12a
0444
(K12a
0444
)
A knot diagram
1
Linearized knot diagam
3 6 10 9 2 12 11 5 4 8 1 7
Solving Sequence
4,9 2,5
6 10 3 1 8 11 7 12
c
4
c
5
c
9
c
3
c
1
c
8
c
10
c
7
c
12
c
2
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
23
− 2u
22
+ ··· + b − 1, u
24
+ 3u
23
+ ··· + 2a + 8u, u
25
+ 3u
24
+ ··· + 8u + 2i
I
u
2
= h2u
20
a − 2u
20
+ ··· + b + 1, −2u
20
a + 2u
20
+ ··· − 2a + 1, u
21
− u
20
+ ··· − u + 1i
I
u
3
= hb − u − 1, 2a + u, u
2
+ 2i
I
v
1
= ha, b + 1, v + 1i
* 4 irreducible components of dim
C
= 0, with total 70 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
23
−2u
22
+· · ·+b−1, u
24
+3u
23
+· · ·+2a+8u, u
25
+3u
24
+· · ·+8u+2i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
2
=
−
1
2
u
24
−
3
2
u
23
+ ··· − 6u
2
− 4u
u
23
+ 2u
22
+ ··· + 3u + 1
a
5
=
1
u
2
a
6
=
−
1
2
u
24
−
1
2
u
23
+ ··· − u + 1
−u
23
− 2u
22
+ ··· − 3u − 1
a
10
=
−u
u
a
3
=
u
2
+ 1
−u
2
a
1
=
−
3
2
u
24
−
9
2
u
23
+ ··· − 14u − 3
2u
23
+ 5u
22
+ ··· + 9u + 3
a
8
=
u
u
3
+ u
a
11
=
u
5
+ 2u
3
− u
u
7
+ 3u
5
+ 2u
3
+ u
a
7
=
u
9
+ 4u
7
+ 3u
5
− 2u
3
+ u
u
11
+ 5u
9
+ 8u
7
+ 5u
5
+ 3u
3
+ u
a
12
=
1
2
u
24
+
3
2
u
23
+ ··· + 3u + 1
−u
23
− 2u
22
+ ··· − 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −8u
24
− 18u
23
− 126u
22
− 242u
21
− 840u
20
− 1374u
19
−
3088u
18
− 4262u
17
− 6822u
16
− 7792u
15
− 9236u
14
− 8416u
13
− 7450u
12
− 5034u
11
−
3220u
10
− 1314u
9
− 430u
8
+ 106u
7
+ 170u
6
+ 114u
5
− 24u
4
− 78u
3
− 80u
2
− 54u − 24
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
25
+ 13u
24
+ ··· + 7u + 1
c
2
, c
5
, c
6
c
12
u
25
+ u
24
+ ··· + u + 1
c
3
, c
4
, c
8
c
9
u
25
+ 3u
24
+ ··· + 8u + 2
c
7
, c
10
u
25
+ 3u
24
+ ··· − 96u
2
+ 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
25
+ 3y
24
+ ··· + 15y −1
c
2
, c
5
, c
6
c
12
y
25
− 13y
24
+ ··· + 7y −1
c
3
, c
4
, c
8
c
9
y
25
+ 27y
24
+ ··· + 8y −4
c
7
, c
10
y
25
+ 19y
24
+ ··· + 3072y −256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.651480 + 0.569397I
a = −1.31726 − 1.62418I
b = −0.115293 + 0.307631I
−8.03966 + 11.38840I −12.3901 − 9.1803I
u = −0.651480 − 0.569397I
a = −1.31726 + 1.62418I
b = −0.115293 − 0.307631I
−8.03966 − 11.38840I −12.3901 + 9.1803I
u = −0.683025 + 0.436288I
a = 0.878062 + 0.233968I
b = −0.828869 − 0.938936I
−8.43694 − 6.90173I −13.44025 + 3.63036I
u = −0.683025 − 0.436288I
a = 0.878062 − 0.233968I
b = −0.828869 + 0.938936I
−8.43694 + 6.90173I −13.44025 − 3.63036I
u = 0.412040 + 0.685850I
a = −0.47621 + 1.83448I
b = 0.072745 − 0.306443I
−0.17864 − 6.77079I −7.18283 + 10.35931I
u = 0.412040 − 0.685850I
a = −0.47621 − 1.83448I
b = 0.072745 + 0.306443I
−0.17864 + 6.77079I −7.18283 − 10.35931I
u = −0.558289 + 0.498098I
a = 0.708276 + 0.173203I
b = 0.206889 + 0.374837I
−1.41690 + 1.92070I −6.23376 − 3.47212I
u = −0.558289 − 0.498098I
a = 0.708276 − 0.173203I
b = 0.206889 − 0.374837I
−1.41690 − 1.92070I −6.23376 + 3.47212I
u = 0.023881 + 0.737446I
a = 0.622644 − 0.953292I
b = 0.315147 + 0.069716I
1.88598 + 1.45733I −1.26812 − 4.21250I
u = 0.023881 − 0.737446I
a = 0.622644 + 0.953292I
b = 0.315147 − 0.069716I
1.88598 − 1.45733I −1.26812 + 4.21250I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.054153 + 1.332330I
a = −0.077174 − 0.173973I
b = 0.971993 + 0.051958I
2.62061 + 1.17903I −5.07770 − 5.84448I
u = 0.054153 − 1.332330I
a = −0.077174 + 0.173973I
b = 0.971993 − 0.051958I
2.62061 − 1.17903I −5.07770 + 5.84448I
u = 0.573959 + 0.177466I
a = 0.874794 − 0.104864I
b = −0.619515 + 0.565743I
−1.76413 + 3.37976I −11.30285 − 5.40492I
u = 0.573959 − 0.177466I
a = 0.874794 + 0.104864I
b = −0.619515 − 0.565743I
−1.76413 − 3.37976I −11.30285 + 5.40492I
u = −0.21436 + 1.46119I
a = 0.144093 + 0.127422I
b = 0.970978 + 0.088622I
−2.31754 − 3.68038I −10.19471 + 3.82630I
u = −0.21436 − 1.46119I
a = 0.144093 − 0.127422I
b = 0.970978 − 0.088622I
−2.31754 + 3.68038I −10.19471 − 3.82630I
u = −0.16059 + 1.52773I
a = −0.753431 − 0.916816I
b = 1.12967 + 1.67442I
5.30944 + 4.47743I −2.55629 − 2.34174I
u = −0.16059 − 1.52773I
a = −0.753431 + 0.916816I
b = 1.12967 − 1.67442I
5.30944 − 4.47743I −2.55629 + 2.34174I
u = −0.20643 + 1.54713I
a = 0.25239 + 1.93004I
b = −0.58907 − 4.16682I
−1.0459 + 14.5269I −8.91507 − 8.56336I
u = −0.20643 − 1.54713I
a = 0.25239 − 1.93004I
b = −0.58907 + 4.16682I
−1.0459 − 14.5269I −8.91507 + 8.56336I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.01772 + 1.57526I
a = −0.91307 + 1.41013I
b = 1.55989 − 2.89337I
9.63847 + 1.22771I −0.35980 − 3.25847I
u = 0.01772 − 1.57526I
a = −0.91307 − 1.41013I
b = 1.55989 + 2.89337I
9.63847 − 1.22771I −0.35980 + 3.25847I
u = 0.10120 + 1.57793I
a = −0.39162 − 1.95619I
b = 0.60654 + 4.12616I
7.45823 − 8.58001I −4.35516 + 8.14193I
u = 0.10120 − 1.57793I
a = −0.39162 + 1.95619I
b = 0.60654 − 4.12616I
7.45823 + 8.58001I −4.35516 − 8.14193I
u = −0.417568
a = 0.897008
b = −0.362213
−0.846371 −11.4470
7
II. I
u
2
=
h2u
20
a −2u
20
+· · ·+b +1, −2u
20
a +2u
20
+· · ·−2a +1, u
21
−u
20
+· · ·−u +1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
2
=
a
−2u
20
a + 2u
20
+ ··· − 3u − 1
a
5
=
1
u
2
a
6
=
−2u
20
a + 2u
20
+ ··· + a − 1
2u
20
a − 2u
20
+ ··· + 3u + 1
a
10
=
−u
u
a
3
=
u
2
+ 1
−u
2
a
1
=
−2u
20
a + 2u
20
+ ··· + a − 1
−2u
20
a + 2u
20
+ ··· − u − 1
a
8
=
u
u
3
+ u
a
11
=
u
5
+ 2u
3
− u
u
7
+ 3u
5
+ 2u
3
+ u
a
7
=
u
9
+ 4u
7
+ 3u
5
− 2u
3
+ u
u
11
+ 5u
9
+ 8u
7
+ 5u
5
+ 3u
3
+ u
a
12
=
−2u
20
a + 2u
20
+ ··· + a − 1
−u
17
a − 9u
15
a + ··· + 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
20
− 4u
19
+ 48u
18
− 40u
17
+ 232u
16
− 156u
15
+ 572u
14
− 292u
13
+ 756u
12
−
256u
11
+ 552u
10
− 88u
9
+ 316u
8
− 24u
7
+ 204u
6
− 8u
5
+ 48u
4
+ 20u
3
+ 16u
2
+ 16u − 10
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
42
+ 25u
41
+ ··· + 52u + 9
c
2
, c
5
, c
6
c
12
u
42
+ u
41
+ ··· + 8u + 3
c
3
, c
4
, c
8
c
9
(u
21
− u
20
+ ··· − u + 1)
2
c
7
, c
10
(u
21
+ 3u
20
+ ··· + 5u + 3)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
42
− 17y
41
+ ··· − 868y + 81
c
2
, c
5
, c
6
c
12
y
42
− 25y
41
+ ··· − 52y + 9
c
3
, c
4
, c
8
c
9
(y
21
+ 23y
20
+ ··· − 5y −1)
2
c
7
, c
10
(y
21
+ 19y
20
+ ··· + 7y −9)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.613284 + 0.552606I
a = 0.663670 − 0.167306I
b = 0.258836 − 0.415847I
−4.68217 − 6.45770I −9.45356 + 6.39068I
u = 0.613284 + 0.552606I
a = −1.31427 + 1.76259I
b = −0.095105 − 0.288417I
−4.68217 − 6.45770I −9.45356 + 6.39068I
u = 0.613284 − 0.552606I
a = 0.663670 + 0.167306I
b = 0.258836 + 0.415847I
−4.68217 + 6.45770I −9.45356 − 6.39068I
u = 0.613284 − 0.552606I
a = −1.31427 − 1.76259I
b = −0.095105 + 0.288417I
−4.68217 + 6.45770I −9.45356 − 6.39068I
u = −0.621912 + 0.497822I
a = 0.918873 + 0.252143I
b = −0.963307 − 0.936491I
−8.79207 + 2.11040I −13.9124 − 3.3898I
u = −0.621912 + 0.497822I
a = −1.51743 − 1.81016I
b = −0.112550 + 0.257173I
−8.79207 + 2.11040I −13.9124 − 3.3898I
u = −0.621912 − 0.497822I
a = 0.918873 − 0.252143I
b = −0.963307 + 0.936491I
−8.79207 − 2.11040I −13.9124 + 3.3898I
u = −0.621912 − 0.497822I
a = −1.51743 + 1.81016I
b = −0.112550 − 0.257173I
−8.79207 − 2.11040I −13.9124 + 3.3898I
u = 0.630060 + 0.435502I
a = 0.902520 − 0.223812I
b = −0.882881 + 0.878598I
−5.02710 + 2.23968I −10.49766 − 0.17506I
u = 0.630060 + 0.435502I
a = 0.706600 − 0.119856I
b = 0.159229 − 0.443860I
−5.02710 + 2.23968I −10.49766 − 0.17506I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.630060 − 0.435502I
a = 0.902520 + 0.223812I
b = −0.882881 − 0.878598I
−5.02710 − 2.23968I −10.49766 + 0.17506I
u = 0.630060 − 0.435502I
a = 0.706600 + 0.119856I
b = 0.159229 + 0.443860I
−5.02710 − 2.23968I −10.49766 + 0.17506I
u = −0.264535 + 0.686798I
a = 0.696463 + 0.484067I
b = 0.329117 + 0.122623I
1.27822 + 2.45481I −3.17392 − 5.13736I
u = −0.264535 + 0.686798I
a = 0.08311 − 1.80534I
b = 0.158741 + 0.236037I
1.27822 + 2.45481I −3.17392 − 5.13736I
u = −0.264535 − 0.686798I
a = 0.696463 − 0.484067I
b = 0.329117 − 0.122623I
1.27822 − 2.45481I −3.17392 + 5.13736I
u = −0.264535 − 0.686798I
a = 0.08311 + 1.80534I
b = 0.158741 − 0.236037I
1.27822 − 2.45481I −3.17392 + 5.13736I
u = 0.17161 + 1.47674I
a = −0.734220 + 0.822924I
b = 1.13685 − 1.39375I
1.167780 − 0.589478I −6.95446 − 0.27365I
u = 0.17161 + 1.47674I
a = 0.126348 − 0.107181I
b = 0.987004 − 0.067695I
1.167780 − 0.589478I −6.95446 − 0.27365I
u = 0.17161 − 1.47674I
a = −0.734220 − 0.822924I
b = 1.13685 + 1.39375I
1.167780 + 0.589478I −6.95446 + 0.27365I
u = 0.17161 − 1.47674I
a = 0.126348 + 0.107181I
b = 0.987004 + 0.067695I
1.167780 + 0.589478I −6.95446 + 0.27365I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.03893 + 1.51037I
a = 0.0956698 − 0.0258264I
b = 1.020350 − 0.014142I
3.29266 − 1.66521I −6.44233 + 3.90994I
u = 0.03893 + 1.51037I
a = −1.43429 − 2.27893I
b = 2.73302 + 4.65122I
3.29266 − 1.66521I −6.44233 + 3.90994I
u = 0.03893 − 1.51037I
a = 0.0956698 + 0.0258264I
b = 1.020350 + 0.014142I
3.29266 + 1.66521I −6.44233 − 3.90994I
u = 0.03893 − 1.51037I
a = −1.43429 + 2.27893I
b = 2.73302 − 4.65122I
3.29266 + 1.66521I −6.44233 − 3.90994I
u = −0.18541 + 1.51409I
a = 0.144635 + 0.094757I
b = 1.004590 + 0.080707I
−2.18398 + 5.00460I −10.15348 − 3.34739I
u = −0.18541 + 1.51409I
a = 0.31289 + 2.15828I
b = −0.68662 − 4.58559I
−2.18398 + 5.00460I −10.15348 − 3.34739I
u = −0.18541 − 1.51409I
a = 0.144635 − 0.094757I
b = 1.004590 − 0.080707I
−2.18398 − 5.00460I −10.15348 + 3.34739I
u = −0.18541 − 1.51409I
a = 0.31289 − 2.15828I
b = −0.68662 + 4.58559I
−2.18398 − 5.00460I −10.15348 + 3.34739I
u = 0.224591 + 0.416086I
a = 1.057800 − 0.095161I
b = −1.241430 + 0.325944I
−3.19863 − 0.86446I −9.82793 + 8.05526I
u = 0.224591 + 0.416086I
a = 0.94469 + 4.04424I
b = 0.0521710 − 0.1030170I
−3.19863 − 0.86446I −9.82793 + 8.05526I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.224591 − 0.416086I
a = 1.057800 + 0.095161I
b = −1.241430 − 0.325944I
−3.19863 + 0.86446I −9.82793 − 8.05526I
u = 0.224591 − 0.416086I
a = 0.94469 − 4.04424I
b = 0.0521710 + 0.1030170I
−3.19863 + 0.86446I −9.82793 − 8.05526I
u = −0.463882
a = 0.882798 + 0.014771I
b = −0.402171 − 0.224592I
−0.823381 −10.2590
u = −0.463882
a = 0.882798 − 0.014771I
b = −0.402171 + 0.224592I
−0.823381 −10.2590
u = 0.18830 + 1.54115I
a = −0.703578 + 0.924146I
b = 0.97403 − 1.67776I
2.24917 − 9.37044I −5.88057 + 5.65030I
u = 0.18830 + 1.54115I
a = 0.20144 − 2.02236I
b = −0.49031 + 4.33008I
2.24917 − 9.37044I −5.88057 + 5.65030I
u = 0.18830 − 1.54115I
a = −0.703578 − 0.924146I
b = 0.97403 + 1.67776I
2.24917 + 9.37044I −5.88057 − 5.65030I
u = 0.18830 − 1.54115I
a = 0.20144 + 2.02236I
b = −0.49031 − 4.33008I
2.24917 + 9.37044I −5.88057 − 5.65030I
u = −0.06297 + 1.57333I
a = −0.87578 − 1.19347I
b = 1.44224 + 2.40342I
8.90560 + 3.59224I −1.57394 − 3.20950I
u = −0.06297 + 1.57333I
a = −0.65794 + 1.88952I
b = 1.11820 − 3.94658I
8.90560 + 3.59224I −1.57394 − 3.20950I
14
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.06297 − 1.57333I
a = −0.87578 + 1.19347I
b = 1.44224 − 2.40342I
8.90560 − 3.59224I −1.57394 + 3.20950I
u = −0.06297 − 1.57333I
a = −0.65794 − 1.88952I
b = 1.11820 + 3.94658I
8.90560 − 3.59224I −1.57394 + 3.20950I
15
III. I
u
3
= hb − u − 1, 2a + u, u
2
+ 2i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
2
=
−
1
2
u
u + 1
a
5
=
1
−2
a
6
=
−
1
2
u + 1
u − 1
a
10
=
−u
u
a
3
=
−1
2
a
1
=
−
1
2
u + 1
u − 1
a
8
=
u
−u
a
11
=
−u
u
a
7
=
u
−u
a
12
=
−
3
2
u + 1
2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
11
c
12
(u − 1)
2
c
2
, c
6
(u + 1)
2
c
3
, c
4
, c
8
c
9
u
2
+ 2
c
7
, c
10
u
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
11
, c
12
(y −1)
2
c
3
, c
4
, c
8
c
9
(y + 2)
2
c
7
, c
10
y
2
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.414210I
a = − 0.707107I
b = 1.00000 + 1.41421I
1.64493 −12.0000
u = − 1.414210I
a = 0.707107I
b = 1.00000 − 1.41421I
1.64493 −12.0000
19
IV. I
v
1
= ha, b + 1, v + 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
−1
0
a
2
=
0
−1
a
5
=
1
0
a
6
=
1
1
a
10
=
−1
0
a
3
=
1
0
a
1
=
−1
−1
a
8
=
−1
0
a
11
=
−1
0
a
7
=
−1
0
a
12
=
−2
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
11
u − 1
c
3
, c
4
, c
7
c
8
, c
9
, c
10
u
c
5
, c
12
u + 1
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
11
, c
12
y − 1
c
3
, c
4
, c
7
c
8
, c
9
, c
10
y
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
23
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
11
((u − 1)
3
)(u
25
+ 13u
24
+ ··· + 7u + 1)(u
42
+ 25u
41
+ ··· + 52u + 9)
c
2
, c
6
(u − 1)(u + 1)
2
(u
25
+ u
24
+ ··· + u + 1)(u
42
+ u
41
+ ··· + 8u + 3)
c
3
, c
4
, c
8
c
9
u(u
2
+ 2)(u
21
− u
20
+ ··· − u + 1)
2
(u
25
+ 3u
24
+ ··· + 8u + 2)
c
5
, c
12
((u − 1)
2
)(u + 1)(u
25
+ u
24
+ ··· + u + 1)(u
42
+ u
41
+ ··· + 8u + 3)
c
7
, c
10
u
3
(u
21
+ 3u
20
+ ··· + 5u + 3)
2
(u
25
+ 3u
24
+ ··· − 96u
2
+ 16)
24
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
11
((y − 1)
3
)(y
25
+ 3y
24
+ ··· + 15y − 1)(y
42
− 17y
41
+ ··· − 868y + 81)
c
2
, c
5
, c
6
c
12
((y − 1)
3
)(y
25
− 13y
24
+ ··· + 7y − 1)(y
42
− 25y
41
+ ··· − 52y + 9)
c
3
, c
4
, c
8
c
9
y(y + 2)
2
(y
21
+ 23y
20
+ ··· − 5y − 1)
2
(y
25
+ 27y
24
+ ··· + 8y − 4)
c
7
, c
10
y
3
(y
21
+ 19y
20
+ ··· + 7y − 9)
2
(y
25
+ 19y
24
+ ··· + 3072y − 256)
25
