10
78
(K10a
17
)
A knot diagram
1
Linearized knot diagam
4 9 5 2 7 1 10 3 8 6
Solving Sequence
6,10 1,4
2 7 8 5 3 9
c
10
c
1
c
6
c
7
c
5
c
3
c
9
c
2
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
11
− u
10
+ 2u
9
+ 3u
8
− 2u
7
− 4u
6
− 2u
5
+ u
4
+ 2u
3
+ u
2
+ b − u − 1,
− u
11
− u
10
+ 2u
9
+ 3u
8
− 2u
7
− 4u
6
− 2u
5
+ u
4
+ u
3
+ u
2
+ a − u − 1,
u
13
+ u
12
− 3u
11
− 4u
10
+ 4u
9
+ 7u
8
− 5u
6
− 3u
5
+ 3u
3
+ 2u
2
− 1i
I
u
2
= hu
20
− 6u
18
+ ··· + b − 2u, 2u
21
+ u
20
+ ··· + a + 1, u
22
+ u
21
+ ··· − 4u
2
+ 1i
I
u
3
= hb + 1, a + 2, u + 1i
* 3 irreducible components of dim
C
= 0, with total 36 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
= h−u
11
−u
10
+· · ·+b−1, −u
11
−u
10
+· · ·+a−1, u
13
+u
12
+· · ·+2u
2
−1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
1
=
1
u
2
a
4
=
u
11
+ u
10
− 2u
9
− 3u
8
+ 2u
7
+ 4u
6
+ 2u
5
− u
4
− u
3
− u
2
+ u + 1
u
11
+ u
10
− 2u
9
− 3u
8
+ 2u
7
+ 4u
6
+ 2u
5
− u
4
− 2u
3
− u
2
+ u + 1
a
2
=
u
12
+ u
11
− 2u
10
− 3u
9
+ 2u
8
+ 4u
7
+ 2u
6
− u
5
− u
4
− u
3
+ u
2
+ u + 1
u
12
+ u
11
− 2u
10
− 3u
9
+ 2u
8
+ 4u
7
+ 2u
6
− u
5
− 2u
4
− u
3
+ 2u
2
+ u
a
7
=
−u
−u
3
+ u
a
8
=
−u
3
−u
3
+ u
a
5
=
u
3
u
5
− u
3
+ u
a
3
=
u
11
+ u
10
− 2u
9
− 3u
8
+ 2u
7
+ 4u
6
+ u
5
− u
4
− u
3
− u
2
+ u + 1
u
11
+ u
10
− 2u
9
− 3u
8
+ u
7
+ 4u
6
+ 3u
5
− u
4
− 3u
3
− u
2
+ u + 1
a
9
=
u
6
− u
4
+ 1
u
6
− 2u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
11
+ 2u
10
+ 8u
9
− 2u
8
− 16u
7
+ 12u
5
+ 10u
4
− 2u
3
− 2u
2
− 8u − 4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
10
u
13
− u
12
− 3u
11
+ 4u
10
+ 4u
9
− 7u
8
+ 5u
6
− 3u
5
+ 3u
3
− 2u
2
+ 1
c
2
, c
8
u
13
+ 3u
12
+ ··· + 4u + 2
c
3
, c
5
u
13
+ 7u
12
+ ··· + 4u + 1
c
7
, c
9
u
13
− 3u
12
+ ··· + 4u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
10
y
13
− 7y
12
+ ··· + 4y −1
c
2
, c
8
y
13
+ 3y
12
+ ··· + 4y −4
c
3
, c
5
y
13
+ y
12
+ ··· + 8y −1
c
7
, c
9
y
13
+ 11y
12
+ ··· + 104y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.915058 + 0.384331I
a = −0.86874 + 2.19716I
b = −1.22946 + 1.28849I
−2.16179 − 3.07776I −9.60750 + 5.91774I
u = 0.915058 − 0.384331I
a = −0.86874 − 2.19716I
b = −1.22946 − 1.28849I
−2.16179 + 3.07776I −9.60750 − 5.91774I
u = −0.992158 + 0.546170I
a = 1.43275 + 1.41238I
b = 1.52152 − 0.03761I
0.33005 + 7.56007I −5.81453 − 9.02411I
u = −0.992158 − 0.546170I
a = 1.43275 − 1.41238I
b = 1.52152 + 0.03761I
0.33005 − 7.56007I −5.81453 + 9.02411I
u = −0.613960 + 0.561299I
a = 0.334868 + 0.840411I
b = −0.013998 + 0.382511I
2.63797 + 1.38269I −0.35464 − 3.62793I
u = −0.613960 − 0.561299I
a = 0.334868 − 0.840411I
b = −0.013998 − 0.382511I
2.63797 − 1.38269I −0.35464 + 3.62793I
u = −0.089121 + 0.795435I
a = 0.065042 + 0.185799I
b = −0.103415 + 0.670130I
−1.44691 − 2.76421I −4.50885 + 2.57748I
u = −0.089121 − 0.795435I
a = 0.065042 − 0.185799I
b = −0.103415 − 0.670130I
−1.44691 + 2.76421I −4.50885 − 2.57748I
u = 1.216140 + 0.467752I
a = −2.46220 + 1.38514I
b = −3.46262 − 0.58793I
−8.78542 − 6.00980I −11.90142 + 4.07839I
u = 1.216140 − 0.467752I
a = −2.46220 − 1.38514I
b = −3.46262 + 0.58793I
−8.78542 + 6.00980I −11.90142 − 4.07839I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.231340 + 0.513532I
a = 2.38620 + 1.20321I
b = 3.27898 − 0.99721I
−8.1203 + 12.5021I −10.75701 − 8.36275I
u = −1.231340 − 0.513532I
a = 2.38620 − 1.20321I
b = 3.27898 + 0.99721I
−8.1203 − 12.5021I −10.75701 + 8.36275I
u = 0.590758
a = 1.22415
b = 1.01798
−1.09585 −8.11210
6
II.
I
u
2
= hu
20
−6u
18
+· · ·+b−2u, 2u
21
+u
20
+· · ·+a+1, u
22
+u
21
+· · ·−4u
2
+1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
1
=
1
u
2
a
4
=
−2u
21
− u
20
+ ··· + 4u − 1
−u
20
+ 6u
18
+ ··· + 2u
2
+ 2u
a
2
=
−2u
21
+ 13u
19
+ ··· + 4u − 1
u
19
− 5u
17
+ ··· + 2u
2
+ u
a
7
=
−u
−u
3
+ u
a
8
=
−u
3
−u
3
+ u
a
5
=
u
3
u
5
− u
3
+ u
a
3
=
−2u
21
+ 12u
19
+ ··· + 3u − 1
−u
16
+ 4u
14
+ ··· + 2u
2
+ u
a
9
=
u
6
− u
4
+ 1
u
6
− 2u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
21
+ 24u
19
+ 4u
18
−64u
17
−20u
16
+ 80u
15
+ 44u
14
−20u
13
−
40u
12
− 72u
11
− 4u
10
+ 76u
9
+ 40u
8
− 8u
7
− 20u
6
− 28u
5
− 4u
4
+ 8u
3
+ 8u
2
− 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
10
u
22
− u
21
+ ··· − 4u
2
+ 1
c
2
, c
8
(u
11
− u
10
+ 2u
9
− u
8
+ 4u
7
− 2u
6
+ 4u
5
− u
4
+ 3u
3
+ u
2
+ 1)
2
c
3
, c
5
u
22
+ 13u
21
+ ··· + 8u + 1
c
7
, c
9
(u
11
− 3u
10
+ ··· − 2u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
10
y
22
− 13y
21
+ ··· − 8y + 1
c
2
, c
8
(y
11
+ 3y
10
+ ··· − 2y −1)
2
c
3
, c
5
y
22
− 9y
21
+ ··· − 32y + 1
c
7
, c
9
(y
11
+ 11y
10
+ ··· + 6y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.878994 + 0.515981I
a = −0.407883 + 0.148860I
b = −0.509746 − 0.200169I
1.89175 + 2.94672I −2.20063 − 4.11787I
u = −0.878994 − 0.515981I
a = −0.407883 − 0.148860I
b = −0.509746 + 0.200169I
1.89175 − 2.94672I −2.20063 + 4.11787I
u = −0.894378 + 0.268842I
a = −2.19177 − 0.42458I
b = −0.632662 + 0.861406I
−2.98514 + 1.13130I −7.98780 − 6.05785I
u = −0.894378 − 0.268842I
a = −2.19177 + 0.42458I
b = −0.632662 − 0.861406I
−2.98514 − 1.13130I −7.98780 + 6.05785I
u = −0.101435 + 0.877274I
a = 1.073150 − 0.632994I
b = −2.13072 − 0.20221I
−4.72165 − 7.47524I −7.77092 + 5.55460I
u = −0.101435 − 0.877274I
a = 1.073150 + 0.632994I
b = −2.13072 + 0.20221I
−4.72165 + 7.47524I −7.77092 − 5.55460I
u = 1.166330 + 0.116345I
a = 1.74332 − 0.35353I
b = 1.54944 + 0.26584I
−2.98514 + 1.13130I −7.98780 − 6.05785I
u = 1.166330 − 0.116345I
a = 1.74332 + 0.35353I
b = 1.54944 − 0.26584I
−2.98514 − 1.13130I −7.98780 + 6.05785I
u = 0.022883 + 0.808487I
a = −1.17422 − 0.82028I
b = 2.00647 + 0.07669I
−5.26692 + 1.41699I −8.79131 − 0.63373I
u = 0.022883 − 0.808487I
a = −1.17422 + 0.82028I
b = 2.00647 − 0.07669I
−5.26692 − 1.41699I −8.79131 + 0.63373I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.438226 + 0.645537I
a = 0.159128 − 0.544432I
b = −1.222780 − 0.483692I
1.89175 − 2.94672I −2.20063 + 4.11787I
u = −0.438226 − 0.645537I
a = 0.159128 + 0.544432I
b = −1.222780 + 0.483692I
1.89175 + 2.94672I −2.20063 − 4.11787I
u = 1.209200 + 0.415611I
a = 0.716733 + 0.554276I
b = 0.389956 + 0.626620I
−5.26692 − 1.41699I −8.79131 + 0.63373I
u = 1.209200 − 0.415611I
a = 0.716733 − 0.554276I
b = 0.389956 − 0.626620I
−5.26692 + 1.41699I −8.79131 − 0.63373I
u = −1.218830 + 0.447288I
a = −1.98611 − 0.66727I
b = −2.01763 + 1.70968I
−8.93247 + 3.04152I −12.06121 − 2.82242I
u = −1.218830 − 0.447288I
a = −1.98611 + 0.66727I
b = −2.01763 − 1.70968I
−8.93247 − 3.04152I −12.06121 + 2.82242I
u = −1.203210 + 0.491862I
a = −0.610676 + 0.586169I
b = −0.143972 + 0.552324I
−4.72165 + 7.47524I −7.77092 − 5.55460I
u = −1.203210 − 0.491862I
a = −0.610676 − 0.586169I
b = −0.143972 − 0.552324I
−4.72165 − 7.47524I −7.77092 + 5.55460I
u = 0.687015
a = 0.995334
b = 0.930026
−1.09450 −8.37630
u = 1.263030 + 0.401917I
a = 1.93778 − 0.67607I
b = 2.24577 + 1.44537I
−8.93247 + 3.04152I −12.06121 − 2.82242I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.263030 − 0.401917I
a = 1.93778 + 0.67607I
b = 2.24577 − 1.44537I
−8.93247 − 3.04152I −12.06121 + 2.82242I
u = 0.460239
a = 1.48577
b = 1.00173
−1.09450 −8.37630
12
III. I
u
3
= hb + 1, a + 2, u + 1i
(i) Arc colorings
a
6
=
0
−1
a
10
=
1
0
a
1
=
1
1
a
4
=
−2
−1
a
2
=
−1
0
a
7
=
1
0
a
8
=
1
0
a
5
=
−1
−1
a
3
=
−1
0
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
6
u − 1
c
2
, c
7
, c
8
c
9
u
c
4
, c
10
u + 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
10
y −1
c
2
, c
7
, c
8
c
9
y
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −2.00000
b = −1.00000
−3.28987 −12.0000
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
(u − 1)(u
13
− u
12
+ ··· − 2u
2
+ 1)
· (u
22
− u
21
+ ··· − 4u
2
+ 1)
c
2
, c
8
u(u
11
− u
10
+ 2u
9
− u
8
+ 4u
7
− 2u
6
+ 4u
5
− u
4
+ 3u
3
+ u
2
+ 1)
2
· (u
13
+ 3u
12
+ ··· + 4u + 2)
c
3
, c
5
(u − 1)(u
13
+ 7u
12
+ ··· + 4u + 1)(u
22
+ 13u
21
+ ··· + 8u + 1)
c
4
, c
10
(u + 1)(u
13
− u
12
+ ··· − 2u
2
+ 1)
· (u
22
− u
21
+ ··· − 4u
2
+ 1)
c
7
, c
9
u(u
11
− 3u
10
+ ··· − 2u + 1)
2
(u
13
− 3u
12
+ ··· + 4u + 4)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
10
(y −1)(y
13
− 7y
12
+ ··· + 4y −1)(y
22
− 13y
21
+ ··· − 8y + 1)
c
2
, c
8
y(y
11
+ 3y
10
+ ··· − 2y −1)
2
(y
13
+ 3y
12
+ ··· + 4y −4)
c
3
, c
5
(y −1)(y
13
+ y
12
+ ··· + 8y −1)(y
22
− 9y
21
+ ··· − 32y + 1)
c
7
, c
9
y(y
11
+ 11y
10
+ ··· + 6y −1)
2
(y
13
+ 11y
12
+ ··· + 104y −16)
18
