10
64
(K10a
122
)
A knot diagram
1
Linearized knot diagam
6 9 7 8 1 10 4 2 3 5
Solving Sequence
3,7
4 8
5,10
6 9 2 1
c
3
c
7
c
4
c
6
c
9
c
2
c
1
c
5
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, −u
10
− u
9
+ 6u
8
+ 5u
7
− 12u
6
− 6u
5
+ 7u
4
− 4u
3
+ u
2
+ 2a + 6u + 1,
u
12
+ u
11
− 7u
10
− 6u
9
+ 18u
8
+ 11u
7
− 19u
6
− 2u
5
+ 6u
4
− 8u
3
+ 1i
I
u
2
= h−79u
15
− 74u
14
+ ··· + 47b − 143, 126u
15
+ 121u
14
+ ··· + 47a + 425, u
16
+ u
15
+ ··· + 6u − 1i
I
u
3
= hb + 1, a, u − 1i
I
u
4
= hb − 1, a
2
− 2, u + 1i
* 4 irreducible components of dim
C
= 0, with total 31 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
10
− u
9
+ · · · + 2a + 1, u
12
+ u
11
+ · · · − 8u
3
+ 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
8
=
−u
−u
3
+ u
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
1
2
u
10
+
1
2
u
9
+ ··· − 3u −
1
2
u
a
6
=
1
2
u
11
−
7
2
u
9
+ ··· −
1
2
u −
1
2
−
1
2
u
10
−
1
2
u
9
+ ··· + u +
1
2
a
9
=
1
2
u
10
+
1
2
u
9
+ ··· − 2u −
1
2
u
a
2
=
−
1
2
u
11
−
1
2
u
10
+ ··· +
1
2
u + 1
−u
2
a
1
=
1
2
u
10
+
1
2
u
9
+ ··· − 2u −
1
2
−
1
2
u
10
+
1
2
u
9
+ ··· + u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
11
− u
10
+ 15u
9
+ 4u
8
− 41u
7
+ 2u
6
+ 46u
5
− 25u
4
− 14u
3
+ 23u
2
− 4u − 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
10
u
12
+ 3u
11
+ ··· + 2u − 2
c
2
, c
3
, c
4
c
7
, c
8
, c
9
u
12
+ u
11
− 7u
10
− 6u
9
+ 18u
8
+ 11u
7
− 19u
6
− 2u
5
+ 6u
4
− 8u
3
+ 1
c
6
u
12
− 9u
11
+ ··· + 102u − 22
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
10
y
12
− 11y
11
+ ··· + 20y + 4
c
2
, c
3
, c
4
c
7
, c
8
, c
9
y
12
− 15y
11
+ ··· + 12y
2
+ 1
c
6
y
12
+ y
11
+ ··· + 1300y + 484
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.298602 + 0.646764I
a = −0.45214 − 1.66459I
b = 0.298602 + 0.646764I
4.65271 − 3.28049I 2.99435 + 5.25300I
u = 0.298602 − 0.646764I
a = −0.45214 + 1.66459I
b = 0.298602 − 0.646764I
4.65271 + 3.28049I 2.99435 − 5.25300I
u = 1.37505
a = 1.71226
b = 1.37505
−1.04846 −6.10990
u = 0.527999
a = −1.99219
b = 0.527999
3.24831 0.826740
u = −1.50349 + 0.33368I
a = −0.268985 − 1.300570I
b = −1.50349 + 0.33368I
−7.04968 + 10.86810I −5.35737 − 5.74032I
u = −1.50349 − 0.33368I
a = −0.268985 + 1.300570I
b = −1.50349 − 0.33368I
−7.04968 − 10.86810I −5.35737 + 5.74032I
u = −1.54202 + 0.13644I
a = −0.585241 − 0.594215I
b = −1.54202 + 0.13644I
−10.10900 + 1.20346I −7.47592 + 0.43067I
u = −1.54202 − 0.13644I
a = −0.585241 + 0.594215I
b = −1.54202 − 0.13644I
−10.10900 − 1.20346I −7.47592 − 0.43067I
u = −0.245576 + 0.368193I
a = 0.577777 − 1.108910I
b = −0.245576 + 0.368193I
−0.111574 + 0.933771I −2.28396 − 7.38290I
u = −0.245576 − 0.368193I
a = 0.577777 + 1.108910I
b = −0.245576 − 0.368193I
−0.111574 − 0.933771I −2.28396 + 7.38290I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.54096 + 0.25161I
a = 0.368549 − 0.997077I
b = 1.54096 + 0.25161I
−12.33390 − 6.28413I −9.23554 + 3.97965I
u = 1.54096 − 0.25161I
a = 0.368549 + 0.997077I
b = 1.54096 − 0.25161I
−12.33390 + 6.28413I −9.23554 − 3.97965I
6
II. I
u
2
= h−79u
15
− 74u
14
+ · · · + 47b − 143, 126u
15
+ 121u
14
+ · · · + 47a +
425, u
16
+ u
15
+ · · · + 6u − 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
4
=
1
u
2
a
8
=
−u
−u
3
+ u
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
−2.68085u
15
− 2.57447u
14
+ ··· + 14.1277u − 9.04255
1.68085u
15
+ 1.57447u
14
+ ··· − 8.12766u + 3.04255
a
6
=
−3.06383u
15
− 4.08511u
14
+ ··· + 23.5745u − 12.1915
0.382979u
15
+ 1.51064u
14
+ ··· − 8.44681u + 3.14894
a
9
=
−u
15
− u
14
+ ··· + 6u − 6
1.68085u
15
+ 1.57447u
14
+ ··· − 8.12766u + 3.04255
a
2
=
3.04255u
15
+ 4.72340u
14
+ ··· − 26.3830u + 11.1277
−0.106383u
15
+ 1.19149u
14
+ ··· − 7.04255u + 0.680851
a
1
=
−1.14894u
15
− 1.53191u
14
+ ··· + 10.3404u − 6.44681
2.04255u
15
+ 2.72340u
14
+ ··· − 12.3830u + 4.12766
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
4
47
u
15
+
120
47
u
14
+
64
47
u
13
−
652
47
u
12
+
36
47
u
11
+ 28u
10
−
856
47
u
9
−
1120
47
u
8
+
1372
47
u
7
+
364
47
u
6
−
708
47
u
5
−
456
47
u
4
+
928
47
u
3
+
412
47
u
2
−
904
47
u +
270
47
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
10
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)
2
c
2
, c
3
, c
4
c
7
, c
8
, c
9
u
16
+ u
15
+ ··· + 6u − 1
c
6
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
10
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
2
c
2
, c
3
, c
4
c
7
, c
8
, c
9
y
16
− 13y
15
+ ··· − 24y + 1
c
6
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.396638 + 0.883588I
a = 1.00561 + 1.17006I
b = −1.42845 − 0.22812I
−0.91019 − 6.44354I −2.57155 + 5.29417I
u = 0.396638 − 0.883588I
a = 1.00561 − 1.17006I
b = −1.42845 + 0.22812I
−0.91019 + 6.44354I −2.57155 − 5.29417I
u = 0.825972 + 0.646815I
a = 0.646365 + 0.503837I
b = −1.396840 + 0.083857I
−2.24921 + 1.13123I −4.58478 − 0.51079I
u = 0.825972 − 0.646815I
a = 0.646365 − 0.503837I
b = −1.396840 − 0.083857I
−2.24921 − 1.13123I −4.58478 + 0.51079I
u = −0.558144 + 0.766237I
a = −0.792286 + 0.953005I
b = 1.41338 − 0.10034I
−5.44928 + 2.57849I −7.72292 − 3.56796I
u = −0.558144 − 0.766237I
a = −0.792286 − 0.953005I
b = 1.41338 + 0.10034I
−5.44928 − 2.57849I −7.72292 + 3.56796I
u = 0.858124
a = −1.40539
b = 0.240055
3.21286 1.86400
u = −1.15431
a = 0.315320
b = 0.551002
−2.44483 −0.105540
u = −1.396840 + 0.083857I
a = −0.112641 − 0.603991I
b = 0.825972 + 0.646815I
−2.24921 + 1.13123I −4.58478 − 0.51079I
u = −1.396840 − 0.083857I
a = −0.112641 + 0.603991I
b = 0.825972 − 0.646815I
−2.24921 − 1.13123I −4.58478 + 0.51079I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41338 + 0.10034I
a = −0.145831 + 0.816217I
b = −0.558144 − 0.766237I
−5.44928 − 2.57849I −7.72292 + 3.56796I
u = 1.41338 − 0.10034I
a = −0.145831 − 0.816217I
b = −0.558144 + 0.766237I
−5.44928 + 2.57849I −7.72292 − 3.56796I
u = −1.42845 + 0.22812I
a = 0.286014 + 0.992605I
b = 0.396638 − 0.883588I
−0.91019 + 6.44354I −2.57155 − 5.29417I
u = −1.42845 − 0.22812I
a = 0.286014 − 0.992605I
b = 0.396638 + 0.883588I
−0.91019 − 6.44354I −2.57155 + 5.29417I
u = 0.551002
a = −0.660569
b = −1.15431
−2.44483 −0.105540
u = 0.240055
a = −5.02383
b = 0.858124
3.21286 1.86400
11
III. I
u
3
= hb + 1, a, u − 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
1
a
4
=
1
1
a
8
=
−1
0
a
5
=
0
1
a
10
=
0
−1
a
6
=
0
1
a
9
=
−1
−1
a
2
=
0
−1
a
1
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
10
u
c
2
, c
7
u + 1
c
3
, c
4
, c
8
c
9
u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
10
y
c
2
, c
3
, c
4
c
7
, c
8
, c
9
y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
15
IV. I
u
4
= hb − 1, a
2
− 2, u + 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
−1
a
4
=
1
1
a
8
=
1
0
a
5
=
0
1
a
10
=
a
1
a
6
=
2
a − 1
a
9
=
a + 1
1
a
2
=
−a
−1
a
1
=
a
−a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
10
u
2
− 2
c
2
, c
7
(u − 1)
2
c
3
, c
4
, c
8
c
9
(u + 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
10
(y −2)
2
c
2
, c
3
, c
4
c
7
, c
8
, c
9
(y −1)
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.41421
b = 1.00000
1.64493 −4.00000
u = −1.00000
a = −1.41421
b = 1.00000
1.64493 −4.00000
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
10
u(u
2
− 2)(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)
2
· (u
12
+ 3u
11
+ ··· + 2u − 2)
c
2
, c
7
(u − 1)
2
(u + 1)
· (u
12
+ u
11
− 7u
10
− 6u
9
+ 18u
8
+ 11u
7
− 19u
6
− 2u
5
+ 6u
4
− 8u
3
+ 1)
· (u
16
+ u
15
+ ··· + 6u − 1)
c
3
, c
4
, c
8
c
9
(u − 1)(u + 1)
2
· (u
12
+ u
11
− 7u
10
− 6u
9
+ 18u
8
+ 11u
7
− 19u
6
− 2u
5
+ 6u
4
− 8u
3
+ 1)
· (u
16
+ u
15
+ ··· + 6u − 1)
c
6
u(u
2
− 2)(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
2
· (u
12
− 9u
11
+ ··· + 102u − 22)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
10
y(y −2)
2
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
2
· (y
12
− 11y
11
+ ··· + 20y + 4)
c
2
, c
3
, c
4
c
7
, c
8
, c
9
((y −1)
3
)(y
12
− 15y
11
+ ··· + 12y
2
+ 1)(y
16
− 13y
15
+ ··· − 24y + 1)
c
6
y(y −2)
2
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
2
· (y
12
+ y
11
+ ··· + 1300y + 484)
21
