11n
84
(K11n
84
)
A knot diagram
1
Linearized knot diagam
6 1 10 7 2 9 11 4 6 1 8
Solving Sequence
1,6
2 3
5,8
11 7 4 10 9
c
1
c
2
c
5
c
11
c
7
c
4
c
10
c
9
c
3
, c
6
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h3u
8
+ 4u
7
+ 24u
6
+ 25u
5
+ 51u
4
+ 41u
3
+ 6u
2
+ 4b + 3u + 5,
− u
8
− 2u
7
− 8u
6
− 13u
5
− 17u
4
− 21u
3
+ 4a + 5u + 3,
u
9
+ 2u
8
+ 9u
7
+ 14u
6
+ 24u
5
+ 27u
4
+ 15u
3
+ 5u
2
+ 3u + 1i
I
u
2
= h−347u
13
− 980u
12
+ ··· + 877b − 863, −2540u
13
− 9801u
12
+ ··· + 877a − 12943,
u
14
+ 4u
13
+ ··· + 19u + 1i
I
u
3
= h−au + b − a − u − 1, a
2
+ 2a + 2, u
2
+ u + 1i
I
u
4
= hb − u, a − u + 1, u
2
− u + 1i
I
u
5
= hau + b − u − 1, a
2
+ 2au − u, u
2
+ u + 1i
* 5 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
= h3u
8
+4u
7
+· · ·+ 4b + 5, −u
8
−2u
7
+· · ·+ 4a + 3, u
9
+2u
8
+· · ·+ 3u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
3
=
u
2
+ 1
−u
2
a
5
=
−u
u
3
+ u
a
8
=
1
4
u
8
+
1
2
u
7
+ ··· −
5
4
u −
3
4
−
3
4
u
8
− u
7
+ ··· −
3
4
u −
5
4
a
11
=
1
4
u
8
+
1
4
u
7
+ ··· +
1
4
u + 2
−u
a
7
=
u
8
+
5
4
u
7
+ ··· − 2u +
1
4
−
1
4
u
8
−
1
4
u
7
+ ··· +
1
4
u −
1
2
a
4
=
−
3
4
u
8
−
3
2
u
7
+ ··· −
5
4
u −
1
4
1
4
u
8
+
1
4
u
7
+ ··· −
3
2
u
2
+
1
4
u
a
10
=
1
4
u
8
+
1
4
u
7
+ ··· +
5
4
u + 2
−u
a
9
=
1
4
u
8
+
1
4
u
7
+ ··· +
5
4
u + 2
−
1
4
u
7
−
1
4
u
6
+ ··· −
3
2
u −
1
4
a
9
=
1
4
u
8
+
1
4
u
7
+ ··· +
5
4
u + 2
−
1
4
u
7
−
1
4
u
6
+ ··· −
3
2
u −
1
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
3
2
u
8
− 10u
6
+
5
2
u
5
−
27
2
u
4
+
19
2
u
3
+ 17u
2
+
3
2
u +
13
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
10
u
9
− 2u
8
+ 9u
7
− 14u
6
+ 24u
5
− 27u
4
+ 15u
3
− 5u
2
+ 3u − 1
c
2
u
9
+ 14u
8
+ 73u
7
+ 158u
6
+ 76u
5
− 99u
4
+ 71u
3
+ 11u
2
− u − 1
c
3
u
9
+ 14u
7
− 26u
6
+ 44u
5
− 169u
4
+ 122u
3
+ 114u
2
− 57u − 31
c
4
u
9
+ 6u
7
− 6u
6
+ 24u
5
− 19u
4
+ 34u
3
− 20u
2
+ 15u + 1
c
6
, c
9
u
9
− 5u
8
+ 14u
7
− 25u
6
+ 35u
5
− 39u
4
+ 38u
3
− 27u
2
+ 16u − 4
c
7
, c
8
, c
11
u
9
− u
7
+ 4u
5
+ u
4
− 3u
3
+ u
2
+ u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
10
y
9
+ 14y
8
+ 73y
7
+ 158y
6
+ 76y
5
− 99y
4
+ 71y
3
+ 11y
2
− y −1
c
2
y
9
− 50y
8
+ ··· + 23y −1
c
3
y
9
+ 28y
8
+ ··· + 10317y −961
c
4
y
9
+ 12y
8
+ ··· + 265y −1
c
6
, c
9
y
9
+ 3y
8
+ ··· + 40y −16
c
7
, c
8
, c
11
y
9
− 2y
8
+ 9y
7
− 14y
6
+ 24y
5
− 27y
4
+ 15y
3
− 5y
2
+ 3y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.727682 + 0.313317I
a = −0.701346 + 1.009510I
b = 0.871765 − 0.179703I
3.76690 − 2.81495I 13.8794 + 4.7349I
u = −0.727682 − 0.313317I
a = −0.701346 − 1.009510I
b = 0.871765 + 0.179703I
3.76690 + 2.81495I 13.8794 − 4.7349I
u = −0.478419
a = −0.563873
b = −0.691679
1.19447 7.73920
u = 0.170878 + 0.444157I
a = −1.35805 − 1.02770I
b = −0.390522 + 0.568670I
0.55350 − 1.83926I 2.90943 + 3.36389I
u = 0.170878 − 0.444157I
a = −1.35805 + 1.02770I
b = −0.390522 − 0.568670I
0.55350 + 1.83926I 2.90943 − 3.36389I
u = 0.14897 + 1.92931I
a = 0.300783 − 0.966751I
b = 0.945009 − 1.020790I
−12.44850 − 2.94293I 2.46663 + 2.24617I
u = 0.14897 − 1.92931I
a = 0.300783 + 0.966751I
b = 0.945009 + 1.020790I
−12.44850 + 2.94293I 2.46663 − 2.24617I
u = −0.35296 + 1.94993I
a = 0.040547 + 1.255940I
b = −1.080410 + 0.902403I
−11.3859 − 11.4316I 3.87498 + 6.27440I
u = −0.35296 − 1.94993I
a = 0.040547 − 1.255940I
b = −1.080410 − 0.902403I
−11.3859 + 11.4316I 3.87498 − 6.27440I
5
II. I
u
2
= h−347u
13
− 980u
12
+ · · · + 877b − 863, −2540u
13
− 9801u
12
+ · · · +
877a − 12943, u
14
+ 4u
13
+ · · · + 19u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
3
=
u
2
+ 1
−u
2
a
5
=
−u
u
3
+ u
a
8
=
2.89624u
13
+ 11.1756u
12
+ ··· + 252.946u + 14.7583
0.395667u
13
+ 1.11745u
12
+ ··· + 20.7822u + 0.984036
a
11
=
−3.98632u
13
− 15.5063u
12
+ ··· − 401.345u − 30.6956
−0.391106u
13
− 1.25884u
12
+ ··· − 39.2098u − 3.02965
a
7
=
−1.95781u
13
− 7.57013u
12
+ ··· − 227.044u − 28.2623
0.101482u
13
+ 0.127708u
12
+ ··· − 27.8883u − 3.48575
a
4
=
3.96807u
13
+ 15.2657u
12
+ ··· + 389.676u + 42.5861
0.438997u
13
+ 1.34436u
12
+ ··· + 45.0445u + 4.98632
a
10
=
−3.59521u
13
− 14.2474u
12
+ ··· − 362.136u − 27.6659
−0.391106u
13
− 1.25884u
12
+ ··· − 39.2098u − 3.02965
a
9
=
−3.59521u
13
− 14.2474u
12
+ ··· − 362.136u − 27.6659
−0.409350u
13
− 1.68415u
12
+ ··· − 40.2702u − 2.89624
a
9
=
−3.59521u
13
− 14.2474u
12
+ ··· − 362.136u − 27.6659
−0.409350u
13
− 1.68415u
12
+ ··· − 40.2702u − 2.89624
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1470
877
u
13
+
4092
877
u
12
+ ··· +
81882
877
u +
15012
877
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
10
u
14
− 4u
13
+ ··· − 19u + 1
c
2
u
14
+ 20u
13
+ ··· − 119u + 1
c
3
u
14
+ 2u
13
+ ··· − 325u + 169
c
4
u
14
+ 2u
13
+ ··· − 35u + 71
c
6
, c
9
(u
7
+ u
6
+ u
5
− u
4
+ 2u
3
− 2u
2
+ u + 1)
2
c
7
, c
8
, c
11
u
14
− 2u
13
+ ··· + 3u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
10
y
14
+ 20y
13
+ ··· − 119y + 1
c
2
y
14
− 36y
13
+ ··· − 3183y + 1
c
3
y
14
+ 24y
13
+ ··· + 94809y + 28561
c
4
y
14
+ 12y
13
+ ··· + 15957y + 5041
c
6
, c
9
(y
7
+ y
6
+ 7y
5
+ 9y
4
+ 2y
2
+ 5y −1)
2
c
7
, c
8
, c
11
y
14
− 4y
13
+ ··· − 19y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.043461 + 1.144030I
a = −0.160092 − 0.469191I
b = 1.114750 − 0.491580I
−1.06225 − 5.14002I 3.39387 + 6.24395I
u = 0.043461 − 1.144030I
a = −0.160092 + 0.469191I
b = 1.114750 + 0.491580I
−1.06225 + 5.14002I 3.39387 − 6.24395I
u = −0.555192 + 1.007120I
a = 0.060823 − 1.111620I
b = 0.332695 − 0.054624I
1.80997 − 2.06468I 8.36726 + 2.56334I
u = −0.555192 − 1.007120I
a = 0.060823 + 1.111620I
b = 0.332695 + 0.054624I
1.80997 + 2.06468I 8.36726 − 2.56334I
u = −0.607165 + 1.075310I
a = 0.087467 − 0.856283I
b = −0.959701 − 0.560232I
−0.224468 2.93248 + 0.I
u = −0.607165 − 1.075310I
a = 0.087467 + 0.856283I
b = −0.959701 + 0.560232I
−0.224468 2.93248 + 0.I
u = −1.00102 + 1.09598I
a = 0.213982 + 0.982289I
b = −0.742091 + 0.770818I
−1.06225 − 5.14002I 3.39387 + 6.24395I
u = −1.00102 − 1.09598I
a = 0.213982 − 0.982289I
b = −0.742091 − 0.770818I
−1.06225 + 5.14002I 3.39387 − 6.24395I
u = 0.36666 + 1.79136I
a = −0.101694 + 1.285900I
b = 1.032640 + 0.962970I
−12.15000 + 4.31290I 2.77263 − 1.98970I
u = 0.36666 − 1.79136I
a = −0.101694 − 1.285900I
b = 1.032640 − 0.962970I
−12.15000 − 4.31290I 2.77263 + 1.98970I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.1077020 + 0.0363463I
a = −7.32712 + 5.78823I
b = −0.923363 + 0.545351I
1.80997 − 2.06468I 8.36726 + 2.56334I
u = −0.1077020 − 0.0363463I
a = −7.32712 − 5.78823I
b = −0.923363 − 0.545351I
1.80997 + 2.06468I 8.36726 − 2.56334I
u = −0.13904 + 1.98881I
a = −0.273361 − 1.021490I
b = −0.854936 − 1.047650I
−12.15000 − 4.31290I 2.77263 + 1.98970I
u = −0.13904 − 1.98881I
a = −0.273361 + 1.021490I
b = −0.854936 + 1.047650I
−12.15000 + 4.31290I 2.77263 − 1.98970I
10
III. I
u
3
= h−au + b − a − u − 1, a
2
+ 2a + 2, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
5
=
−u
u + 1
a
8
=
a
au + a + u + 1
a
11
=
−au − a − 2u − 1
−u
a
7
=
−u − 1
au + u
a
4
=
−au − 2u + 1
−au − a + 1
a
10
=
−au − a − u − 1
−u
a
9
=
−au − a − u − 1
−au − 2u
a
9
=
−au − a − u − 1
−au − 2u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u + 12
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
10
(u
2
+ u + 1)
2
c
3
, c
4
u
4
+ 2u
3
+ 2u
2
− 2u + 1
c
5
(u
2
− u + 1)
2
c
6
, c
9
(u
2
+ 1)
2
c
7
, c
8
, c
11
u
4
− u
2
+ 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
10
(y
2
+ y + 1)
2
c
3
, c
4
y
4
+ 14y
2
+ 1
c
6
, c
9
(y + 1)
4
c
7
, c
8
, c
11
(y
2
− y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −1.00000 + 1.00000I
b = −0.866025 + 0.500000I
1.64493 − 4.05977I 8.00000 + 6.92820I
u = −0.500000 + 0.866025I
a = −1.00000 − 1.00000I
b = 0.866025 − 0.500000I
1.64493 − 4.05977I 8.00000 + 6.92820I
u = −0.500000 − 0.866025I
a = −1.00000 + 1.00000I
b = 0.866025 + 0.500000I
1.64493 + 4.05977I 8.00000 − 6.92820I
u = −0.500000 − 0.866025I
a = −1.00000 − 1.00000I
b = −0.866025 − 0.500000I
1.64493 + 4.05977I 8.00000 − 6.92820I
14
IV. I
u
4
= hb − u, a − u + 1, u
2
− u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
−u + 1
a
3
=
u
−u + 1
a
5
=
−u
u − 1
a
8
=
u − 1
u
a
11
=
0
u − 1
a
7
=
u − 1
u + 1
a
4
=
0
1
a
10
=
−u + 1
u − 1
a
9
=
−u + 1
−1
a
9
=
−u + 1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
8
, c
10
, c
11
u
2
+ u + 1
c
6
, c
9
(u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
7
c
8
, c
10
, c
11
y
2
+ y + 1
c
6
, c
9
(y −1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = 0.500000 + 0.866025I
−3.28987 0
u = 0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = 0.500000 − 0.866025I
−3.28987 0
18
V. I
u
5
= hau + b − u − 1, a
2
+ 2au − u, u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u + 1
a
3
=
−u
u + 1
a
5
=
−u
u + 1
a
8
=
a
−au + u + 1
a
11
=
−au − a + u + 2
u + 1
a
7
=
−u − 1
−au − a + 1
a
4
=
au + a − u
a + u
a
10
=
−au − a + 1
u + 1
a
9
=
−au − a + 1
−au + 2u + 2
a
9
=
−au − a + 1
−au + 2u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
10
(u
2
+ u + 1)
2
c
3
u
4
− 4u
3
+ 5u
2
− 2u + 1
c
4
u
4
+ 2u
3
+ 5u
2
+ 4u + 1
c
5
(u
2
− u + 1)
2
c
6
, c
9
(u
2
+ 1)
2
c
7
, c
8
, c
11
u
4
− u
2
+ 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
10
(y
2
+ y + 1)
2
c
3
y
4
− 6y
3
+ 11y
2
+ 6y + 1
c
4
y
4
+ 6y
3
+ 11y
2
− 6y + 1
c
6
, c
9
(y + 1)
4
c
7
, c
8
, c
11
(y
2
− y + 1)
2
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.500000 + 0.133975I
b = 0.866025 + 0.500000I
1.64493 8.00000
u = −0.500000 + 0.866025I
a = 0.50000 − 1.86603I
b = −0.866025 − 0.500000I
1.64493 8.00000
u = −0.500000 − 0.866025I
a = 0.500000 − 0.133975I
b = 0.866025 − 0.500000I
1.64493 8.00000
u = −0.500000 − 0.866025I
a = 0.50000 + 1.86603I
b = −0.866025 + 0.500000I
1.64493 8.00000
22
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
10
(u
2
+ u + 1)
5
· (u
9
− 2u
8
+ 9u
7
− 14u
6
+ 24u
5
− 27u
4
+ 15u
3
− 5u
2
+ 3u − 1)
· (u
14
− 4u
13
+ ··· − 19u + 1)
c
2
(u
2
+ u + 1)
5
· (u
9
+ 14u
8
+ 73u
7
+ 158u
6
+ 76u
5
− 99u
4
+ 71u
3
+ 11u
2
− u − 1)
· (u
14
+ 20u
13
+ ··· − 119u + 1)
c
3
(u
2
+ u + 1)(u
4
− 4u
3
+ 5u
2
− 2u + 1)(u
4
+ 2u
3
+ 2u
2
− 2u + 1)
· (u
9
+ 14u
7
− 26u
6
+ 44u
5
− 169u
4
+ 122u
3
+ 114u
2
− 57u − 31)
· (u
14
+ 2u
13
+ ··· − 325u + 169)
c
4
(u
2
+ u + 1)(u
4
+ 2u
3
+ 2u
2
− 2u + 1)(u
4
+ 2u
3
+ 5u
2
+ 4u + 1)
· (u
9
+ 6u
7
− 6u
6
+ 24u
5
− 19u
4
+ 34u
3
− 20u
2
+ 15u + 1)
· (u
14
+ 2u
13
+ ··· − 35u + 71)
c
5
(u
2
− u + 1)
4
(u
2
+ u + 1)
· (u
9
− 2u
8
+ 9u
7
− 14u
6
+ 24u
5
− 27u
4
+ 15u
3
− 5u
2
+ 3u − 1)
· (u
14
− 4u
13
+ ··· − 19u + 1)
c
6
, c
9
(u + 1)
2
(u
2
+ 1)
4
(u
7
+ u
6
+ u
5
− u
4
+ 2u
3
− 2u
2
+ u + 1)
2
· (u
9
− 5u
8
+ 14u
7
− 25u
6
+ 35u
5
− 39u
4
+ 38u
3
− 27u
2
+ 16u − 4)
c
7
, c
8
, c
11
(u
2
+ u + 1)(u
4
− u
2
+ 1)
2
(u
9
− u
7
+ 4u
5
+ u
4
− 3u
3
+ u
2
+ u − 1)
· (u
14
− 2u
13
+ ··· + 3u + 1)
23
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
10
(y
2
+ y + 1)
5
· (y
9
+ 14y
8
+ 73y
7
+ 158y
6
+ 76y
5
− 99y
4
+ 71y
3
+ 11y
2
− y −1)
· (y
14
+ 20y
13
+ ··· − 119y + 1)
c
2
((y
2
+ y + 1)
5
)(y
9
− 50y
8
+ ··· + 23y −1)
· (y
14
− 36y
13
+ ··· − 3183y + 1)
c
3
(y
2
+ y + 1)(y
4
+ 14y
2
+ 1)(y
4
− 6y
3
+ 11y
2
+ 6y + 1)
· (y
9
+ 28y
8
+ ··· + 10317y −961)
· (y
14
+ 24y
13
+ ··· + 94809y + 28561)
c
4
(y
2
+ y + 1)(y
4
+ 14y
2
+ 1)(y
4
+ 6y
3
+ 11y
2
− 6y + 1)
· (y
9
+ 12y
8
+ ··· + 265y −1)(y
14
+ 12y
13
+ ··· + 15957y + 5041)
c
6
, c
9
(y −1)
2
(y + 1)
8
(y
7
+ y
6
+ 7y
5
+ 9y
4
+ 2y
2
+ 5y −1)
2
· (y
9
+ 3y
8
+ ··· + 40y −16)
c
7
, c
8
, c
11
(y
2
− y + 1)
4
(y
2
+ y + 1)
· (y
9
− 2y
8
+ 9y
7
− 14y
6
+ 24y
5
− 27y
4
+ 15y
3
− 5y
2
+ 3y −1)
· (y
14
− 4y
13
+ ··· − 19y + 1)
24
