10
67
(K10a
37
)
A knot diagram
1
Linearized knot diagam
8 10 9 6 1 4 2 7 3 5
Solving Sequence
4,9
3 10
2,7
6 5 8 1
c
3
c
9
c
2
c
6
c
4
c
8
c
1
c
5
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h54165895u
25
− 37175946u
24
+ ··· + 40233748b − 246895131,
25054784u
25
− 23549273u
24
+ ··· + 10058437a − 172885695, u
26
− u
25
+ ··· − 10u + 1i
I
u
2
= h−u
3
+ b − u, a + u, u
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u
2
+ 2u + 1i
I
u
3
= hb
2
− b + 1, a + u, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 39 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
= h5.42 × 10
7
u
25
− 3.72 × 10
7
u
24
+ · · · + 4.02 × 10
7
b − 2.47 × 10
8
, 2.51 ×
10
7
u
25
−2.35×10
7
u
24
+· · · +1.01×10
7
a−1.73×10
8
, u
26
−u
25
+· · ·−10u +1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
10
=
−u
u
3
+ u
a
2
=
u
2
+ 1
−u
4
− 2u
2
a
7
=
−2.49092u
25
+ 2.34125u
24
+ ··· − 66.6876u + 17.1881
−1.34628u
25
+ 0.923999u
24
+ ··· − 34.6377u + 6.13652
a
6
=
−3.83720u
25
+ 3.26524u
24
+ ··· − 101.325u + 23.3246
−1.34628u
25
+ 0.923999u
24
+ ··· − 34.6377u + 6.13652
a
5
=
−4.25185u
25
+ 3.16846u
24
+ ··· − 103.985u + 15.7177
1.25534u
25
− 1.00540u
24
+ ··· + 37.9947u − 7.88999
a
8
=
−4.11984u
25
+ 3.49052u
24
+ ··· − 105.381u + 24.5259
−1.33333u
25
+ 0.849641u
24
+ ··· − 31.4642u + 5.50719
a
1
=
−6.70848u
25
+ 5.09250u
24
+ ··· − 173.700u + 30.5647
1.11302u
25
− 1.09292u
24
+ ··· + 24.4986u − 5.45317
(ii) Obstruction class = −1
(iii) Cusp Shapes =
46161492
10058437
u
25
−
42859051
10058437
u
24
+ ··· +
1130296717
10058437
u −
328648814
10058437
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
26
− u
25
+ ··· − 8u + 1
c
2
, c
3
, c
9
u
26
− u
25
+ ··· − 10u + 1
c
4
, c
6
u
26
+ 8u
25
+ ··· − 19u + 4
c
5
, c
10
u
26
+ 2u
25
+ ··· + 3u + 2
c
8
u
26
+ 9u
25
+ ··· − 10u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
26
+ 9y
25
+ ··· − 10y + 1
c
2
, c
3
, c
9
y
26
+ 29y
25
+ ··· − 42y + 1
c
4
, c
6
y
26
+ 20y
25
+ ··· − 289y + 16
c
5
, c
10
y
26
− 8y
25
+ ··· + 19y + 4
c
8
y
26
+ 21y
25
+ ··· + 102y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.142027 + 0.957282I
a = −0.0519905 + 0.0539815I
b = −0.331584 + 0.800882I
1.78211 + 2.09880I 0.37548 − 4.34730I
u = −0.142027 − 0.957282I
a = −0.0519905 − 0.0539815I
b = −0.331584 − 0.800882I
1.78211 − 2.09880I 0.37548 + 4.34730I
u = 0.867268 + 0.410491I
a = 1.15659 + 0.97828I
b = −0.39998 − 1.44546I
1.87436 − 8.06990I −5.21390 + 7.61920I
u = 0.867268 − 0.410491I
a = 1.15659 − 0.97828I
b = −0.39998 + 1.44546I
1.87436 + 8.06990I −5.21390 − 7.61920I
u = −0.805271 + 0.489072I
a = −1.057530 + 0.864294I
b = 0.050804 − 1.204580I
2.52884 + 2.28245I −3.60243 − 2.76883I
u = −0.805271 − 0.489072I
a = −1.057530 − 0.864294I
b = 0.050804 + 1.204580I
2.52884 − 2.28245I −3.60243 + 2.76883I
u = 0.005357 + 1.342280I
a = 0.598391 − 0.082953I
b = −1.020530 + 0.849992I
2.37838 + 1.33649I −3.37936 − 0.64092I
u = 0.005357 − 1.342280I
a = 0.598391 + 0.082953I
b = −1.020530 − 0.849992I
2.37838 − 1.33649I −3.37936 + 0.64092I
u = 0.620125 + 0.190982I
a = 1.70404 + 0.76762I
b = −0.941748 − 0.311004I
−3.68014 − 3.21386I −12.77386 + 5.40899I
u = 0.620125 − 0.190982I
a = 1.70404 − 0.76762I
b = −0.941748 + 0.311004I
−3.68014 + 3.21386I −12.77386 − 5.40899I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.203679 + 1.390330I
a = 0.917399 − 0.307104I
b = −1.237320 − 0.529870I
1.38330 − 6.14693I −5.92538 + 6.07633I
u = 0.203679 − 1.390330I
a = 0.917399 + 0.307104I
b = −1.237320 + 0.529870I
1.38330 + 6.14693I −5.92538 − 6.07633I
u = −0.09127 + 1.45225I
a = −0.858293 − 0.088928I
b = 0.632773 + 0.158193I
5.55732 + 2.75521I 0.17738 − 3.03141I
u = −0.09127 − 1.45225I
a = −0.858293 + 0.088928I
b = 0.632773 − 0.158193I
5.55732 − 2.75521I 0.17738 + 3.03141I
u = −0.345805 + 0.407161I
a = −1.277870 − 0.234930I
b = 0.165885 + 0.110267I
−0.439201 + 1.278560I −4.70955 − 5.15889I
u = −0.345805 − 0.407161I
a = −1.277870 + 0.234930I
b = 0.165885 − 0.110267I
−0.439201 − 1.278560I −4.70955 + 5.15889I
u = 0.32724 + 1.50601I
a = 1.191910 − 0.296128I
b = −0.49080 − 1.60111I
8.0695 − 12.4216I −2.63275 + 7.54670I
u = 0.32724 − 1.50601I
a = 1.191910 + 0.296128I
b = −0.49080 + 1.60111I
8.0695 + 12.4216I −2.63275 − 7.54670I
u = −0.28252 + 1.52523I
a = −1.160510 − 0.229643I
b = 0.300935 − 1.274240I
9.09667 + 6.25190I −1.00234 − 2.90360I
u = −0.28252 − 1.52523I
a = −1.160510 + 0.229643I
b = 0.300935 + 1.274240I
9.09667 − 6.25190I −1.00234 + 2.90360I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.21524 + 1.54201I
a = 0.711239 + 0.409154I
b = −0.33688 + 1.58737I
10.14430 + 6.11174I 0. − 3.37506I
u = −0.21524 − 1.54201I
a = 0.711239 − 0.409154I
b = −0.33688 − 1.58737I
10.14430 − 6.11174I 0. + 3.37506I
u = 0.15473 + 1.55892I
a = −0.778021 + 0.341219I
b = 0.146371 + 1.395820I
10.76070 + 0.12219I 0.65472 − 1.66625I
u = 0.15473 − 1.55892I
a = −0.778021 − 0.341219I
b = 0.146371 − 1.395820I
10.76070 − 0.12219I 0.65472 + 1.66625I
u = 0.203738 + 0.030867I
a = 4.90464 − 1.72660I
b = −0.537925 − 0.970190I
−1.75304 − 2.04961I −11.77790 + 2.96215I
u = 0.203738 − 0.030867I
a = 4.90464 + 1.72660I
b = −0.537925 + 0.970190I
−1.75304 + 2.04961I −11.77790 − 2.96215I
7
II. I
u
2
= h−u
3
+ b − u, a + u, u
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u
2
+ 2u + 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
10
=
−u
u
3
+ u
a
2
=
u
2
+ 1
−u
4
− 2u
2
a
7
=
−u
u
3
+ u
a
6
=
u
3
u
3
+ u
a
5
=
u
6
+ u
4
+ 1
u
6
+ 2u
4
+ u
2
a
8
=
u
3
−u
5
− u
3
+ u
a
1
=
u
4
+ u
2
+ 1
−u
6
− 2u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
6
− 8u
4
− 4u
3
− 4u
2
− 4u − 10
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
9
u
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u
2
+ 2u + 1
c
4
, c
6
(u
3
+ u
2
+ 2u + 1)
3
c
5
, c
10
(u
3
− u
2
+ 1)
3
c
8
u
9
+ 6u
8
+ 15u
7
+ 23u
6
+ 27u
5
+ 24u
4
+ 15u
3
+ 7u
2
+ 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
9
y
9
+ 6y
8
+ 15y
7
+ 23y
6
+ 27y
5
+ 24y
4
+ 15y
3
+ 7y
2
+ 2y −1
c
4
, c
6
(y
3
+ 3y
2
+ 2y −1)
3
c
5
, c
10
(y
3
− y
2
+ 2y −1)
3
c
8
y
9
− 6y
8
+ 3y
7
+ 23y
6
− 5y
5
− 16y
4
+ 43y
3
+ 59y
2
+ 18y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.656619 + 0.765660I
a = −0.656619 − 0.765660I
b = −0.215080 + 1.307140I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = 0.656619 − 0.765660I
a = −0.656619 + 0.765660I
b = −0.215080 − 1.307140I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = −0.701160 + 0.628458I
a = 0.701160 − 0.628458I
b = −0.215080 + 1.307140I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = −0.701160 − 0.628458I
a = 0.701160 + 0.628458I
b = −0.215080 − 1.307140I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = 0.233800 + 1.078880I
a = −0.233800 − 1.078880I
b = −0.569840
−1.11345 −9.01951 + 0.I
u = 0.233800 − 1.078880I
a = −0.233800 + 1.078880I
b = −0.569840
−1.11345 −9.01951 + 0.I
u = 0.044542 + 1.394120I
a = −0.044542 − 1.394120I
b = −0.215080 − 1.307140I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = 0.044542 − 1.394120I
a = −0.044542 + 1.394120I
b = −0.215080 + 1.307140I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = −0.467600
a = 0.467600
b = −0.569840
−1.11345 −9.01950
11
III. I
u
3
= hb
2
− b + 1, a + u, u
2
+ 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
3
=
1
1
a
10
=
−u
0
a
2
=
0
1
a
7
=
−u
b
a
6
=
b − u
b
a
5
=
−bu + b
b − 1
a
8
=
−u
b + u
a
1
=
1
bu
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b − 8
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
9
(u
2
+ 1)
2
c
4
(u
2
− u + 1)
2
c
5
, c
10
u
4
− u
2
+ 1
c
6
(u
2
+ u + 1)
2
c
8
(u + 1)
4
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
9
(y + 1)
4
c
4
, c
6
(y
2
+ y + 1)
2
c
5
, c
10
(y
2
− y + 1)
2
c
8
(y −1)
4
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = − 1.000000I
b = 0.500000 + 0.866025I
− 2.02988I −6.00000 + 3.46410I
u = 1.000000I
a = − 1.000000I
b = 0.500000 − 0.866025I
2.02988I −6.00000 − 3.46410I
u = − 1.000000I
a = 1.000000I
b = 0.500000 + 0.866025I
− 2.02988I −6.00000 + 3.46410I
u = − 1.000000I
a = 1.000000I
b = 0.500000 − 0.866025I
2.02988I −6.00000 − 3.46410I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
2
+ 1)
2
(u
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u
2
+ 2u + 1)
· (u
26
− u
25
+ ··· − 8u + 1)
c
2
, c
3
, c
9
(u
2
+ 1)
2
(u
9
+ 3u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ u
2
+ 2u + 1)
· (u
26
− u
25
+ ··· − 10u + 1)
c
4
((u
2
− u + 1)
2
)(u
3
+ u
2
+ 2u + 1)
3
(u
26
+ 8u
25
+ ··· − 19u + 4)
c
5
, c
10
((u
3
− u
2
+ 1)
3
)(u
4
− u
2
+ 1)(u
26
+ 2u
25
+ ··· + 3u + 2)
c
6
((u
2
+ u + 1)
2
)(u
3
+ u
2
+ 2u + 1)
3
(u
26
+ 8u
25
+ ··· − 19u + 4)
c
8
((u + 1)
4
)(u
9
+ 6u
8
+ ··· + 2u − 1)
· (u
26
+ 9u
25
+ ··· − 10u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
((y + 1)
4
)(y
9
+ 6y
8
+ ··· + 2y −1)
· (y
26
+ 9y
25
+ ··· − 10y + 1)
c
2
, c
3
, c
9
((y + 1)
4
)(y
9
+ 6y
8
+ ··· + 2y −1)
· (y
26
+ 29y
25
+ ··· − 42y + 1)
c
4
, c
6
((y
2
+ y + 1)
2
)(y
3
+ 3y
2
+ 2y −1)
3
(y
26
+ 20y
25
+ ··· − 289y + 16)
c
5
, c
10
((y
2
− y + 1)
2
)(y
3
− y
2
+ 2y −1)
3
(y
26
− 8y
25
+ ··· + 19y + 4)
c
8
((y −1)
4
)(y
9
− 6y
8
+ ··· + 18y −1)
· (y
26
+ 21y
25
+ ··· + 102y + 1)
17
