12n
0821
(K12n
0821
)
A knot diagram
1
Linearized knot diagam
4 5 11 9 3 12 1 4 6 3 6 7
Solving Sequence
6,11
12 7
1,4
3 5 2 10 9 8
c
11
c
6
c
12
c
3
c
5
c
2
c
10
c
9
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h5u
12
+ 17u
11
− 8u
10
− 70u
9
− 18u
8
+ 113u
7
+ 91u
6
− 46u
5
− 122u
4
− 31u
3
+ 55u
2
+ 2b + 23u + 12,
− 11u
12
− 36u
11
+ ··· + 2a − 27,
u
13
+ 5u
12
+ 4u
11
− 16u
10
− 26u
9
+ 15u
8
+ 53u
7
+ 22u
6
− 36u
5
− 45u
4
− u
3
+ 21u
2
+ 10u + 4i
I
u
2
= h−u
7
+ 5u
5
− u
4
− 7u
3
+ 3u
2
+ b + 2u − 1, u
7
− 5u
5
+ u
4
+ 7u
3
− 4u
2
+ a − 2u + 3,
u
8
− 6u
6
+ u
5
+ 11u
4
− 4u
3
− 6u
2
+ 3u + 1i
I
u
3
= h−a
3
u
2
− 10a
3
u + 2a
2
u
2
+ 8a
3
− 9a
2
u + 16u
2
a + 13a
2
− 14au + 19u
2
+ 29b − 12a − 13u − 36,
− 2a
3
u
2
+ a
4
+ a
3
u + 3a
2
u
2
+ 4a
3
− a
2
u + 12u
2
a − 5a
2
− 7au − 27a + u + 1, u
3
− u
2
− 2u + 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
= h5u
12
+ 17u
11
+ · · · + 2b + 12, −11u
12
− 36u
11
+ · · · + 2a − 27, u
13
+
5u
12
+ · · · + 10u + 4i
(i) Arc colorings
a
6
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
7
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
11
2
u
12
+ 18u
11
+ ··· + 24u +
27
2
−
5
2
u
12
−
17
2
u
11
+ ··· −
23
2
u − 6
a
3
=
3u
12
+
19
2
u
11
+ ··· +
25
2
u +
15
2
−
5
2
u
12
−
17
2
u
11
+ ··· −
23
2
u − 6
a
5
=
11
4
u
12
+
37
4
u
11
+ ··· +
49
4
u + 7
−
1
2
u
12
−
3
2
u
11
+ ··· −
3
2
u − 1
a
2
=
−
11
4
u
12
−
37
4
u
11
+ ··· −
53
4
u − 6
−
1
2
u
12
−
3
2
u
11
+ ··· −
3
2
u − 1
a
10
=
1
4
u
12
+
3
4
u
11
+ ··· −
1
4
u + 1
1
2
u
12
+
3
2
u
11
+ ··· +
3
2
u + 1
a
9
=
1
4
u
12
+
3
4
u
11
+ ··· −
1
4
u + 1
−
1
2
u
12
−
3
2
u
11
+ ··· −
5
2
u − 1
a
8
=
u
3
− 2u
−u
5
+ 3u
3
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −17u
12
− 55u
11
+ 30u
10
+ 220u
9
+ 44u
8
− 346u
7
− 267u
6
+
135u
5
+ 358u
4
+ 81u
3
− 157u
2
− 56u − 34
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
13
− u
12
+ ··· + 15u + 1
c
2
, c
5
u
13
+ 6u
12
+ ··· + 12u + 8
c
3
, c
4
, c
8
c
10
u
13
− u
12
+ ··· + u − 1
c
6
, c
7
, c
11
c
12
u
13
− 5u
12
+ ··· + 10u − 4
c
9
u
13
+ 15u
11
+ ··· − 15u
2
− 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
13
+ 39y
12
+ ··· + 155y −1
c
2
, c
5
y
13
− 14y
12
+ ··· + 80y −64
c
3
, c
4
, c
8
c
10
y
13
+ 7y
12
+ ··· − y −1
c
6
, c
7
, c
11
c
12
y
13
− 17y
12
+ ··· − 68y −16
c
9
y
13
+ 30y
12
+ ··· − 30y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.497615 + 0.876393I
a = −0.520432 − 0.143375I
b = −0.286884 − 1.048300I
5.56724 − 2.86079I 6.58762 + 4.73580I
u = −0.497615 − 0.876393I
a = −0.520432 + 0.143375I
b = −0.286884 + 1.048300I
5.56724 + 2.86079I 6.58762 − 4.73580I
u = 0.977918 + 0.258584I
a = −0.50072 − 1.42125I
b = −0.432880 + 0.770070I
3.49671 + 3.30133I 5.66986 − 7.29619I
u = 0.977918 − 0.258584I
a = −0.50072 + 1.42125I
b = −0.432880 − 0.770070I
3.49671 − 3.30133I 5.66986 + 7.29619I
u = −1.15240
a = −0.0663024
b = −0.413299
2.44636 4.36790
u = 1.276260 + 0.459752I
a = 0.350061 + 1.031200I
b = 0.82833 − 1.24672I
11.16010 + 7.44705I 5.81652 − 5.20775I
u = 1.276260 − 0.459752I
a = 0.350061 − 1.031200I
b = 0.82833 + 1.24672I
11.16010 − 7.44705I 5.81652 + 5.20775I
u = −0.158213 + 0.403429I
a = 0.826836 − 0.260199I
b = 0.295622 + 0.443698I
−0.016018 − 0.973727I −0.21220 + 6.98709I
u = −0.158213 − 0.403429I
a = 0.826836 + 0.260199I
b = 0.295622 − 0.443698I
−0.016018 + 0.973727I −0.21220 − 6.98709I
u = −1.71570 + 0.06763I
a = 0.12881 − 1.62879I
b = 0.519770 + 0.949390I
13.07970 − 4.61256I 6.03428 + 7.03944I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.71570 − 0.06763I
a = 0.12881 + 1.62879I
b = 0.519770 − 0.949390I
13.07970 + 4.61256I 6.03428 − 7.03944I
u = −1.80645 + 0.12280I
a = 0.24859 + 1.52769I
b = −1.21731 − 1.42694I
−17.2391 − 10.0928I 5.92000 + 4.03274I
u = −1.80645 − 0.12280I
a = 0.24859 − 1.52769I
b = −1.21731 + 1.42694I
−17.2391 + 10.0928I 5.92000 − 4.03274I
6
II. I
u
2
= h−u
7
+ 5u
5
− u
4
− 7u
3
+ 3u
2
+ b + 2u − 1, u
7
− 5u
5
+ u
4
+ 7u
3
−
4u
2
+ a − 2u + 3, u
8
− 6u
6
+ u
5
+ 11u
4
− 4u
3
− 6u
2
+ 3u + 1i
(i) Arc colorings
a
6
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
7
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
−u
7
+ 5u
5
− u
4
− 7u
3
+ 4u
2
+ 2u − 3
u
7
− 5u
5
+ u
4
+ 7u
3
− 3u
2
− 2u + 1
a
3
=
u
2
− 2
u
7
− 5u
5
+ u
4
+ 7u
3
− 3u
2
− 2u + 1
a
5
=
−u
5
+ 4u
3
− 4u
−u
4
+ 3u
2
− 1
a
2
=
u
5
− u
4
− 4u
3
+ 3u
2
+ 3u − 1
u
4
− 3u
2
+ 1
a
10
=
u
7
− 6u
5
+ u
4
+ 10u
3
− 4u
2
− 3u + 3
−u
3
+ 2u − 1
a
9
=
u
7
− 6u
5
+ u
4
+ 10u
3
− 4u
2
− 3u + 3
u
5
− 4u
3
+ 3u − 1
a
8
=
−u
3
+ 2u
u
5
− 3u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
7
+ 2u
6
+ 21u
5
− 12u
4
− 42u
3
+ 24u
2
+ 17u − 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ 3u
7
+ 6u
6
+ 4u
5
− 7u
4
− 12u
3
− u
2
+ 6u + 1
c
2
u
8
+ 3u
7
− u
6
− 11u
5
− 6u
4
+ 11u
3
+ 8u
2
− 3u − 1
c
3
, c
8
u
8
+ u
7
− 2u
6
− 2u
5
− u
4
+ u
2
+ 1
c
4
, c
10
u
8
− u
7
− 2u
6
+ 2u
5
− u
4
+ u
2
+ 1
c
5
u
8
− 3u
7
− u
6
+ 11u
5
− 6u
4
− 11u
3
+ 8u
2
+ 3u − 1
c
6
, c
7
u
8
− 6u
6
− u
5
+ 11u
4
+ 4u
3
− 6u
2
− 3u + 1
c
9
u
8
+ u
6
− u
4
− 2u
3
− 2u
2
+ u + 1
c
11
, c
12
u
8
− 6u
6
+ u
5
+ 11u
4
− 4u
3
− 6u
2
+ 3u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
+ 3y
7
− 2y
6
− 30y
5
+ 99y
4
− 166y
3
+ 131y
2
− 38y + 1
c
2
, c
5
y
8
− 11y
7
+ 55y
6
− 159y
5
+ 278y
4
− 281y
3
+ 142y
2
− 25y + 1
c
3
, c
4
, c
8
c
10
y
8
− 5y
7
+ 6y
6
+ 2y
5
− y
4
− 6y
3
− y
2
+ 2y + 1
c
6
, c
7
, c
11
c
12
y
8
− 12y
7
+ 58y
6
− 145y
5
+ 203y
4
− 166y
3
+ 82y
2
− 21y + 1
c
9
y
8
+ 2y
7
− y
6
− 6y
5
− y
4
+ 2y
3
+ 6y
2
− 5y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.868162
a = 0.196614
b = −1.44291
−1.71749 5.61020
u = 0.733070 + 0.412657I
a = −1.46575 − 0.08392I
b = −0.167149 + 0.688931I
4.59844 + 1.46844I 5.92040 − 3.51787I
u = 0.733070 − 0.412657I
a = −1.46575 + 0.08392I
b = −0.167149 − 0.688931I
4.59844 − 1.46844I 5.92040 + 3.51787I
u = 1.35093
a = 0.698866
b = −0.873848
1.54653 −4.73980
u = 1.69498
a = −0.701727
b = 1.57470
7.46249 4.83590
u = −1.69932 + 0.10356I
a = 0.446197 − 1.151580I
b = 0.430778 + 0.799616I
13.38720 − 3.48023I 8.31022 + 1.19329I
u = −1.69932 − 0.10356I
a = 0.446197 + 1.151580I
b = 0.430778 − 0.799616I
13.38720 + 3.48023I 8.31022 − 1.19329I
u = −0.245247
a = −3.15466
b = 1.21480
−3.78433 −12.1680
10
III. I
u
3
=
h−a
3
u
2
+2a
2
u
2
+· · ·−12a−36, −2a
3
u
2
+3a
2
u
2
+· · ·−27a+1, u
3
−u
2
−2u+1i
(i) Arc colorings
a
6
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
7
=
u
−u
2
− u + 1
a
1
=
−u
2
+ 1
u
2
+ u − 1
a
4
=
a
0.0344828a
3
u
2
− 0.0689655a
2
u
2
+ ··· + 0.413793a + 1.24138
a
3
=
0.0344828a
3
u
2
− 0.0689655a
2
u
2
+ ··· + 1.41379a + 1.24138
0.0344828a
3
u
2
− 0.0689655a
2
u
2
+ ··· + 0.413793a + 1.24138
a
5
=
0.0344828a
3
u
2
− 0.0689655a
2
u
2
+ ··· + 1.41379a − 0.758621
0.379310a
3
u
2
− 0.758621a
2
u
2
+ ··· + 0.551724a − 0.344828
a
2
=
−0.0344828a
3
u
2
+ 0.0689655a
2
u
2
+ ··· − 1.41379a + 0.758621
−0.206897a
3
u
2
+ 0.413793a
2
u
2
+ ··· − 0.482759a + 0.551724
a
10
=
0.0344828a
3
u
2
− 0.0689655a
2
u
2
+ ··· + 1.41379a − 0.758621
−0.206897a
3
u
2
− 0.586207a
2
u
2
+ ··· − 0.482759a − 1.44828
a
9
=
0.0344828a
3
u
2
− 0.0689655a
2
u
2
+ ··· + 1.41379a − 0.758621
−0.379310a
3
u
2
− 0.241379a
2
u
2
+ ··· − 0.551724a − 1.65517
a
8
=
u
2
− 1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
12
− u
11
+ ··· − 42u − 1
c
2
, c
5
(u
2
− u − 1)
6
c
3
, c
4
, c
8
c
10
u
12
− u
11
+ u
9
+ 8u
8
+ u
7
− 7u
6
− 3u
5
− 6u
4
+ 10u
3
− 18u
2
+ 12u + 1
c
6
, c
7
, c
11
c
12
(u
3
+ u
2
− 2u − 1)
4
c
9
u
12
+ u
11
+ ··· + 84u − 29
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
12
+ 15y
11
+ ··· − 1900y + 1
c
2
, c
5
(y
2
− 3y + 1)
6
c
3
, c
4
, c
8
c
10
y
12
− y
11
+ ··· − 180y + 1
c
6
, c
7
, c
11
c
12
(y
3
− 5y
2
+ 6y −1)
4
c
9
y
12
+ 11y
11
+ ··· − 7288y + 841
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = 0.288735 + 1.074830I
b = 1.00883 − 1.07483I
10.2926 6.00000
u = −1.24698
a = 0.288735 − 1.074830I
b = 1.00883 + 1.07483I
10.2926 6.00000
u = −1.24698
a = −0.570245
b = 0.0746199
2.39690 6.00000
u = −1.24698
a = 0.349671
b = −0.845296
2.39690 6.00000
u = 0.445042
a = 0.0516489
b = 1.33706
−3.24287 6.00000
u = 0.445042
a = 2.45072
b = −1.06201
−3.24287 6.00000
u = 0.445042
a = −3.27564 + 0.82853I
b = −0.360046 − 0.828531I
4.65281 6.00000
u = 0.445042
a = −3.27564 − 0.82853I
b = −0.360046 + 0.828531I
4.65281 6.00000
u = 1.80194
a = −0.213846 + 1.148430I
b = 0.556829 − 1.148430I
13.6765 6.00000
u = 1.80194
a = −0.213846 − 1.148430I
b = 0.556829 + 1.148430I
13.6765 6.00000
14
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.80194
a = 0.55986 + 1.31903I
b = −1.45780 − 1.31903I
−17.9063 6.00000
u = 1.80194
a = 0.55986 − 1.31903I
b = −1.45780 + 1.31903I
−17.9063 6.00000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
8
+ 3u
7
+ 6u
6
+ 4u
5
− 7u
4
− 12u
3
− u
2
+ 6u + 1)
· (u
12
− u
11
+ ··· − 42u − 1)(u
13
− u
12
+ ··· + 15u + 1)
c
2
(u
2
− u − 1)
6
(u
8
+ 3u
7
− u
6
− 11u
5
− 6u
4
+ 11u
3
+ 8u
2
− 3u − 1)
· (u
13
+ 6u
12
+ ··· + 12u + 8)
c
3
, c
8
(u
8
+ u
7
− 2u
6
− 2u
5
− u
4
+ u
2
+ 1)
· (u
12
− u
11
+ u
9
+ 8u
8
+ u
7
− 7u
6
− 3u
5
− 6u
4
+ 10u
3
− 18u
2
+ 12u + 1)
· (u
13
− u
12
+ ··· + u − 1)
c
4
, c
10
(u
8
− u
7
− 2u
6
+ 2u
5
− u
4
+ u
2
+ 1)
· (u
12
− u
11
+ u
9
+ 8u
8
+ u
7
− 7u
6
− 3u
5
− 6u
4
+ 10u
3
− 18u
2
+ 12u + 1)
· (u
13
− u
12
+ ··· + u − 1)
c
5
(u
2
− u − 1)
6
(u
8
− 3u
7
− u
6
+ 11u
5
− 6u
4
− 11u
3
+ 8u
2
+ 3u − 1)
· (u
13
+ 6u
12
+ ··· + 12u + 8)
c
6
, c
7
(u
3
+ u
2
− 2u − 1)
4
(u
8
− 6u
6
− u
5
+ 11u
4
+ 4u
3
− 6u
2
− 3u + 1)
· (u
13
− 5u
12
+ ··· + 10u − 4)
c
9
(u
8
+ u
6
− u
4
− 2u
3
− 2u
2
+ u + 1)(u
12
+ u
11
+ ··· + 84u − 29)
· (u
13
+ 15u
11
+ ··· − 15u
2
− 1)
c
11
, c
12
(u
3
+ u
2
− 2u − 1)
4
(u
8
− 6u
6
+ u
5
+ 11u
4
− 4u
3
− 6u
2
+ 3u + 1)
· (u
13
− 5u
12
+ ··· + 10u − 4)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
8
+ 3y
7
− 2y
6
− 30y
5
+ 99y
4
− 166y
3
+ 131y
2
− 38y + 1)
· (y
12
+ 15y
11
+ ··· − 1900y + 1)(y
13
+ 39y
12
+ ··· + 155y −1)
c
2
, c
5
(y
2
− 3y + 1)
6
· (y
8
− 11y
7
+ 55y
6
− 159y
5
+ 278y
4
− 281y
3
+ 142y
2
− 25y + 1)
· (y
13
− 14y
12
+ ··· + 80y −64)
c
3
, c
4
, c
8
c
10
(y
8
− 5y
7
+ 6y
6
+ 2y
5
− y
4
− 6y
3
− y
2
+ 2y + 1)
· (y
12
− y
11
+ ··· − 180y + 1)(y
13
+ 7y
12
+ ··· − y −1)
c
6
, c
7
, c
11
c
12
(y
3
− 5y
2
+ 6y −1)
4
· (y
8
− 12y
7
+ 58y
6
− 145y
5
+ 203y
4
− 166y
3
+ 82y
2
− 21y + 1)
· (y
13
− 17y
12
+ ··· − 68y −16)
c
9
(y
8
+ 2y
7
− y
6
− 6y
5
− y
4
+ 2y
3
+ 6y
2
− 5y + 1)
· (y
12
+ 11y
11
+ ··· − 7288y + 841)(y
13
+ 30y
12
+ ··· − 30y −1)
17
