9
y
12
− 9y
11
+ ··· + 4y + 1
c
3
, c
8
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
c
4
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.895235 + 0.524661I
a = −0.831450 + 0.487279I
b = −1.323480 + 0.139870I
−5.18047 + 0.92430I −15.7167 − 0.7942I
u = 0.895235 − 0.524661I
a = −0.831450 − 0.487279I
b = −1.323480 − 0.139870I
−5.18047 − 0.92430I −15.7167 + 0.7942I
u = 0.282166 + 0.828798I
a = −0.368111 + 1.081240I
b = −1.356120 − 0.270046I
−3.28987 − 5.69302I −12.00000 + 5.51057I
u = 0.282166 − 0.828798I
a = −0.368111 − 1.081240I
b = −1.356120 + 0.270046I
−3.28987 + 5.69302I −12.00000 − 5.51057I
u = 1.155020 + 0.191936I
a = −0.842520 + 0.140006I
b = −0.152828 − 0.487477I
−1.39926 − 0.92430I −8.28328 + 0.79423I
u = 1.155020 − 0.191936I
a = −0.842520 − 0.140006I
b = −0.152828 + 0.487477I
−1.39926 + 0.92430I −8.28328 − 0.79423I
u = −1.323480 + 0.139870I
a = 0.747239 + 0.078971I
b = 0.895235 + 0.524661I
−5.18047 + 0.92430I −15.7167 − 0.7942I
u = −1.323480 − 0.139870I
a = 0.747239 − 0.078971I
b = 0.895235 − 0.524661I
−5.18047 − 0.92430I −15.7167 + 0.7942I
u = −1.356120 + 0.270046I
a = 0.709275 + 0.141239I
b = 0.282166 − 0.828798I
−3.28987 + 5.69302I −12.00000 − 5.51057I
u = −1.356120 − 0.270046I
a = 0.709275 − 0.141239I
b = 0.282166 + 0.828798I
−3.28987 − 5.69302I −12.00000 + 5.51057I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.152828 + 0.487477I
a = 0.58557 + 1.86780I
b = 1.155020 − 0.191936I
−1.39926 + 0.92430I −8.28328 − 0.79423I
u = −0.152828 − 0.487477I
a = 0.58557 − 1.86780I
b = 1.155020 + 0.191936I
−1.39926 − 0.92430I −8.28328 + 0.79423I
10
III. I
u
3
= hb + 1, a + 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
6
=
−1
0
a
3
=
0
1
a
7
=
−1
−1
a
8
=
0
−1
a
4
=
0
1
a
9
=
0
−1
a
9
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
u − 1
c
3
, c
4
, c
8
u
c
5
, c
9
u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
9
y −1
c
3
, c
4
, c
8
y
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −1.00000
−3.28987 −12.0000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
(u − 1)(u
8
− u
7
− 4u
6
+ 3u
5
+ 5u
4
− u
3
− u
2
− 3u − 1)
· (u
12
− u
11
− 4u
10
+ 2u
9
+ 7u
8
+ u
7
− 5u
6
− 5u
5
− u
4
+ 3u
3
+ 2u
2
+ 1)
c
3
, c
8
u(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
· (u
8
− 3u
7
+ 3u
6
+ 2u
5
− 8u
4
+ 9u
3
− 3u
2
− 2u + 2)
c
4
u(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
2
· (u
8
+ 3u
7
+ 5u
6
+ 4u
5
+ 2u
4
+ 13u
3
+ 13u
2
+ 16u + 4)
c
5
, c
9
(u + 1)(u
8
− u
7
− 4u
6
+ 3u
5
+ 5u
4
− u
3
− u
2
− 3u − 1)
· (u
12
− u
11
− 4u
10
+ 2u
9
+ 7u
8
+ u
7
− 5u
6
− 5u
5
− u
4
+ 3u
3
+ 2u
2
+ 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
9
(y −1)(y
8
− 9y
7
+ 32y
6
− 53y
5
+ 31y
4
+ 15y
3
− 15y
2
− 7y + 1)
· (y
12
− 9y
11
+ ··· + 4y + 1)
c
3
, c
8
y(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
· (y
8
− 3y
7
+ 5y
6
− 4y
5
+ 2y
4
− 13y
3
+ 13y
2
− 16y + 4)
c
4
y(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
· (y
8
+ y
7
+ 5y
6
− 48y
5
− 58y
4
− 205y
3
− 231y
2
− 152y + 16)
16
