10
112
(K10a
76
)
A knot diagram
1
Linearized knot diagam
5 6 10 7 8 9 1 2 3 4
Solving Sequence
3,9
10 4
1,7
6 2 8 5
c
9
c
3
c
10
c
6
c
2
c
8
c
5
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−11u
18
+ 43u
17
+ ··· + 3b − 32, −95u
18
+ 507u
17
+ ··· + 21a − 761, u
19
− 6u
18
+ ··· − 11u − 7i
I
u
2
= hu
14
+ 2u
13
+ ··· + b + 2, −2u
14
a − 2u
14
+ ··· − 4a − 4,
u
15
+ 2u
14
− 6u
13
− 11u
12
+ 16u
11
+ 19u
10
− 30u
9
− 7u
8
+ 38u
7
− 12u
6
− 20u
5
+ 14u
4
− 3u
2
+ 2u + 1i
I
u
3
= h−u
4
− u
3
+ 2u
2
+ b + 2u + 1, 2u
4
+ u
3
− 5u
2
+ a − u, u
5
− u
4
− 3u
3
+ 3u
2
+ 1i
I
u
4
= hb + 1, a
2
− a − 1, u + 1i
I
v
1
= ha, b + 1, v + 1i
* 5 irreducible components of dim
C
= 0, with total 57 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−11u
18
+ 43u
17
+ · · · + 3b − 32, −95u
18
+ 507u
17
+ · · · + 21a −
761, u
19
− 6u
18
+ · · · − 11u − 7i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
4
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
7
=
4.52381u
18
− 24.1429u
17
+ ··· + 85.8571u + 36.2381
11
3
u
18
−
43
3
u
17
+ ··· +
64
3
u +
32
3
a
6
=
8.19048u
18
− 38.4762u
17
+ ··· + 107.190u + 46.9048
11
3
u
18
−
43
3
u
17
+ ··· +
64
3
u +
32
3
a
2
=
−8.47619u
18
+ 39.5238u
17
+ ··· − 97.4762u − 43.0952
25
3
u
18
−
116
3
u
17
+ ··· + 99u + 41
a
8
=
−0.809524u
18
+ 2.85714u
17
+ ··· + 1.52381u + 0.238095
−
28
3
u
18
+ 43u
17
+ ··· −
323
3
u −
143
3
a
5
=
6.19048u
18
− 30.1429u
17
+ ··· + 87.5238u + 39.2381
−
1
3
u
18
+
10
3
u
17
+ ··· − 12u −
17
3
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
59
3
u
18
−
281
3
u
17
+ 23u
16
+ 434u
15
−
847
3
u
14
−
2660
3
u
13
− 22u
12
+
5324
3
u
11
+ 1228u
10
−
6821
3
u
9
− 2347u
8
+ 388u
7
+
9250
3
u
6
+
2645
3
u
5
− 1147u
4
− 1166u
3
− 10u
2
+
733
3
u + 99
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
19
− 2u
18
+ ··· − 12u
2
+ 1
c
2
, c
7
u
19
− 2u
18
+ ··· + 2u + 1
c
3
, c
9
, c
10
u
19
− 6u
18
+ ··· − 11u − 7
c
4
, c
6
u
19
+ 2u
18
+ ··· − 2u + 1
c
5
u
19
+ 11u
18
+ ··· − 22u − 7
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
19
− 12y
18
+ ··· + 24y −1
c
2
, c
7
y
19
− 6y
18
+ ··· + 6y −1
c
3
, c
9
, c
10
y
19
− 22y
18
+ ··· + 331y −49
c
4
, c
6
y
19
− 2y
18
+ ··· + 18y −1
c
5
y
19
− y
18
+ ··· + 316y −49
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.650742 + 0.795961I
a = 0.354047 − 0.615330I
b = 0.971206 + 0.919721I
0.21692 − 10.80920I 2.85095 + 8.95586I
u = −0.650742 − 0.795961I
a = 0.354047 + 0.615330I
b = 0.971206 − 0.919721I
0.21692 + 10.80920I 2.85095 − 8.95586I
u = −0.438994 + 0.966374I
a = −0.257963 − 0.341691I
b = 0.542166 − 0.571410I
−0.47646 + 5.13597I 2.04643 − 8.91772I
u = −0.438994 − 0.966374I
a = −0.257963 + 0.341691I
b = 0.542166 + 0.571410I
−0.47646 − 5.13597I 2.04643 + 8.91772I
u = −0.500281 + 0.484136I
a = −0.276043 + 1.168470I
b = −0.989225 − 0.870492I
−1.71464 − 3.32825I −3.18882 + 7.99623I
u = −0.500281 − 0.484136I
a = −0.276043 − 1.168470I
b = −0.989225 + 0.870492I
−1.71464 + 3.32825I −3.18882 − 7.99623I
u = 1.320090 + 0.044695I
a = 0.592095 + 1.229420I
b = −0.158877 − 0.560433I
2.62449 − 0.38341I 2.28736 + 1.27302I
u = 1.320090 − 0.044695I
a = 0.592095 − 1.229420I
b = −0.158877 + 0.560433I
2.62449 + 0.38341I 2.28736 − 1.27302I
u = 0.612375
a = 0.930010
b = 0.220758
1.15807 8.48700
u = −1.43114
a = 0.461846
b = −1.60691
3.44527 2.15800
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.397187 + 0.334084I
a = 0.957646 + 0.804912I
b = −0.907078 + 0.217237I
−1.91282 + 0.23550I −3.85755 + 0.64166I
u = −0.397187 − 0.334084I
a = 0.957646 − 0.804912I
b = −0.907078 − 0.217237I
−1.91282 − 0.23550I −3.85755 − 0.64166I
u = 1.52853 + 0.13991I
a = 0.49931 − 2.07085I
b = −0.98962 + 1.48876I
5.05013 + 5.56057I −1.07165 − 5.51845I
u = 1.52853 − 0.13991I
a = 0.49931 + 2.07085I
b = −0.98962 − 1.48876I
5.05013 − 5.56057I −1.07165 + 5.51845I
u = −1.55827
a = −0.243774
b = 0.971797
8.51485 10.8570
u = 1.58255 + 0.26743I
a = −0.25739 + 1.68674I
b = 1.19555 − 1.28537I
7.5458 + 14.7559I 5.72071 − 7.88264I
u = 1.58255 − 0.26743I
a = −0.25739 − 1.68674I
b = 1.19555 + 1.28537I
7.5458 − 14.7559I 5.72071 + 7.88264I
u = 1.74455 + 0.26523I
a = 0.099974 − 0.506108I
b = −0.456945 + 0.555778I
6.78146 + 0.51735I 9.96153 − 9.39104I
u = 1.74455 − 0.26523I
a = 0.099974 + 0.506108I
b = −0.456945 − 0.555778I
6.78146 − 0.51735I 9.96153 + 9.39104I
6
II.
I
u
2
= hu
14
+2u
13
+· · ·+b+2, −2u
14
a−2u
14
+· · ·−4a−4, u
15
+2u
14
+· · ·+2u+1i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
4
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
7
=
a
−u
14
− 2u
13
+ ··· − au − 2
a
6
=
−u
14
− 2u
13
+ ··· + a − 2
−u
14
− 2u
13
+ ··· − au − 2
a
2
=
−u
13
− u
12
+ ··· + a − 3u
−u
14
a − u
13
a + ··· − a − 1
a
8
=
−u
13
− u
12
+ ··· + a − 1
−u
14
− u
13
+ ··· − u − 2
a
5
=
u
14
a − u
14
+ ··· + 2a − u
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 11u
14
+ 9u
13
−77u
12
−34u
11
+ 218u
10
−21u
9
−319u
8
+ 224u
7
+
214u
6
− 298u
5
+ 14u
4
+ 132u
3
− 60u
2
− 5u + 23
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
30
+ 2u
28
+ ··· + 7u + 1
c
2
, c
7
u
30
− 4u
28
+ ··· − 37u + 43
c
3
, c
9
, c
10
(u
15
+ 2u
14
+ ··· + 2u + 1)
2
c
4
, c
6
u
30
− 3u
29
+ ··· − 42u + 7
c
5
(u
15
− 7u
14
+ ··· + 3u − 2)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
30
+ 4y
29
+ ··· − 19y + 1
c
2
, c
7
y
30
− 8y
29
+ ··· − 32587y + 1849
c
3
, c
9
, c
10
(y
15
− 16y
14
+ ··· + 10y −1)
2
c
4
, c
6
y
30
+ 13y
29
+ ··· + 182y + 49
c
5
(y
15
− 3y
14
+ ··· + 37y −4)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.564527 + 0.799929I
a = 0.618356 + 0.354320I
b = 0.554999 − 0.686515I
2.03837 + 2.66927I 9.65376 − 4.84373I
u = 0.564527 + 0.799929I
a = −0.180396 − 0.172783I
b = −0.148347 + 0.802094I
2.03837 + 2.66927I 9.65376 − 4.84373I
u = 0.564527 − 0.799929I
a = 0.618356 − 0.354320I
b = 0.554999 + 0.686515I
2.03837 − 2.66927I 9.65376 + 4.84373I
u = 0.564527 − 0.799929I
a = −0.180396 + 0.172783I
b = −0.148347 − 0.802094I
2.03837 − 2.66927I 9.65376 + 4.84373I
u = 0.860038 + 0.294980I
a = 1.344360 − 0.145933I
b = −0.576437 − 0.370669I
0.620973 − 0.239040I 7.64024 + 3.49944I
u = 0.860038 + 0.294980I
a = 0.467288 + 0.091114I
b = 0.940505 − 0.025509I
0.620973 − 0.239040I 7.64024 + 3.49944I
u = 0.860038 − 0.294980I
a = 1.344360 + 0.145933I
b = −0.576437 + 0.370669I
0.620973 + 0.239040I 7.64024 − 3.49944I
u = 0.860038 − 0.294980I
a = 0.467288 − 0.091114I
b = 0.940505 + 0.025509I
0.620973 + 0.239040I 7.64024 − 3.49944I
u = 0.239953 + 0.580457I
a = −0.446760 − 0.059168I
b = −0.908941 + 1.005900I
−1.25960 + 3.60373I −3.55671 − 7.52468I
u = 0.239953 + 0.580457I
a = 0.85432 + 1.67566I
b = 0.632582 − 0.043404I
−1.25960 + 3.60373I −3.55671 − 7.52468I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.239953 − 0.580457I
a = −0.446760 + 0.059168I
b = −0.908941 − 1.005900I
−1.25960 − 3.60373I −3.55671 + 7.52468I
u = 0.239953 − 0.580457I
a = 0.85432 − 1.67566I
b = 0.632582 + 0.043404I
−1.25960 − 3.60373I −3.55671 + 7.52468I
u = −1.42712 + 0.14742I
a = −0.17362 − 1.66530I
b = 0.195570 + 0.362588I
4.10336 − 6.07313I 1.68774 + 6.92177I
u = −1.42712 + 0.14742I
a = 0.38365 + 2.08559I
b = −1.15734 − 1.68991I
4.10336 − 6.07313I 1.68774 + 6.92177I
u = −1.42712 − 0.14742I
a = −0.17362 + 1.66530I
b = 0.195570 − 0.362588I
4.10336 + 6.07313I 1.68774 − 6.92177I
u = −1.42712 − 0.14742I
a = 0.38365 − 2.08559I
b = −1.15734 + 1.68991I
4.10336 + 6.07313I 1.68774 − 6.92177I
u = 1.49768 + 0.04419I
a = −0.20828 − 1.77267I
b = −0.809632 + 1.029070I
7.81267 + 4.54595I 9.44858 − 4.92517I
u = 1.49768 + 0.04419I
a = −0.75346 − 1.96166I
b = 1.19533 + 1.83190I
7.81267 + 4.54595I 9.44858 − 4.92517I
u = 1.49768 − 0.04419I
a = −0.20828 + 1.77267I
b = −0.809632 − 1.029070I
7.81267 − 4.54595I 9.44858 + 4.92517I
u = 1.49768 − 0.04419I
a = −0.75346 + 1.96166I
b = 1.19533 − 1.83190I
7.81267 − 4.54595I 9.44858 + 4.92517I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.54349
a = −0.239030 + 0.599706I
b = 0.938402 − 0.503082I
8.47953 10.2010
u = −1.54349
a = −0.239030 − 0.599706I
b = 0.938402 + 0.503082I
8.47953 10.2010
u = −0.406537 + 0.119542I
a = 1.03173 + 0.97861I
b = 0.502233 − 1.320460I
1.41571 − 3.90370I 10.38515 + 7.89648I
u = −0.406537 + 0.119542I
a = −2.55257 + 2.38070I
b = −0.382424 − 0.882051I
1.41571 − 3.90370I 10.38515 + 7.89648I
u = −0.406537 − 0.119542I
a = 1.03173 − 0.97861I
b = 0.502233 + 1.320460I
1.41571 + 3.90370I 10.38515 − 7.89648I
u = −0.406537 − 0.119542I
a = −2.55257 − 2.38070I
b = −0.382424 + 0.882051I
1.41571 + 3.90370I 10.38515 − 7.89648I
u = −1.55680 + 0.27188I
a = 0.037420 − 1.346330I
b = 0.959638 + 0.986410I
8.99262 − 6.60915I 9.14063 + 5.69443I
u = −1.55680 + 0.27188I
a = −0.183014 + 1.386800I
b = −0.436143 − 1.307360I
8.99262 − 6.60915I 9.14063 + 5.69443I
u = −1.55680 − 0.27188I
a = 0.037420 + 1.346330I
b = 0.959638 − 0.986410I
8.99262 + 6.60915I 9.14063 − 5.69443I
u = −1.55680 − 0.27188I
a = −0.183014 − 1.386800I
b = −0.436143 + 1.307360I
8.99262 + 6.60915I 9.14063 − 5.69443I
12
III.
I
u
3
= h−u
4
−u
3
+2u
2
+b+2u+1, 2u
4
+u
3
−5u
2
+a−u, u
5
−u
4
−3u
3
+3u
2
+1i
(i) Arc colorings
a
3
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
4
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
7
=
−2u
4
− u
3
+ 5u
2
+ u
u
4
+ u
3
− 2u
2
− 2u − 1
a
6
=
−u
4
+ 3u
2
− u − 1
u
4
+ u
3
− 2u
2
− 2u − 1
a
2
=
−u
4
+ 2u
2
− u + 2
u
4
− 3u
2
+ u
a
8
=
−u
4
+ 3u
2
− u
u
3
− 2u
a
5
=
−u
4
+ 2u
2
− u
u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
4
− 7u
3
+ 10u
2
+ 14u + 6
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
5
+ u
4
+ u
2
+ u + 1
c
2
, c
7
u
5
− u
4
+ u
3
+ u − 1
c
3
u
5
+ u
4
− 3u
3
− 3u
2
− 1
c
4
, c
6
u
5
− u
4
+ 3u
3
+ u + 1
c
5
u
5
+ 4u
4
+ 9u
3
+ 13u
2
+ 11u + 5
c
9
, c
10
u
5
− u
4
− 3u
3
+ 3u
2
+ 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
5
− y
4
− 3y
2
− y −1
c
2
, c
7
y
5
+ y
4
+ 3y
3
+ y −1
c
3
, c
9
, c
10
y
5
− 7y
4
+ 15y
3
− 7y
2
− 6y −1
c
4
, c
6
y
5
+ 5y
4
+ 11y
3
+ 8y
2
+ y −1
c
5
y
5
+ 2y
4
− y
3
− 11y
2
− 9y −25
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.48162 + 0.12936I
a = −0.00174 − 2.14399I
b = −0.54328 + 1.49449I
6.00251 + 5.77307I 7.88552 − 6.98438I
u = 1.48162 − 0.12936I
a = −0.00174 + 2.14399I
b = −0.54328 − 1.49449I
6.00251 − 5.77307I 7.88552 + 6.98438I
u = −0.099006 + 0.496292I
a = −1.44626 + 0.01961I
b = −0.210516 − 0.857202I
0.38751 − 3.74061I 1.55846 + 6.53295I
u = −0.099006 − 0.496292I
a = −1.44626 − 0.01961I
b = −0.210516 + 0.857202I
0.38751 + 3.74061I 1.55846 − 6.53295I
u = −1.76524
a = −0.103987
b = 0.507589
6.95916 12.1120
16
IV. I
u
4
= hb + 1, a
2
− a − 1, u + 1i
(i) Arc colorings
a
3
=
0
−1
a
9
=
1
0
a
10
=
1
−1
a
4
=
−1
0
a
1
=
0
−1
a
7
=
a
−1
a
6
=
a − 1
−1
a
2
=
−a + 2
−a
a
8
=
a
−a − 1
a
5
=
a − 1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
8
u
2
+ u − 1
c
3
, c
4
, c
6
(u − 1)
2
c
5
u
2
c
9
, c
10
(u + 1)
2
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
c
8
y
2
− 3y + 1
c
3
, c
4
, c
6
c
9
, c
10
(y −1)
2
c
5
y
2
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.618034
b = −1.00000
0 −5.00000
u = −1.00000
a = 1.61803
b = −1.00000
0 −5.00000
20
V. I
v
1
= ha, b + 1, v + 1i
(i) Arc colorings
a
3
=
−1
0
a
9
=
1
0
a
10
=
1
0
a
4
=
−1
0
a
1
=
1
0
a
7
=
0
−1
a
6
=
−1
−1
a
2
=
−2
−1
a
8
=
−1
−1
a
5
=
−1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
7
, c
8
u + 1
c
3
, c
5
, c
9
c
10
u
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
7
, c
8
y −1
c
3
, c
5
, c
9
c
10
y
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = −1.00000
−1.64493 −6.00000
24
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
8
(u + 1)(u
2
+ u − 1)(u
5
+ u
4
+ ··· + u + 1)(u
19
− 2u
18
+ ··· − 12u
2
+ 1)
· (u
30
+ 2u
28
+ ··· + 7u + 1)
c
2
, c
7
(u + 1)(u
2
+ u − 1)(u
5
− u
4
+ ··· + u − 1)(u
19
− 2u
18
+ ··· + 2u + 1)
· (u
30
− 4u
28
+ ··· − 37u + 43)
c
3
u(u − 1)
2
(u
5
+ u
4
+ ··· − 3u
2
− 1)(u
15
+ 2u
14
+ ··· + 2u + 1)
2
· (u
19
− 6u
18
+ ··· − 11u − 7)
c
4
, c
6
((u − 1)
2
)(u + 1)(u
5
− u
4
+ ··· + u + 1)(u
19
+ 2u
18
+ ··· − 2u + 1)
· (u
30
− 3u
29
+ ··· − 42u + 7)
c
5
u
3
(u
5
+ 4u
4
+ ··· + 11u + 5)(u
15
− 7u
14
+ ··· + 3u − 2)
2
· (u
19
+ 11u
18
+ ··· − 22u − 7)
c
9
, c
10
u(u + 1)
2
(u
5
− u
4
+ ··· + 3u
2
+ 1)(u
15
+ 2u
14
+ ··· + 2u + 1)
2
· (u
19
− 6u
18
+ ··· − 11u − 7)
25
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y −1)(y
2
− 3y + 1)(y
5
− y
4
+ ··· − y −1)(y
19
− 12y
18
+ ··· + 24y −1)
· (y
30
+ 4y
29
+ ··· − 19y + 1)
c
2
, c
7
(y −1)(y
2
− 3y + 1)(y
5
+ y
4
+ ··· + y −1)(y
19
− 6y
18
+ ··· + 6y −1)
· (y
30
− 8y
29
+ ··· − 32587y + 1849)
c
3
, c
9
, c
10
y(y −1)
2
(y
5
− 7y
4
+ 15y
3
− 7y
2
− 6y −1)
· ((y
15
− 16y
14
+ ··· + 10y −1)
2
)(y
19
− 22y
18
+ ··· + 331y −49)
c
4
, c
6
((y −1)
3
)(y
5
+ 5y
4
+ ··· + y −1)(y
19
− 2y
18
+ ··· + 18y −1)
· (y
30
+ 13y
29
+ ··· + 182y + 49)
c
5
y
3
(y
5
+ 2y
4
+ ··· − 9y −25)(y
15
− 3y
14
+ ··· + 37y −4)
2
· (y
19
− y
18
+ ··· + 316y −49)
26
