12n
0547
(K12n
0547
)
A knot diagram
1
Linearized knot diagam
3 5 10 9 2 11 5 12 4 7 3 8
Solving Sequence
5,9
4 10 3 2
1,12
8 7 11 6
c
4
c
9
c
3
c
2
c
1
c
8
c
7
c
11
c
6
c
5
, c
10
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
18
+ u
17
+ ··· + b − 1, u
21
− 3u
20
+ ··· + 2a − 5, u
22
− 3u
21
+ ··· − 7u + 2i
I
u
2
= hu
9
a − u
10
+ ··· + b − 1,
2u
10
+ u
9
+ 11u
8
+ 4u
7
+ 20u
6
+ 4u
5
− u
3
a + 12u
4
− 2u
3
+ a
2
− 2au + 3u
2
− 3u + 5,
u
11
+ u
10
+ 6u
9
+ 5u
8
+ 12u
7
+ 8u
6
+ 8u
5
+ 3u
4
+ u
3
− u
2
+ 2u + 1i
I
u
3
= h−u
9
+ u
8
− 4u
7
+ 3u
6
− 5u
5
− u
3
− 4u
2
+ b + 1, −u
8
− 4u
6
− 5u
4
− 2u
2
+ a − 1,
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 54 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
18
+u
17
+· · ·+b−1, u
21
−3u
20
+· · ·+2a−5, u
22
−3u
21
+· · ·−7u+2i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
10
=
u
u
3
+ u
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
−u
4
− u
2
+ 1
u
4
+ 2u
2
a
1
=
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
u
12
+ 6u
10
+ 12u
8
+ 8u
6
+ u
4
+ 2u
2
a
12
=
−
1
2
u
21
+
3
2
u
20
+ ··· − 4u +
5
2
u
18
− u
17
+ ··· − u + 1
a
8
=
−
1
2
u
21
+
1
2
u
20
+ ··· − u −
1
2
−u
21
+ 2u
20
+ ··· − 3u + 1
a
7
=
1
2
u
21
−
3
2
u
20
+ ··· + 2u −
3
2
−u
21
+ 2u
20
+ ··· − 3u + 1
a
11
=
1
2
u
21
−
3
2
u
20
+ ··· + 4u −
1
2
−u
18
+ u
17
+ ··· + 2u − 1
a
6
=
−u
8
− 3u
6
− u
4
+ 2u
2
− 1
u
8
+ 4u
6
+ 4u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
21
+10u
20
−54u
19
+108u
18
−298u
17
+482u
16
−868u
15
+1122u
14
−1404u
13
+1378u
12
−
1148u
11
+ 700u
10
−226u
9
−136u
8
+ 252u
7
−198u
6
+ 68u
5
+ 44u
4
−74u
3
+ 56u
2
−34u +14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
22
+ 21u
21
+ ··· + 2639u + 576
c
2
, c
5
u
22
+ 3u
21
+ ··· + 55u + 24
c
3
, c
4
, c
9
u
22
− 3u
21
+ ··· − 7u + 2
c
6
, c
8
, c
10
c
12
u
22
+ 2u
20
+ ··· + u + 1
c
7
, c
11
u
22
+ 4u
21
+ ··· − 111u + 79
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
22
− 39y
21
+ ··· − 969313y + 331776
c
2
, c
5
y
22
+ 21y
21
+ ··· + 2639y + 576
c
3
, c
4
, c
9
y
22
+ 21y
21
+ ··· + 7y + 4
c
6
, c
8
, c
10
c
12
y
22
+ 4y
21
+ ··· + 11y + 1
c
7
, c
11
y
22
+ 36y
21
+ ··· + 103651y + 6241
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.010224 + 1.078500I
a = 0.551130 + 0.398952I
b = −1.328730 − 0.305483I
−1.00862 + 1.46670I 0.87483 − 4.74244I
u = 0.010224 − 1.078500I
a = 0.551130 − 0.398952I
b = −1.328730 + 0.305483I
−1.00862 − 1.46670I 0.87483 + 4.74244I
u = 0.645447 + 0.555123I
a = −1.67610 − 0.24813I
b = −0.075830 − 0.925613I
−7.27791 − 5.01880I 0.42259 + 1.50295I
u = 0.645447 − 0.555123I
a = −1.67610 + 0.24813I
b = −0.075830 + 0.925613I
−7.27791 + 5.01880I 0.42259 − 1.50295I
u = 0.721588 + 0.445135I
a = 0.20635 − 1.70531I
b = 0.57553 − 1.93709I
−6.88868 + 9.58963I 1.41263 − 7.09128I
u = 0.721588 − 0.445135I
a = 0.20635 + 1.70531I
b = 0.57553 + 1.93709I
−6.88868 − 9.58963I 1.41263 + 7.09128I
u = −0.674386 + 0.184008I
a = −0.568999 − 1.273370I
b = −0.64562 − 1.59538I
1.10963 − 4.53074I 4.38661 + 7.87378I
u = −0.674386 − 0.184008I
a = −0.568999 + 1.273370I
b = −0.64562 + 1.59538I
1.10963 + 4.53074I 4.38661 − 7.87378I
u = −0.157344 + 0.613642I
a = 0.998496 + 0.716026I
b = −0.337730 − 0.250909I
−0.85003 + 1.30702I −1.78276 − 3.10332I
u = −0.157344 − 0.613642I
a = 0.998496 − 0.716026I
b = −0.337730 + 0.250909I
−0.85003 − 1.30702I −1.78276 + 3.10332I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.261209 + 1.342740I
a = −0.524408 + 0.595373I
b = 2.12772 + 1.40190I
−3.69180 − 7.92861I −0.13268 + 8.49431I
u = −0.261209 − 1.342740I
a = −0.524408 − 0.595373I
b = 2.12772 − 1.40190I
−3.69180 + 7.92861I −0.13268 − 8.49431I
u = 0.587900 + 0.230513I
a = 0.564070 + 0.683652I
b = 0.173424 + 0.964850I
1.14825 + 1.09478I 3.74609 − 1.33769I
u = 0.587900 − 0.230513I
a = 0.564070 − 0.683652I
b = 0.173424 − 0.964850I
1.14825 − 1.09478I 3.74609 + 1.33769I
u = 0.22679 + 1.39926I
a = −0.435721 + 0.005462I
b = 0.901316 − 0.971219I
−4.08990 + 4.07620I −2.65002 − 1.63109I
u = 0.22679 − 1.39926I
a = −0.435721 − 0.005462I
b = 0.901316 + 0.971219I
−4.08990 − 4.07620I −2.65002 + 1.63109I
u = −0.06984 + 1.45133I
a = −0.328600 − 0.625258I
b = −0.008270 − 0.590017I
−7.21159 + 0.38553I −5.41030 − 1.50832I
u = −0.06984 − 1.45133I
a = −0.328600 + 0.625258I
b = −0.008270 + 0.590017I
−7.21159 − 0.38553I −5.41030 + 1.50832I
u = 0.26357 + 1.48721I
a = 0.667824 + 0.729458I
b = −1.59693 + 2.46692I
−13.1389 + 13.1870I −1.85056 − 6.97436I
u = 0.26357 − 1.48721I
a = 0.667824 − 0.729458I
b = −1.59693 − 2.46692I
−13.1389 − 13.1870I −1.85056 + 6.97436I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.20725 + 1.51244I
a = 0.795966 − 0.560386I
b = 0.215117 − 0.125032I
−14.0282 − 1.9396I −3.01643 + 1.43017I
u = 0.20725 − 1.51244I
a = 0.795966 + 0.560386I
b = 0.215117 + 0.125032I
−14.0282 + 1.9396I −3.01643 − 1.43017I
7
II.
I
u
2
= hu
9
a − u
10
+ · · · + b − 1, 2u
10
+ u
9
+ · · · + a
2
+ 5, u
11
+ u
10
+ · · · + 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
10
=
u
u
3
+ u
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
−u
4
− u
2
+ 1
u
4
+ 2u
2
a
1
=
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
u
10
+ u
9
+ 5u
8
+ 4u
7
+ 8u
6
+ 5u
5
+ 3u
4
+ 2u
3
− u
2
+ u + 1
a
12
=
a
−u
9
a + u
10
+ ··· − au + 1
a
8
=
u
10
+ u
9
+ ··· − u + 2
−u
9
a − u
8
a + ··· − a + 1
a
7
=
u
9
a + u
10
+ ··· + a + 1
−u
9
a − u
8
a + ··· − a + 1
a
11
=
u
9
a − u
10
+ ··· + a − 1
−u
9
a + u
10
+ ··· + u + 1
a
6
=
−u
8
− 3u
6
− u
4
+ 2u
2
− 1
u
8
+ 4u
6
+ 4u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
10
+ 4u
9
+ 24u
8
+ 16u
7
+ 44u
6
+ 16u
5
+ 20u
4
− 4u
3
− 4u
2
− 4u + 10
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
11
+ 15u
10
+ ··· + 6u − 1)
2
c
2
, c
5
(u
11
+ u
10
+ 8u
9
+ 7u
8
+ 22u
7
+ 16u
6
+ 24u
5
+ 13u
4
+ 9u
3
+ 3u
2
− 1)
2
c
3
, c
4
, c
9
(u
11
+ u
10
+ 6u
9
+ 5u
8
+ 12u
7
+ 8u
6
+ 8u
5
+ 3u
4
+ u
3
− u
2
+ 2u + 1)
2
c
6
, c
8
, c
10
c
12
u
22
+ u
21
+ ··· + 20u
3
+ 1
c
7
, c
11
u
22
+ u
21
+ ··· + 3324u + 5777
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
11
− 37y
10
+ ··· + 70y −1)
2
c
2
, c
5
(y
11
+ 15y
10
+ ··· + 6y −1)
2
c
3
, c
4
, c
9
(y
11
+ 11y
10
+ ··· + 6y −1)
2
c
6
, c
8
, c
10
c
12
y
22
+ 7y
21
+ ··· + 84y
2
+ 1
c
7
, c
11
y
22
+ 11y
21
+ ··· + 10487680y + 33373729
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.691368 + 0.499908I
a = −1.59688 + 0.02132I
b = −0.638758 + 0.617238I
−7.95553 − 2.30219I −0.32022 + 2.86330I
u = −0.691368 + 0.499908I
a = 0.40201 + 1.57042I
b = 0.73852 + 1.27823I
−7.95553 − 2.30219I −0.32022 + 2.86330I
u = −0.691368 − 0.499908I
a = −1.59688 − 0.02132I
b = −0.638758 − 0.617238I
−7.95553 + 2.30219I −0.32022 − 2.86330I
u = −0.691368 − 0.499908I
a = 0.40201 − 1.57042I
b = 0.73852 − 1.27823I
−7.95553 + 2.30219I −0.32022 − 2.86330I
u = −0.081634 + 1.321480I
a = 0.925705 + 0.046443I
b = −2.41105 − 0.00160I
−0.18031 − 1.62554I 1.42199 + 3.91435I
u = −0.081634 + 1.321480I
a = −0.661845 + 0.315235I
b = 0.12364 + 2.65077I
−0.18031 − 1.62554I 1.42199 + 3.91435I
u = −0.081634 − 1.321480I
a = 0.925705 − 0.046443I
b = −2.41105 + 0.00160I
−0.18031 + 1.62554I 1.42199 − 3.91435I
u = −0.081634 − 1.321480I
a = −0.661845 − 0.315235I
b = 0.12364 − 2.65077I
−0.18031 + 1.62554I 1.42199 − 3.91435I
u = 0.525209 + 0.369457I
a = 0.073929 − 0.488809I
b = −0.422255 + 0.248502I
1.26759 + 1.65848I 0.54419 − 4.72916I
u = 0.525209 + 0.369457I
a = 0.90630 + 1.48303I
b = −0.128441 + 1.396340I
1.26759 + 1.65848I 0.54419 − 4.72916I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.525209 − 0.369457I
a = 0.073929 + 0.488809I
b = −0.422255 − 0.248502I
1.26759 − 1.65848I 0.54419 + 4.72916I
u = 0.525209 − 0.369457I
a = 0.90630 − 1.48303I
b = −0.128441 − 1.396340I
1.26759 − 1.65848I 0.54419 + 4.72916I
u = 0.18554 + 1.42716I
a = −0.752259 − 0.261413I
b = 1.03684 − 1.52064I
−4.47712 + 4.26374I −2.95029 − 4.02329I
u = 0.18554 + 1.42716I
a = −0.003996 + 0.356321I
b = 0.624973 + 0.515135I
−4.47712 + 4.26374I −2.95029 − 4.02329I
u = 0.18554 − 1.42716I
a = −0.752259 + 0.261413I
b = 1.03684 + 1.52064I
−4.47712 − 4.26374I −2.95029 + 4.02329I
u = 0.18554 − 1.42716I
a = −0.003996 − 0.356321I
b = 0.624973 − 0.515135I
−4.47712 − 4.26374I −2.95029 + 4.02329I
u = −0.23988 + 1.50376I
a = 0.504184 − 0.804775I
b = −1.21612 − 1.84606I
−14.4695 − 5.6984I −3.54476 + 2.83577I
u = −0.23988 + 1.50376I
a = 0.629578 + 0.671456I
b = 0.849788 + 0.004381I
−14.4695 − 5.6984I −3.54476 + 2.83577I
u = −0.23988 − 1.50376I
a = 0.504184 + 0.804775I
b = −1.21612 + 1.84606I
−14.4695 + 5.6984I −3.54476 − 2.83577I
u = −0.23988 − 1.50376I
a = 0.629578 − 0.671456I
b = 0.849788 − 0.004381I
−14.4695 + 5.6984I −3.54476 − 2.83577I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.395736
a = −0.42672 + 2.63281I
b = 0.94286 + 1.41200I
3.92670 11.6980
u = −0.395736
a = −0.42672 − 2.63281I
b = 0.94286 − 1.41200I
3.92670 11.6980
13
III. I
u
3
=
h−u
9
+u
8
+· · ·+b+1, −u
8
−4u
6
−5u
4
−2u
2
+a−1, u
10
+5u
8
+8u
6
+3u
4
−u
2
+1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
10
=
u
u
3
+ u
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
−u
4
− u
2
+ 1
u
4
+ 2u
2
a
1
=
0
−u
8
− 3u
6
− u
4
+ 2u
2
− 1
a
12
=
u
8
+ 4u
6
+ 5u
4
+ 2u
2
+ 1
u
9
− u
8
+ 4u
7
− 3u
6
+ 5u
5
+ u
3
+ 4u
2
− 1
a
8
=
u
9
+ 5u
7
+ 8u
5
+ 3u
3
− u
−u
8
+ u
7
− 4u
6
+ 3u
5
− 4u
4
+ 2u
3
− u − 1
a
7
=
u
9
+ u
8
+ 4u
7
+ 4u
6
+ 5u
5
+ 4u
4
+ u
3
+ 1
−u
8
+ u
7
− 4u
6
+ 3u
5
− 4u
4
+ 2u
3
− u − 1
a
11
=
−u
9
+ u
8
− 4u
7
+ 4u
6
− 5u
5
+ 4u
4
− u
3
+ u + 1
u
9
+ 4u
7
+ 5u
5
+ u
4
+ 2u
3
+ 2u
2
+ u
a
6
=
u
8
+ 3u
6
+ u
4
− 2u
2
+ 1
−u
8
− 4u
6
− 4u
4
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
6
− 12u
4
− 8u
2
+ 8
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
2
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
3
, c
4
, c
9
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
c
5
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
6
, c
8
, c
10
c
12
(u
2
+ 1)
5
c
7
u
10
− 4u
9
+ ··· − 74u + 29
c
11
u
10
+ 4u
9
+ ··· + 74u + 29
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
c
2
, c
5
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
3
, c
4
, c
9
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
c
6
, c
8
, c
10
c
12
(y + 1)
10
c
7
, c
11
y
10
− 6y
9
+ ··· − 546y + 841
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.217740I
a = 0.821196
b = −1.98456 + 1.58802I
0.888787 6.51890
u = − 1.217740I
a = 0.821196
b = −1.98456 − 1.58802I
0.888787 6.51890
u = 0.549911 + 0.309916I
a = 0.77780 + 1.38013I
b = −0.52856 + 1.81098I
2.96077 + 1.53058I 7.48489 − 4.43065I
u = 0.549911 − 0.309916I
a = 0.77780 − 1.38013I
b = −0.52856 − 1.81098I
2.96077 − 1.53058I 7.48489 + 4.43065I
u = −0.549911 + 0.309916I
a = 0.77780 − 1.38013I
b = 0.586946 − 0.933592I
2.96077 − 1.53058I 7.48489 + 4.43065I
u = −0.549911 − 0.309916I
a = 0.77780 + 1.38013I
b = 0.586946 + 0.933592I
2.96077 + 1.53058I 7.48489 − 4.43065I
u = −0.21917 + 1.41878I
a = −0.688402 + 0.106340I
b = −0.13073 + 1.65202I
−2.58269 − 4.40083I 3.25569 + 3.49859I
u = −0.21917 − 1.41878I
a = −0.688402 − 0.106340I
b = −0.13073 − 1.65202I
−2.58269 + 4.40083I 3.25569 − 3.49859I
u = 0.21917 + 1.41878I
a = −0.688402 − 0.106340I
b = 2.05690 − 1.18661I
−2.58269 + 4.40083I 3.25569 − 3.49859I
u = 0.21917 − 1.41878I
a = −0.688402 + 0.106340I
b = 2.05690 + 1.18661I
−2.58269 − 4.40083I 3.25569 + 3.49859I
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
)(u
11
+ 15u
10
+ ··· + 6u − 1)
2
· (u
22
+ 21u
21
+ ··· + 2639u + 576)
c
2
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
· (u
11
+ u
10
+ 8u
9
+ 7u
8
+ 22u
7
+ 16u
6
+ 24u
5
+ 13u
4
+ 9u
3
+ 3u
2
− 1)
2
· (u
22
+ 3u
21
+ ··· + 55u + 24)
c
3
, c
4
, c
9
(u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1)
· (u
11
+ u
10
+ 6u
9
+ 5u
8
+ 12u
7
+ 8u
6
+ 8u
5
+ 3u
4
+ u
3
− u
2
+ 2u + 1)
2
· (u
22
− 3u
21
+ ··· − 7u + 2)
c
5
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
· (u
11
+ u
10
+ 8u
9
+ 7u
8
+ 22u
7
+ 16u
6
+ 24u
5
+ 13u
4
+ 9u
3
+ 3u
2
− 1)
2
· (u
22
+ 3u
21
+ ··· + 55u + 24)
c
6
, c
8
, c
10
c
12
((u
2
+ 1)
5
)(u
22
+ 2u
20
+ ··· + u + 1)(u
22
+ u
21
+ ··· + 20u
3
+ 1)
c
7
(u
10
− 4u
9
+ ··· − 74u + 29)(u
22
+ u
21
+ ··· + 3324u + 5777)
· (u
22
+ 4u
21
+ ··· − 111u + 79)
c
11
(u
10
+ 4u
9
+ ··· + 74u + 29)(u
22
+ u
21
+ ··· + 3324u + 5777)
· (u
22
+ 4u
21
+ ··· − 111u + 79)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
)(y
11
− 37y
10
+ ··· + 70y −1)
2
· (y
22
− 39y
21
+ ··· − 969313y + 331776)
c
2
, c
5
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
11
+ 15y
10
+ ··· + 6y −1)
2
· (y
22
+ 21y
21
+ ··· + 2639y + 576)
c
3
, c
4
, c
9
((y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
)(y
11
+ 11y
10
+ ··· + 6y −1)
2
· (y
22
+ 21y
21
+ ··· + 7y + 4)
c
6
, c
8
, c
10
c
12
((y + 1)
10
)(y
22
+ 4y
21
+ ··· + 11y + 1)(y
22
+ 7y
21
+ ··· + 84y
2
+ 1)
c
7
, c
11
(y
10
− 6y
9
+ ··· − 546y + 841)
· (y
22
+ 11y
21
+ ··· + 10487680y + 33373729)
· (y
22
+ 36y
21
+ ··· + 103651y + 6241)
19
