10
98
(K10a
96
)
A knot diagram
1
Linearized knot diagam
6 9 7 10 8 1 4 2 5 3
Solving Sequence
4,10 5,7
8 3 1 6 9 2
c
4
c
7
c
3
c
10
c
6
c
9
c
2
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= ⟨−4789953u
11
+ 15314376u
10
+ ··· + 9342488b + 9873021,
− 16446192u
11
+ 38491161u
10
+ ··· + 23356220a + 2924955,
3u
12
− 9u
11
+ 22u
10
− 42u
9
+ 70u
8
− 110u
7
+ 139u
6
− 157u
5
+ 149u
4
− 106u
3
+ 64u
2
− 20u + 5⟩
I
u
2
= ⟨−u
9
− u
8
− 3u
7
− 3u
6
− 5u
5
− 5u
4
− u
2
a − 4u
3
− 4u
2
+ b − a − 3u − 2, 6u
9
a − 3u
9
+ ··· + 6a − 7,
u
10
+ u
9
+ 3u
8
+ 3u
7
+ 5u
6
+ 5u
5
+ 4u
4
+ 4u
3
+ 3u
2
+ 2u + 1⟩
I
u
3
= ⟨b + u, 2a − u − 1, u
2
+ 1⟩
I
u
4
= ⟨b, a − 1, u
3
+ u − 1⟩
I
u
5
= ⟨b − 1, a − u − 1, u
3
+ u − 1⟩
I
u
6
= ⟨b − 1, u
3
a − u
3
+ au − 2u − 1⟩
* 5 irreducible components of dim
C
= 0, with total 40 representations.
* 1 irreducible components of dim
C
= 1
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
= ⟨−4.79 × 10
6
u
11
+ 1.53 × 10
7
u
10
+ · · · + 9.34 × 10
6
b + 9.87 × 10
6
, −1.64 ×
10
7
u
11
+3.85×10
7
u
10
+· · ·+2.34×10
7
a+2.92×10
6
, 3u
12
−9u
11
+· · ·−20u+5⟩
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
7
=
0.704146u
11
− 1.64800u
10
+ ··· + 5.67136u − 0.125232
0.512706u
11
− 1.63922u
10
+ ··· + 5.56042u − 1.05679
a
8
=
0.191440u
11
− 0.00878657u
10
+ ··· + 0.110936u + 0.931555
0.512706u
11
− 1.63922u
10
+ ··· + 5.56042u − 1.05679
a
3
=
0.0685395u
11
− 0.0742645u
10
+ ··· − 2.54876u + 1.65234
−0.524747u
11
+ 1.36714u
10
+ ··· − 1.44636u + 0.100143
a
1
=
−0.159552u
11
+ 0.582496u
10
+ ··· − 1.27370u − 0.346141
0.514632u
11
− 1.19725u
10
+ ··· + 3.19361u − 0.462234
a
6
=
0.173499u
11
− 0.281506u
10
+ ··· + 1.01679u + 1.07875
0.0278643u
11
− 0.396636u
10
+ ··· + 2.71393u − 0.664239
a
9
=
u
u
3
+ u
a
2
=
0.169639u
11
− 0.151768u
10
+ ··· − 4.91002u + 2.50685
−0.477984u
11
+ 1.17054u
10
+ ··· − 2.47082u + 0.578331
(ii) Obstruction class = −1
(iii) Cusp Shapes =
4958289
2335622
u
11
−
5676297
1167811
u
10
+ ··· +
80416785
2335622
u −
35331855
2335622
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
3(3u
12
+ 9u
11
+ ··· + 26u + 5)
c
2
, c
3
, c
7
c
8
u
12
− 2u
11
+ u
9
+ 3u
7
+ u
6
− 15u
5
+ 21u
4
− 14u
3
+ 3u
2
+ u + 2
c
4
, c
9
3(3u
12
+ 9u
11
+ ··· + 20u + 5)
c
5
, c
10
2(2u
12
− 4u
11
+ ··· + 12u + 3)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
9(9y
12
+ 87y
11
+ ··· − 16y + 25)
c
2
, c
3
, c
7
c
8
y
12
− 4y
11
+ ··· + 11y + 4
c
4
, c
9
9(9y
12
+ 51y
11
+ ··· + 240y + 25)
c
5
, c
10
4(4y
12
− 20y
11
+ ··· − 30y + 9)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.294163 + 0.893263I
a = −0.392558 + 0.428484I
b = 0.312276 − 1.374800I
0.93953 − 1.28267I −9.97159 + 5.49540I
u = 0.294163 − 0.893263I
a = −0.392558 − 0.428484I
b = 0.312276 + 1.374800I
0.93953 + 1.28267I −9.97159 − 5.49540I
u = −0.132310 + 1.149860I
a = −0.019852 + 0.332765I
b = −0.435810 − 0.817904I
3.72744 − 0.65506I 0.96539 + 2.36054I
u = −0.132310 − 1.149860I
a = −0.019852 − 0.332765I
b = −0.435810 + 0.817904I
3.72744 + 0.65506I 0.96539 − 2.36054I
u = 1.258330 + 0.213822I
a = −1.45041 + 0.16116I
b = −1.325110 + 0.371579I
−10.52620 + 7.73722I −13.2705 − 5.1580I
u = 1.258330 − 0.213822I
a = −1.45041 − 0.16116I
b = −1.325110 − 0.371579I
−10.52620 − 7.73722I −13.2705 + 5.1580I
u = −0.77981 + 1.24219I
a = −1.28771 − 0.68411I
b = −1.170890 + 0.448059I
−1.38664 + 8.65525I −8.05360 − 7.75821I
u = −0.77981 − 1.24219I
a = −1.28771 + 0.68411I
b = −1.170890 − 0.448059I
−1.38664 − 8.65525I −8.05360 + 7.75821I
u = 0.66776 + 1.32565I
a = 1.16439 − 0.95583I
b = 1.36378 + 0.57208I
−7.0244 − 14.4129I −10.01184 + 7.82077I
u = 0.66776 − 1.32565I
a = 1.16439 + 0.95583I
b = 1.36378 − 0.57208I
−7.0244 + 14.4129I −10.01184 − 7.82077I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.191864 + 0.381263I
a = 0.986139 + 0.917994I
b = 0.255755 + 0.338417I
−0.534161 − 1.008590I −7.65789 + 6.71362I
u = 0.191864 − 0.381263I
a = 0.986139 − 0.917994I
b = 0.255755 − 0.338417I
−0.534161 + 1.008590I −7.65789 − 6.71362I
6
II.
I
u
2
= ⟨−u
9
− u
8
+ · · · − a − 2, 6u
9
a − 3u
9
+ · · · + 6a − 7, u
10
+ u
9
+ · · · + 2u + 1⟩
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
7
=
a
u
9
+ u
8
+ 3u
7
+ 3u
6
+ 5u
5
+ 5u
4
+ u
2
a + 4u
3
+ 4u
2
+ a + 3u + 2
a
8
=
−u
9
− u
8
− 3u
7
− 3u
6
− 5u
5
− 5u
4
− u
2
a − 4u
3
− 4u
2
− 3u − 2
u
9
+ u
8
+ 3u
7
+ 3u
6
+ 5u
5
+ 5u
4
+ u
2
a + 4u
3
+ 4u
2
+ a + 3u + 2
a
3
=
u
9
a + u
9
+ ··· + 2a − u
−u
5
a + u
6
− 2u
3
a + 2u
4
− au + 2u
2
a
1
=
−u
9
a −
1
2
u
9
+ ··· − 2a +
5
2
−u
9
a − 2u
9
+ ··· − 2a − 1
a
6
=
−2u
9
a −
1
2
u
9
+ ··· − a +
1
2
u
9
+ u
8
+ ··· + 3u + 2
a
9
=
u
u
3
+ u
a
2
=
u
9
a + u
9
+ ··· + 2a − 1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
9
− 4u
8
− 8u
7
− 8u
6
− 8u
5
− 12u
4
− 4u
2
− 4u − 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
10
− u
9
+ 5u
8
− 5u
7
+ 9u
6
− 9u
5
+ 6u
4
− 6u
3
+ u
2
+ 1)
2
c
2
, c
3
, c
7
c
8
u
20
− 2u
19
+ ··· − 58u + 31
c
4
, c
9
(u
10
− u
9
+ 3u
8
− 3u
7
+ 5u
6
− 5u
5
+ 4u
4
− 4u
3
+ 3u
2
− 2u + 1)
2
c
5
, c
10
2(2u
20
− 13u
18
+ ··· − 518u + 121)
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
10
+ 9y
9
+ 33y
8
+ 59y
7
+ 41y
6
− 21y
5
− 44y
4
− 6y
3
+ 13y
2
+ 2y + 1)
2
c
2
, c
3
, c
7
c
8
y
20
− 14y
19
+ ··· − 388y + 961
c
4
, c
9
(y
10
+ 5y
9
+ 13y
8
+ 19y
7
+ 17y
6
+ 7y
5
− 2y
3
+ y
2
+ 2y + 1)
2
c
5
, c
10
4(4y
20
− 52y
19
+ ··· − 56574y + 14641)
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.584958 + 0.771492I
a = −0.759755 + 0.346084I
b = −1.50393 + 0.39581I
−8.22706 − 2.31006I −12.86369 + 3.52133I
u = 0.584958 + 0.771492I
a = 0.94152 − 1.29105I
b = 1.24454 + 0.70845I
−8.22706 − 2.31006I −12.86369 + 3.52133I
u = 0.584958 − 0.771492I
a = −0.759755 − 0.346084I
b = −1.50393 − 0.39581I
−8.22706 + 2.31006I −12.86369 − 3.52133I
u = 0.584958 − 0.771492I
a = 0.94152 + 1.29105I
b = 1.24454 − 0.70845I
−8.22706 + 2.31006I −12.86369 − 3.52133I
u = −0.248527 + 0.782547I
a = 2.10247 − 0.40028I
b = 1.157780 + 0.163121I
−2.84181 + 1.23169I −7.09823 − 5.44908I
u = −0.248527 + 0.782547I
a = −0.73260 − 2.58251I
b = −0.965077 + 0.285214I
−2.84181 + 1.23169I −7.09823 − 5.44908I
u = −0.248527 − 0.782547I
a = 2.10247 + 0.40028I
b = 1.157780 − 0.163121I
−2.84181 − 1.23169I −7.09823 + 5.44908I
u = −0.248527 − 0.782547I
a = −0.73260 + 2.58251I
b = −0.965077 − 0.285214I
−2.84181 − 1.23169I −7.09823 + 5.44908I
u = −0.761643 + 0.208049I
a = −0.670419 + 0.201639I
b = 0.255380 + 0.856092I
−5.70347 − 3.47839I −11.19503 + 2.79515I
u = −0.761643 + 0.208049I
a = −1.53048 − 0.72472I
b = −1.359950 − 0.294980I
−5.70347 − 3.47839I −11.19503 + 2.79515I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.761643 − 0.208049I
a = −0.670419 − 0.201639I
b = 0.255380 − 0.856092I
−5.70347 + 3.47839I −11.19503 − 2.79515I
u = −0.761643 − 0.208049I
a = −1.53048 + 0.72472I
b = −1.359950 + 0.294980I
−5.70347 + 3.47839I −11.19503 − 2.79515I
u = 0.449566 + 1.164790I
a = −1.032640 + 0.923920I
b = −1.096360 − 0.477116I
1.58679 − 4.14585I −3.01866 + 3.97600I
u = 0.449566 + 1.164790I
a = 0.0061280 − 0.0919696I
b = −0.193027 + 0.767853I
1.58679 − 4.14585I −3.01866 + 3.97600I
u = 0.449566 − 1.164790I
a = −1.032640 − 0.923920I
b = −1.096360 + 0.477116I
1.58679 + 4.14585I −3.01866 − 3.97600I
u = 0.449566 − 1.164790I
a = 0.0061280 + 0.0919696I
b = −0.193027 − 0.767853I
1.58679 + 4.14585I −3.01866 − 3.97600I
u = −0.524355 + 1.163410I
a = 1.040500 + 0.946543I
b = 1.39510 − 0.62944I
−2.90872 + 8.28632I −7.82440 − 6.14881I
u = −0.524355 + 1.163410I
a = −0.364738 − 0.233686I
b = 0.065535 + 1.177790I
−2.90872 + 8.28632I −7.82440 − 6.14881I
u = −0.524355 − 1.163410I
a = 1.040500 − 0.946543I
b = 1.39510 + 0.62944I
−2.90872 − 8.28632I −7.82440 + 6.14881I
u = −0.524355 − 1.163410I
a = −0.364738 + 0.233686I
b = 0.065535 − 1.177790I
−2.90872 − 8.28632I −7.82440 + 6.14881I
11
III. I
u
3
= ⟨b + u, 2a − u − 1, u
2
+ 1⟩
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
−1
a
7
=
1
2
u +
1
2
−u
a
8
=
3
2
u +
1
2
−u
a
3
=
1
2
u +
1
2
1
a
1
=
0.5
1
2
u +
1
2
a
6
=
u +
1
2
−
1
2
u −
1
2
a
9
=
u
0
a
2
=
1
2
u −
1
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
9
u
2
+ 1
c
5
, c
10
2(2u
2
− 2u + 1)
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
9
(y + 1)
2
c
5
, c
10
4(4y
2
+ 1)
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 0.500000 + 0.500000I
b = − 1.000000I
1.64493 −4.00000
u = − 1.000000I
a = 0.500000 − 0.500000I
b = 1.000000I
1.64493 −4.00000
15
IV. I
u
4
= ⟨b, a − 1, u
3
+ u − 1⟩
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
7
=
1
0
a
8
=
1
0
a
3
=
1
0
a
1
=
−u
u
a
6
=
−u
2
+ 1
u
2
a
9
=
u
1
a
2
=
−u + 1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
9
, c
10
u
3
+ u + 1
c
2
, c
8
(u + 1)
3
c
3
, c
7
u
3
c
5
u
3
+ 2u
2
+ u − 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
9
, c
10
y
3
+ 2y
2
+ y −1
c
2
, c
8
(y −1)
3
c
3
, c
7
y
3
c
5
y
3
− 2y
2
+ 5y −1
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.341164 + 1.161540I
a = 1.00000
b = 0
−1.64493 −6.00000
u = −0.341164 − 1.161540I
a = 1.00000
b = 0
−1.64493 −6.00000
u = 0.682328
a = 1.00000
b = 0
−1.64493 −6.00000
19
V. I
u
5
= ⟨b − 1, a − u − 1, u
3
+ u − 1⟩
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
7
=
u + 1
1
a
8
=
u
1
a
3
=
−u
−1
a
1
=
u − 1
−u
2
+ u
a
6
=
u
2
+ 1
u
2
+ u
a
9
=
u
1
a
2
=
−u
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
u
3
+ u + 1
c
2
, c
8
u
3
c
3
, c
7
(u + 1)
3
c
10
u
3
+ 2u
2
+ u − 1
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
9
y
3
+ 2y
2
+ y −1
c
2
, c
8
y
3
c
3
, c
7
(y −1)
3
c
10
y
3
− 2y
2
+ 5y −1
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.341164 + 1.161540I
a = 0.658836 + 1.161540I
b = 1.00000
−1.64493 −6.00000
u = −0.341164 − 1.161540I
a = 0.658836 − 1.161540I
b = 1.00000
−1.64493 −6.00000
u = 0.682328
a = 1.68233
b = 1.00000
−1.64493 −6.00000
23
VI. I
u
6
= ⟨b − 1, u
3
a − u
3
+ au − 2u − 1⟩
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
7
=
a
1
a
8
=
a − 1
1
a
3
=
−a + 1
−1
a
1
=
−a
2
u + 2au − u
−au + 2u
a
6
=
−a
2
u
2
+ 2u
2
a − u
2
+ a
−u
2
a + 2u
2
+ 1
a
9
=
u
u
3
+ u
a
2
=
−a + u + 1
u
3
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
24
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
−3.28987 −12.0000
25
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
3(u
2
+ 1)(u
3
+ u + 1)
2
· (u
10
− u
9
+ 5u
8
− 5u
7
+ 9u
6
− 9u
5
+ 6u
4
− 6u
3
+ u
2
+ 1)
2
· (3u
12
+ 9u
11
+ ··· + 26u + 5)
c
2
, c
3
, c
7
c
8
u
3
(u + 1)
3
(u
2
+ 1)
· (u
12
− 2u
11
+ u
9
+ 3u
7
+ u
6
− 15u
5
+ 21u
4
− 14u
3
+ 3u
2
+ u + 2)
· (u
20
− 2u
19
+ ··· − 58u + 31)
c
4
, c
9
3(u
2
+ 1)(u
3
+ u + 1)
2
· (u
10
− u
9
+ 3u
8
− 3u
7
+ 5u
6
− 5u
5
+ 4u
4
− 4u
3
+ 3u
2
− 2u + 1)
2
· (3u
12
+ 9u
11
+ ··· + 20u + 5)
c
5
, c
10
8(2u
2
− 2u + 1)(u
3
+ u + 1)(u
3
+ 2u
2
+ u − 1)(2u
12
− 4u
11
+ ··· + 12u + 3)
· (2u
20
− 13u
18
+ ··· − 518u + 121)
26
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
9(y + 1)
2
(y
3
+ 2y
2
+ y −1)
2
· (y
10
+ 9y
9
+ 33y
8
+ 59y
7
+ 41y
6
− 21y
5
− 44y
4
− 6y
3
+ 13y
2
+ 2y + 1)
2
· (9y
12
+ 87y
11
+ ··· − 16y + 25)
c
2
, c
3
, c
7
c
8
y
3
(y −1)
3
(y + 1)
2
(y
12
− 4y
11
+ ··· + 11y + 4)
· (y
20
− 14y
19
+ ··· − 388y + 961)
c
4
, c
9
9(y + 1)
2
(y
3
+ 2y
2
+ y −1)
2
· (y
10
+ 5y
9
+ 13y
8
+ 19y
7
+ 17y
6
+ 7y
5
− 2y
3
+ y
2
+ 2y + 1)
2
· (9y
12
+ 51y
11
+ ··· + 240y + 25)
c
5
, c
10
64(4y
2
+ 1)(y
3
− 2y
2
+ 5y −1)(y
3
+ 2y
2
+ y −1)
· (4y
12
− 20y
11
+ ··· − 30y + 9)(4y
20
− 52y
19
+ ··· − 56574y + 14641)
27
