10
160
(K10n
33
)
A knot diagram
1
Linearized knot diagam
7 9 7 2 8 1 3 5 4 2
Solving Sequence
2,9 3,7
4 5 1 6 8 10
c
2
c
3
c
4
c
1
c
6
c
8
c
10
c
5
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
6
− 2u
5
+ 3u
4
− 2u
3
+ 3u
2
+ b − 1, −u
8
+ 3u
7
− 5u
6
+ 4u
5
− 4u
4
+ 2u
3
+ u
2
+ 2a − 4u − 1,
u
9
− 5u
8
+ 13u
7
− 20u
6
+ 22u
5
− 18u
4
+ 11u
3
− 5u + 2i
I
u
2
= hu
2
+ b + u + 1, u
3
+ 2u
2
+ a + 2u + 2, u
5
+ 2u
4
+ 3u
3
+ 2u
2
− 1i
I
u
3
= h−u
2
a + b − 1, a
2
+ 2au − 2u
2
+ 3a − 3u − 2, u
3
+ u
2
− 1i
* 3 irreducible components of dim
C
= 0, with total 20 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
=
hu
6
−2u
5
+3u
4
−2u
3
+3u
2
+b−1, −u
8
+3u
7
+· · ·+2a−1, u
9
−5u
8
+· · ·−5u+2i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
7
=
1
2
u
8
−
3
2
u
7
+ ··· + 2u +
1
2
−u
6
+ 2u
5
− 3u
4
+ 2u
3
− 3u
2
+ 1
a
4
=
−
5
2
u
8
+
21
2
u
7
+ ··· − 8u +
13
2
u
6
− 2u
5
+ 3u
4
− 2u
3
+ 3u
2
− 1
a
5
=
−
5
2
u
8
+
21
2
u
7
+ ··· − 8u +
15
2
u
6
− 2u
5
+ 3u
4
− 2u
3
+ 3u
2
− 1
a
1
=
1
2
u
8
−
5
2
u
7
+ ··· + 2u −
3
2
u
7
− 3u
6
+ 5u
5
− 4u
4
+ 4u
3
− 2u
2
− u + 1
a
6
=
−u
8
+ 4u
7
− 9u
6
+ 12u
5
− 12u
4
+ 9u
3
− 4u
2
− u + 3
−2u
8
+ 9u
7
− 19u
6
+ 24u
5
− 22u
4
+ 17u
3
− 6u
2
− 6u + 4
a
8
=
3
2
u
8
−
13
2
u
7
+ ··· + 6u −
5
2
−u
8
+ 4u
7
− 8u
6
+ 9u
5
− 8u
4
+ 6u
3
− u
2
− 2u + 1
a
10
=
1
2
u
8
−
3
2
u
7
+ ··· + u −
1
2
u
7
− 3u
6
+ 5u
5
− 4u
4
+ 4u
3
− 2u
2
− u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
8
− 5u
7
+ 12u
6
− 15u
5
+ 12u
4
− 4u
3
+ 8u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
u
9
− u
8
− 7u
7
+ 7u
6
+ 13u
5
− 13u
4
+ 2u
3
+ u
2
+ 1
c
2
u
9
− 5u
8
+ 13u
7
− 20u
6
+ 22u
5
− 18u
4
+ 11u
3
− 5u + 2
c
3
, c
7
u
9
+ 7u
8
+ 26u
7
+ 61u
6
+ 103u
5
+ 129u
4
+ 125u
3
+ 86u
2
+ 40u + 8
c
5
, c
8
u
9
+ 6u
7
− 3u
6
+ 14u
5
− 10u
4
+ 13u
3
− 7u
2
− u + 1
c
9
u
9
− 11u
7
+ 2u
6
+ 35u
5
− 32u
4
+ 47u
3
+ 8u
2
+ u + 13
c
10
u
9
+ 15u
8
+ 89u
7
+ 253u
6
+ 325u
5
+ 129u
4
+ 16u
3
− 25u
2
− 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
y
9
− 15y
8
+ 89y
7
− 253y
6
+ 325y
5
− 129y
4
+ 16y
3
+ 25y
2
− 2y −1
c
2
y
9
+ y
8
+ 13y
7
+ 14y
6
+ 40y
5
+ 50y
4
− 19y
3
− 38y
2
+ 25y −4
c
3
, c
7
y
9
+ 3y
8
+ ··· + 224y −64
c
5
, c
8
y
9
+ 12y
8
+ 64y
7
+ 185y
6
+ 290y
5
+ 210y
4
+ 7y
3
− 55y
2
+ 15y −1
c
9
y
9
− 22y
8
+ ··· − 207y −169
c
10
y
9
− 47y
8
+ ··· + 54y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.204797 + 1.087900I
a = −0.055258 + 1.397040I
b = 0.689596 − 0.376245I
2.64060 + 1.65275I −0.59079 − 4.28210I
u = −0.204797 − 1.087900I
a = −0.055258 − 1.397040I
b = 0.689596 + 0.376245I
2.64060 − 1.65275I −0.59079 + 4.28210I
u = 0.647333 + 0.135453I
a = 1.54477 + 0.21297I
b = −0.017613 − 0.474078I
−0.87559 − 2.35950I −4.89060 + 1.18144I
u = 0.647333 − 0.135453I
a = 1.54477 − 0.21297I
b = −0.017613 + 0.474078I
−0.87559 + 2.35950I −4.89060 − 1.18144I
u = −0.531326
a = −0.232368
b = −0.493195
−0.846327 −11.7230
u = 1.20035 + 1.05816I
a = 0.77288 − 1.22009I
b = 1.96815 + 0.34791I
−14.0726 − 9.2039I −9.16258 + 4.28229I
u = 1.20035 − 1.05816I
a = 0.77288 + 1.22009I
b = 1.96815 − 0.34791I
−14.0726 + 9.2039I −9.16258 − 4.28229I
u = 1.12278 + 1.21739I
a = −0.396214 + 0.245006I
b = −1.89353 + 0.26305I
−13.58820 + 0.68871I −9.49454 − 0.10018I
u = 1.12278 − 1.21739I
a = −0.396214 − 0.245006I
b = −1.89353 − 0.26305I
−13.58820 − 0.68871I −9.49454 + 0.10018I
5
II. I
u
2
= hu
2
+ b + u + 1, u
3
+ 2u
2
+ a + 2u + 2, u
5
+ 2u
4
+ 3u
3
+ 2u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
7
=
−u
3
− 2u
2
− 2u − 2
−u
2
− u − 1
a
4
=
u
3
+ 2u
2
+ 3u + 2
u
2
+ u + 1
a
5
=
u
3
+ u
2
+ 2u + 1
u
2
+ u + 1
a
1
=
−u
4
− 2u
3
− 4u
2
− 4u − 2
−u
4
− 2u
3
− 3u
2
− 2u − 1
a
6
=
2u
4
+ 4u
3
+ 6u
2
+ 5u + 1
u
4
+ 2u
3
+ 3u
2
+ 3u + 1
a
8
=
−u
2
− u − 2
−u
4
− 2u
3
− 3u
2
− u
a
10
=
−2u
4
− 4u
3
− 7u
2
− 6u − 3
−u
4
− 2u
3
− 3u
2
− 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
4
+ 3u
3
+ 7u
2
+ u − 8
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
5
+ u
4
− 2u
3
− u
2
+ u + 1
c
2
u
5
+ 2u
4
+ 3u
3
+ 2u
2
− 1
c
3
u
5
+ 2u
3
+ u
2
+ 1
c
5
u
5
+ u
3
− 2u
2
− 1
c
6
u
5
− u
4
− 2u
3
+ u
2
+ u − 1
c
7
u
5
+ 2u
3
− u
2
− 1
c
8
u
5
+ u
3
+ 2u
2
+ 1
c
9
u
5
− 2u
3
+ 3u
2
− 2u + 1
c
10
u
5
− 5u
4
+ 8u
3
− 7u
2
+ 3u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
y
5
− 5y
4
+ 8y
3
− 7y
2
+ 3y −1
c
2
y
5
+ 2y
4
+ y
3
+ 4y −1
c
3
, c
7
y
5
+ 4y
4
+ 4y
3
− y
2
− 2y −1
c
5
, c
8
y
5
+ 2y
4
+ y
3
− 4y
2
− 4y −1
c
9
y
5
− 4y
4
− y
2
− 2y −1
c
10
y
5
− 9y
4
− 11y
2
− 5y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.885210 + 0.546617I
a = −1.299020 − 0.279409I
b = −0.599596 + 0.421125I
−1.44657 + 3.45949I −7.29654 − 5.67761I
u = −0.885210 − 0.546617I
a = −1.299020 + 0.279409I
b = −0.599596 − 0.421125I
−1.44657 − 3.45949I −7.29654 + 5.67761I
u = −0.361950 + 1.318330I
a = 0.098088 + 1.045130I
b = 0.968932 − 0.363992I
1.57933 + 1.42206I −9.07660 − 1.47974I
u = −0.361950 − 1.318330I
a = 0.098088 − 1.045130I
b = 0.968932 + 0.363992I
1.57933 − 1.42206I −9.07660 + 1.47974I
u = 0.494320
a = −3.59813
b = −1.73867
−6.84525 −5.25370
9
III. I
u
3
= h−u
2
a + b − 1, a
2
+ 2au − 2u
2
+ 3a − 3u − 2, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
7
=
a
u
2
a + 1
a
4
=
a + 1
u
2
a + u
2
+ 1
a
5
=
−u
2
a − u
2
+ a
u
2
a + u
2
+ 1
a
1
=
u
2
a − u
2
+ a − 2u
u
2
a + au − a − u − 2
a
6
=
−2u
2
a − 2au + u + 2
−2au − u
a
8
=
a − 1
u
2
a − u
2
+ 1
a
10
=
2u
2
a + au − u
2
− 3u − 2
u
2
a + au − a − u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u − 14
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
u
6
+ u
5
− 4u
4
− 4u
3
+ 4u
2
+ 8u − 7
c
2
(u
3
+ u
2
− 1)
2
c
3
, c
7
(u − 1)
6
c
5
, c
8
u
6
+ 3u
5
+ 6u
4
+ 12u
3
+ 10u
2
+ 10u + 1
c
9
u
6
+ u
5
− 6u
4
+ 8u
3
+ 2u
2
− 6u − 11
c
10
u
6
+ 9u
5
+ 32u
4
+ 78u
3
+ 136u
2
+ 120u + 49
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
y
6
− 9y
5
+ 32y
4
− 78y
3
+ 136y
2
− 120y + 49
c
2
(y
3
− y
2
+ 2y −1)
2
c
3
, c
7
(y −1)
6
c
5
, c
8
y
6
+ 3y
5
− 16y
4
− 82y
3
− 128y
2
− 80y + 1
c
9
y
6
− 13y
5
+ 24y
4
− 98y
3
+ 232y
2
− 80y + 121
c
10
y
6
− 17y
5
− 108y
4
+ 558y
3
+ 2912y
2
− 1072y + 2401
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.240939 − 0.027206I
b = 0.912616 + 0.309089I
−3.55561 + 2.82812I −10.49024 − 2.97945I
u = −0.877439 + 0.744862I
a = −1.00418 − 1.46252I
b = −1.12770 + 0.99805I
−3.55561 + 2.82812I −10.49024 − 2.97945I
u = −0.877439 − 0.744862I
a = −0.240939 + 0.027206I
b = 0.912616 − 0.309089I
−3.55561 − 2.82812I −10.49024 + 2.97945I
u = −0.877439 − 0.744862I
a = −1.00418 + 1.46252I
b = −1.12770 − 0.99805I
−3.55561 − 2.82812I −10.49024 + 2.97945I
u = 0.754878
a = 0.983762
b = 1.56059
−7.69319 −17.0200
u = 0.754878
a = −5.49352
b = −2.13043
−7.69319 −17.0200
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
5
+ u
4
− 2u
3
− u
2
+ u + 1)(u
6
+ u
5
− 4u
4
− 4u
3
+ 4u
2
+ 8u − 7)
· (u
9
− u
8
− 7u
7
+ 7u
6
+ 13u
5
− 13u
4
+ 2u
3
+ u
2
+ 1)
c
2
(u
3
+ u
2
− 1)
2
(u
5
+ 2u
4
+ 3u
3
+ 2u
2
− 1)
· (u
9
− 5u
8
+ 13u
7
− 20u
6
+ 22u
5
− 18u
4
+ 11u
3
− 5u + 2)
c
3
(u − 1)
6
(u
5
+ 2u
3
+ u
2
+ 1)
· (u
9
+ 7u
8
+ 26u
7
+ 61u
6
+ 103u
5
+ 129u
4
+ 125u
3
+ 86u
2
+ 40u + 8)
c
5
(u
5
+ u
3
− 2u
2
− 1)(u
6
+ 3u
5
+ 6u
4
+ 12u
3
+ 10u
2
+ 10u + 1)
· (u
9
+ 6u
7
− 3u
6
+ 14u
5
− 10u
4
+ 13u
3
− 7u
2
− u + 1)
c
6
(u
5
− u
4
− 2u
3
+ u
2
+ u − 1)(u
6
+ u
5
− 4u
4
− 4u
3
+ 4u
2
+ 8u − 7)
· (u
9
− u
8
− 7u
7
+ 7u
6
+ 13u
5
− 13u
4
+ 2u
3
+ u
2
+ 1)
c
7
(u − 1)
6
(u
5
+ 2u
3
− u
2
− 1)
· (u
9
+ 7u
8
+ 26u
7
+ 61u
6
+ 103u
5
+ 129u
4
+ 125u
3
+ 86u
2
+ 40u + 8)
c
8
(u
5
+ u
3
+ 2u
2
+ 1)(u
6
+ 3u
5
+ 6u
4
+ 12u
3
+ 10u
2
+ 10u + 1)
· (u
9
+ 6u
7
− 3u
6
+ 14u
5
− 10u
4
+ 13u
3
− 7u
2
− u + 1)
c
9
(u
5
− 2u
3
+ 3u
2
− 2u + 1)(u
6
+ u
5
− 6u
4
+ 8u
3
+ 2u
2
− 6u − 11)
· (u
9
− 11u
7
+ 2u
6
+ 35u
5
− 32u
4
+ 47u
3
+ 8u
2
+ u + 13)
c
10
(u
5
− 5u
4
+ 8u
3
− 7u
2
+ 3u − 1)
· (u
6
+ 9u
5
+ 32u
4
+ 78u
3
+ 136u
2
+ 120u + 49)
· (u
9
+ 15u
8
+ 89u
7
+ 253u
6
+ 325u
5
+ 129u
4
+ 16u
3
− 25u
2
− 2u + 1)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
(y
5
− 5y
4
+ 8y
3
− 7y
2
+ 3y −1)
· (y
6
− 9y
5
+ 32y
4
− 78y
3
+ 136y
2
− 120y + 49)
· (y
9
− 15y
8
+ 89y
7
− 253y
6
+ 325y
5
− 129y
4
+ 16y
3
+ 25y
2
− 2y −1)
c
2
(y
3
− y
2
+ 2y −1)
2
(y
5
+ 2y
4
+ y
3
+ 4y −1)
· (y
9
+ y
8
+ 13y
7
+ 14y
6
+ 40y
5
+ 50y
4
− 19y
3
− 38y
2
+ 25y −4)
c
3
, c
7
((y −1)
6
)(y
5
+ 4y
4
+ ··· − 2y −1)(y
9
+ 3y
8
+ ··· + 224y −64)
c
5
, c
8
(y
5
+ 2y
4
+ y
3
− 4y
2
− 4y −1)
· (y
6
+ 3y
5
− 16y
4
− 82y
3
− 128y
2
− 80y + 1)
· (y
9
+ 12y
8
+ 64y
7
+ 185y
6
+ 290y
5
+ 210y
4
+ 7y
3
− 55y
2
+ 15y −1)
c
9
(y
5
− 4y
4
− y
2
− 2y −1)(y
6
− 13y
5
+ ··· − 80y + 121)
· (y
9
− 22y
8
+ ··· − 207y −169)
c
10
(y
5
− 9y
4
− 11y
2
− 5y −1)
· (y
6
− 17y
5
− 108y
4
+ 558y
3
+ 2912y
2
− 1072y + 2401)
· (y
9
− 47y
8
+ ··· + 54y −1)
15
