11n
169
(K11n
169
)
A knot diagram
1
Linearized knot diagam
5 8 1 9 1 4 11 2 4 7 8
Solving Sequence
2,5
1
6,9
4 3 8 11 7 10
c
1
c
5
c
4
c
3
c
8
c
11
c
7
c
10
c
2
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
12
+ 8u
11
+ ··· + 4b − 12, 3u
12
− 22u
11
+ ··· + 8a + 36, u
13
− 8u
12
+ ··· + 56u − 8i
I
u
2
= hu
5
+ u
4
+ u
3
− u
2
+ b − 1, u
7
+ u
6
+ 2u
5
+ 2u
3
− 2u
2
+ a − 2, u
8
+ u
7
+ 2u
6
− u
5
+ u
4
− 3u
3
+ u
2
− 2u + 1i
I
u
3
= h−5a
5
u − 4a
5
+ 7a
4
+ 5a
3
u + 4a
3
+ 8a
2
u − 2a
2
− 6au + 7b + 5a − 3u − 1,
a
6
+ a
5
u − a
4
− 3a
3
u − 2a
3
+ 3a
2
u + a
2
+ 2au + 3a − 3u − 1, u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 33 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−u
12
+8u
11
+· · ·+4b−12, 3u
12
−22u
11
+· · ·+8a+36, u
13
−8u
12
+· · ·+56u−8i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
1
=
1
−u
2
a
6
=
−u
u
3
+ u
a
9
=
−
3
8
u
12
+
11
4
u
11
+ ··· + 20u −
9
2
1
4
u
12
− 2u
11
+ ··· −
33
2
u + 3
a
4
=
1
4
u
12
− 2u
11
+ ··· − 13u +
5
2
1
2
u
11
− 3u
10
+ ··· +
25
2
u − 2
a
3
=
−
1
4
u
12
+
3
2
u
11
+ ··· +
3
2
u +
1
2
−
1
2
u
11
+ 3u
10
+ ··· −
23
2
u + 2
a
8
=
−
1
8
u
12
+
3
4
u
11
+ ··· +
7
2
u −
3
2
1
4
u
12
− 2u
11
+ ··· −
33
2
u + 3
a
11
=
1
4
u
12
− 2u
11
+ ··· − 14u +
7
2
1
2
u
12
−
7
2
u
11
+ ··· −
29
2
u + 2
a
7
=
u
12
−
27
4
u
11
+ ··· −
73
4
u + 2
−
3
4
u
12
+
11
2
u
11
+ ··· + 29u − 4
a
10
=
−
1
4
u
12
+
5
4
u
11
+ ··· −
55
4
u + 4
3
4
u
12
−
11
2
u
11
+ ··· − 29u + 4
a
10
=
−
1
4
u
12
+
5
4
u
11
+ ··· −
55
4
u + 4
3
4
u
12
−
11
2
u
11
+ ··· − 29u + 4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −u
12
+5u
11
−15u
10
+31u
9
−51u
8
+70u
7
−85u
6
+89u
5
−83u
4
+63u
3
−43u
2
+24u −22
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
13
+ 8u
12
+ ··· + 56u + 8
c
2
, c
4
, c
8
c
9
u
13
+ 3u
11
+ ··· + 2u + 1
c
3
, c
6
u
13
− 2u
12
+ ··· − 4u + 1
c
7
, c
10
, c
11
u
13
+ 6u
12
+ ··· + 14u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
13
+ 4y
12
+ ··· + 480y −64
c
2
, c
4
, c
8
c
9
y
13
+ 6y
12
+ ··· + 2y −1
c
3
, c
6
y
13
− 26y
12
+ ··· + 42y −1
c
7
, c
10
, c
11
y
13
− 16y
12
+ ··· + 204y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.326542 + 1.020100I
a = −0.503264 + 0.483746I
b = 0.329129 + 0.671341I
−1.07131 + 1.81105I −12.48073 − 2.50977I
u = −0.326542 − 1.020100I
a = −0.503264 − 0.483746I
b = 0.329129 − 0.671341I
−1.07131 − 1.81105I −12.48073 + 2.50977I
u = 1.161480 + 0.385396I
a = 0.216539 − 0.581687I
b = −0.475687 + 0.592166I
−3.17225 + 0.94602I −12.22572 − 6.14642I
u = 1.161480 − 0.385396I
a = 0.216539 + 0.581687I
b = −0.475687 − 0.592166I
−3.17225 − 0.94602I −12.22572 + 6.14642I
u = 0.270743 + 1.206070I
a = 0.733128 − 0.331803I
b = −0.598666 − 0.794370I
2.89440 − 2.21633I −9.35734 + 3.25180I
u = 0.270743 − 1.206070I
a = 0.733128 + 0.331803I
b = −0.598666 + 0.794370I
2.89440 + 2.21633I −9.35734 − 3.25180I
u = 0.63465 + 1.27236I
a = −0.928004 + 0.162795I
b = 0.796089 + 1.077430I
−0.15730 − 7.29804I −11.37128 + 6.48312I
u = 0.63465 − 1.27236I
a = −0.928004 − 0.162795I
b = 0.796089 − 1.077430I
−0.15730 + 7.29804I −11.37128 − 6.48312I
u = 1.26554 + 0.69993I
a = −0.057804 + 0.864578I
b = 0.678296 − 1.053700I
−11.99570 + 3.34885I −12.69425 − 2.28469I
u = 1.26554 − 0.69993I
a = −0.057804 − 0.864578I
b = 0.678296 + 1.053700I
−11.99570 − 3.34885I −12.69425 + 2.28469I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.83155 + 1.23498I
a = 1.136480 + 0.017597I
b = −0.92331 − 1.41816I
−10.0617 − 10.8173I −12.24076 + 5.20880I
u = 0.83155 − 1.23498I
a = 1.136480 − 0.017597I
b = −0.92331 + 1.41816I
−10.0617 + 10.8173I −12.24076 − 5.20880I
u = 0.325158
a = −1.19415
b = 0.388289
−0.575325 −17.2600
6
II. I
u
2
= hu
5
+ u
4
+ u
3
− u
2
+ b − 1, u
7
+ u
6
+ 2u
5
+ 2u
3
− 2u
2
+ a − 2, u
8
+
u
7
+ 2u
6
− u
5
+ u
4
− 3u
3
+ u
2
− 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
1
=
1
−u
2
a
6
=
−u
u
3
+ u
a
9
=
−u
7
− u
6
− 2u
5
− 2u
3
+ 2u
2
+ 2
−u
5
− u
4
− u
3
+ u
2
+ 1
a
4
=
−u
7
− 2u
6
− 3u
5
− u
4
+ 2u
2
+ 2u + 1
−u
7
− u
6
− 2u
5
+ u
4
− u
3
+ 3u
2
+ 1
a
3
=
−2u
7
− 3u
6
− 5u
5
− u
3
+ 5u
2
+ u + 3
−u
7
− u
6
− 2u
5
+ u
4
− u
3
+ 3u
2
− u + 1
a
8
=
−u
7
− u
6
− 3u
5
− u
4
− 3u
3
+ 3u
2
+ 3
−u
5
− u
4
− u
3
+ u
2
+ 1
a
11
=
3u
7
+ 4u
6
+ 7u
5
− u
4
+ 2u
3
− 7u
2
+ u − 4
u
7
+ u
6
+ 2u
5
− u
4
+ u
3
− 3u
2
+ u − 2
a
7
=
u
7
+ 2u
6
+ 4u
5
+ 2u
4
+ 2u
3
− 3u
2
− 2u − 4
u
7
+ 2u
6
+ 3u
5
+ u
4
+ u
3
− 2u
2
− u − 2
a
10
=
−2u
7
− 3u
6
− 5u
5
− 2u
3
+ 4u
2
+ 4
−u
7
− 2u
6
− 3u
5
− u
4
− u
3
+ 2u
2
+ u + 2
a
10
=
−2u
7
− 3u
6
− 5u
5
− 2u
3
+ 4u
2
+ 4
−u
7
− 2u
6
− 3u
5
− u
4
− u
3
+ 2u
2
+ u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
7
+ 2u
6
+ 4u
5
− 3u
4
+ 3u
3
− 7u
2
− u − 11
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ u
7
+ 2u
6
− u
5
+ u
4
− 3u
3
+ u
2
− 2u + 1
c
2
, c
9
u
8
+ 3u
6
− u
5
+ 3u
4
− 3u
3
− 3u − 1
c
3
, c
6
u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ u
3
+ 2u
2
− u + 1
c
4
, c
8
u
8
+ 3u
6
+ u
5
+ 3u
4
+ 3u
3
+ 3u − 1
c
5
u
8
− u
7
+ 2u
6
+ u
5
+ u
4
+ 3u
3
+ u
2
+ 2u + 1
c
7
u
8
+ u
7
− 5u
6
− 4u
5
+ 8u
4
+ 5u
3
− 3u
2
− u − 1
c
10
, c
11
u
8
− u
7
− 5u
6
+ 4u
5
+ 8u
4
− 5u
3
− 3u
2
+ u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
8
+ 3y
7
+ 8y
6
+ 11y
5
+ 5y
4
− 7y
3
− 9y
2
− 2y + 1
c
2
, c
4
, c
8
c
9
y
8
+ 6y
7
+ 15y
6
+ 17y
5
+ y
4
− 21y
3
− 24y
2
− 9y + 1
c
3
, c
6
y
8
− 2y
7
− 9y
6
− 7y
5
+ 5y
4
+ 11y
3
+ 8y
2
+ 3y + 1
c
7
, c
10
, c
11
y
8
− 11y
7
+ 49y
6
− 112y
5
+ 134y
4
− 71y
3
+ 3y
2
+ 5y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.163169 + 0.915412I
a = 1.000040 + 0.736649I
b = −0.511162 + 1.035650I
−0.155635 − 0.787051I −8.59786 − 1.33483I
u = 0.163169 − 0.915412I
a = 1.000040 − 0.736649I
b = −0.511162 − 1.035650I
−0.155635 + 0.787051I −8.59786 + 1.33483I
u = 0.918626
a = −0.323992
b = −0.297628
−2.98361 −12.1620
u = −0.404913 + 1.017880I
a = −1.143500 + 0.110127I
b = 0.350924 − 1.208540I
5.21920 + 1.77211I −2.23409 − 0.85548I
u = −0.404913 − 1.017880I
a = −1.143500 − 0.110127I
b = 0.350924 + 1.208540I
5.21920 − 1.77211I −2.23409 + 0.85548I
u = −0.95744 + 1.12705I
a = 0.720153 − 0.424011I
b = −0.211625 + 1.217620I
1.24083 + 3.75870I −10.69968 − 3.38204I
u = −0.95744 − 1.12705I
a = 0.720153 + 0.424011I
b = −0.211625 − 1.217620I
1.24083 − 3.75870I −10.69968 + 3.38204I
u = 0.479751
a = 2.17061
b = 1.04135
−12.9150 −12.7750
10
III. I
u
3
= h−5a
5
u + 5a
3
u + · · · + 5a − 1, a
5
u − 3a
3
u + · · · + 3a − 1, u
2
+ u + 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
1
=
1
u + 1
a
6
=
−u
u + 1
a
9
=
a
5
7
a
5
u −
5
7
a
3
u + ··· −
5
7
a +
1
7
a
4
=
a
2
u
−
3
7
a
5
u −
4
7
a
3
u + ··· +
10
7
a −
2
7
a
3
=
−
3
7
a
5
u −
4
7
a
3
u + ··· +
10
7
a −
2
7
−
4
7
a
5
u −
3
7
a
3
u + ··· +
11
7
a +
2
7
a
8
=
5
7
a
5
u −
5
7
a
3
u + ··· +
2
7
a +
1
7
5
7
a
5
u −
5
7
a
3
u + ··· −
5
7
a +
1
7
a
11
=
a
2
u − 2u
3
7
a
5
u +
4
7
a
3
u + ··· −
10
7
a +
2
7
a
7
=
−
2
7
a
5
u + a
4
u + ··· +
2
7
a +
8
7
−a
4
u − a
4
+ a
3
+ au + 2
a
10
=
a
3
u + a
3
− a
−
4
7
a
5
u − a
4
u + ··· +
4
7
a +
2
7
a
10
=
a
3
u + a
3
− a
−
4
7
a
5
u − a
4
u + ··· +
4
7
a +
2
7
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u − 14
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
2
− u + 1)
6
c
2
, c
4
, c
8
c
9
u
12
− u
11
+ ··· + 14u + 7
c
3
, c
6
u
12
− u
11
+ ··· − 56u + 13
c
7
, c
10
, c
11
(u
3
− u
2
− 2u + 1)
4
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
2
+ y + 1)
6
c
2
, c
4
, c
8
c
9
y
12
+ 3y
11
+ ··· − 140y
2
+ 49
c
3
, c
6
y
12
− 13y
11
+ ··· − 432y + 169
c
7
, c
10
, c
11
(y
3
− 5y
2
+ 6y −1)
4
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −1.100660 + 0.111510I
b = 0.231240 − 1.394380I
4.22983 + 2.02988I −12.00000 − 3.46410I
u = −0.500000 + 0.866025I
a = 1.091430 + 0.189362I
b = −1.61068 − 0.71000I
−12.68950 + 2.02988I −12.00000 − 3.46410I
u = −0.500000 + 0.866025I
a = −1.045080 + 0.441811I
b = 0.763411 − 0.046058I
−1.40994 + 2.02988I −12.00000 − 3.46410I
u = −0.500000 + 0.866025I
a = 0.421593 + 0.638105I
b = −0.139922 + 1.125970I
−1.40994 + 2.02988I −12.00000 − 3.46410I
u = −0.500000 + 0.866025I
a = 1.323190 − 0.496928I
b = −0.453761 + 1.008960I
4.22983 + 2.02988I −12.00000 − 3.46410I
u = −0.500000 + 0.866025I
a = −0.19046 − 1.74989I
b = 0.709707 − 0.850523I
−12.68950 + 2.02988I −12.00000 − 3.46410I
u = −0.500000 − 0.866025I
a = −1.100660 − 0.111510I
b = 0.231240 + 1.394380I
4.22983 − 2.02988I −12.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 1.091430 − 0.189362I
b = −1.61068 + 0.71000I
−12.68950 − 2.02988I −12.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = −1.045080 − 0.441811I
b = 0.763411 + 0.046058I
−1.40994 − 2.02988I −12.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 0.421593 − 0.638105I
b = −0.139922 − 1.125970I
−1.40994 − 2.02988I −12.00000 + 3.46410I
14
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 − 0.866025I
a = 1.323190 + 0.496928I
b = −0.453761 − 1.008960I
4.22983 − 2.02988I −12.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = −0.19046 + 1.74989I
b = 0.709707 + 0.850523I
−12.68950 − 2.02988I −12.00000 + 3.46410I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)
6
(u
8
+ u
7
+ 2u
6
− u
5
+ u
4
− 3u
3
+ u
2
− 2u + 1)
· (u
13
+ 8u
12
+ ··· + 56u + 8)
c
2
, c
9
(u
8
+ 3u
6
− u
5
+ 3u
4
− 3u
3
− 3u − 1)(u
12
− u
11
+ ··· + 14u + 7)
· (u
13
+ 3u
11
+ ··· + 2u + 1)
c
3
, c
6
(u
8
+ 2u
7
+ ··· − u + 1)(u
12
− u
11
+ ··· − 56u + 13)
· (u
13
− 2u
12
+ ··· − 4u + 1)
c
4
, c
8
(u
8
+ 3u
6
+ u
5
+ 3u
4
+ 3u
3
+ 3u − 1)(u
12
− u
11
+ ··· + 14u + 7)
· (u
13
+ 3u
11
+ ··· + 2u + 1)
c
5
(u
2
− u + 1)
6
(u
8
− u
7
+ 2u
6
+ u
5
+ u
4
+ 3u
3
+ u
2
+ 2u + 1)
· (u
13
+ 8u
12
+ ··· + 56u + 8)
c
7
(u
3
− u
2
− 2u + 1)
4
(u
8
+ u
7
− 5u
6
− 4u
5
+ 8u
4
+ 5u
3
− 3u
2
− u − 1)
· (u
13
+ 6u
12
+ ··· + 14u + 4)
c
10
, c
11
(u
3
− u
2
− 2u + 1)
4
(u
8
− u
7
− 5u
6
+ 4u
5
+ 8u
4
− 5u
3
− 3u
2
+ u − 1)
· (u
13
+ 6u
12
+ ··· + 14u + 4)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
2
+ y + 1)
6
(y
8
+ 3y
7
+ 8y
6
+ 11y
5
+ 5y
4
− 7y
3
− 9y
2
− 2y + 1)
· (y
13
+ 4y
12
+ ··· + 480y −64)
c
2
, c
4
, c
8
c
9
(y
8
+ 6y
7
+ 15y
6
+ 17y
5
+ y
4
− 21y
3
− 24y
2
− 9y + 1)
· (y
12
+ 3y
11
+ ··· − 140y
2
+ 49)(y
13
+ 6y
12
+ ··· + 2y −1)
c
3
, c
6
(y
8
− 2y
7
− 9y
6
− 7y
5
+ 5y
4
+ 11y
3
+ 8y
2
+ 3y + 1)
· (y
12
− 13y
11
+ ··· − 432y + 169)(y
13
− 26y
12
+ ··· + 42y −1)
c
7
, c
10
, c
11
(y
3
− 5y
2
+ 6y −1)
4
· (y
8
− 11y
7
+ 49y
6
− 112y
5
+ 134y
4
− 71y
3
+ 3y
2
+ 5y + 1)
· (y
13
− 16y
12
+ ··· + 204y −16)
17
