12a
0757
(K12a
0757
)
A knot diagram
1
Linearized knot diagam
3 8 10 11 12 9 2 6 7 1 5 4
Solving Sequence
4,11
5 12
6,9
7 1 8 10 3 2
c
4
c
11
c
5
c
6
c
12
c
8
c
10
c
3
c
2
c
1
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
85
− 38u
83
+ ··· + b + 1, u
84
− u
83
+ ··· + a + 2, u
86
− 2u
85
+ ··· − 4u − 1i
I
u
2
= hb, −u
4
− u
3
+ u
2
+ a + u, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 91 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
= hu
85
−38u
83
+· · ·+b +1, u
84
−u
83
+· · ·+a +2, u
86
−2u
85
+· · ·−4u −1i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
5
=
1
−u
2
a
12
=
u
−u
3
+ u
a
6
=
−u
2
+ 1
u
4
− 2u
2
a
9
=
−u
84
+ u
83
+ ··· − 3u − 2
−u
85
+ 38u
83
+ ··· − 3u − 1
a
7
=
u
84
− u
83
+ ··· + u
2
+ 2
−u
51
+ 23u
49
+ ··· − 2u
2
− u
a
1
=
−u
3
+ 2u
−u
3
+ u
a
8
=
−u
84
+ u
83
+ ··· − 5u − 3
−2u
85
+ 76u
83
+ ··· − 8u − 2
a
10
=
u
7
− 4u
5
+ 4u
3
u
7
− 3u
5
+ 2u
3
+ u
a
3
=
−u
14
+ 7u
12
− 18u
10
+ 19u
8
− 4u
6
− 4u
4
+ 1
−u
14
+ 6u
12
− 13u
10
+ 10u
8
+ 2u
6
− 4u
4
− u
2
a
2
=
−u
25
+ 12u
23
+ ··· − 2u
3
+ u
−u
25
+ 11u
23
+ ··· + 5u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6u
85
+ 4u
84
+ ··· − 25u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
86
+ 33u
85
+ ··· + 7680u + 1024
c
2
, c
7
u
86
+ u
85
+ ··· + 64u + 32
c
3
u
86
+ 2u
85
+ ··· − 3956u − 757
c
4
, c
5
, c
11
u
86
− 2u
85
+ ··· − 4u − 1
c
6
, c
8
, c
9
u
86
− 6u
85
+ ··· + 6u − 1
c
10
u
86
− 18u
85
+ ··· − 31954u + 1153
c
12
u
86
+ 6u
85
+ ··· + 48u + 5
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
86
+ 31y
85
+ ··· − 28966912y + 1048576
c
2
, c
7
y
86
− 33y
85
+ ··· − 7680y + 1024
c
3
y
86
− 6y
85
+ ··· − 13406188y + 573049
c
4
, c
5
, c
11
y
86
− 78y
85
+ ··· − 8y + 1
c
6
, c
8
, c
9
y
86
− 74y
85
+ ··· − 18y + 1
c
10
y
86
+ 30y
85
+ ··· − 95840184y + 1329409
c
12
y
86
− 6y
85
+ ··· − 324y + 25
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.091010 + 0.090090I
a = −0.956544 + 0.537574I
b = −0.493444 − 0.302436I
−0.575276 − 0.519687I 0
u = −1.091010 − 0.090090I
a = −0.956544 − 0.537574I
b = −0.493444 + 0.302436I
−0.575276 + 0.519687I 0
u = 1.094950 + 0.151003I
a = −1.23286 − 0.93748I
b = −2.64637 − 1.01452I
−1.57274 + 2.87671I 0
u = 1.094950 − 0.151003I
a = −1.23286 + 0.93748I
b = −2.64637 + 1.01452I
−1.57274 − 2.87671I 0
u = −0.855622 + 0.201501I
a = 0.719578 − 0.625350I
b = 2.47832 − 0.23964I
−8.00416 + 0.05306I −10.12496 + 0.65924I
u = −0.855622 − 0.201501I
a = 0.719578 + 0.625350I
b = 2.47832 + 0.23964I
−8.00416 − 0.05306I −10.12496 − 0.65924I
u = −1.110730 + 0.237283I
a = 0.953610 − 0.941499I
b = 2.19457 − 0.80602I
−4.00442 − 8.63773I 0
u = −1.110730 − 0.237283I
a = 0.953610 + 0.941499I
b = 2.19457 + 0.80602I
−4.00442 + 8.63773I 0
u = −1.121590 + 0.183537I
a = −0.313878 − 0.392217I
b = −0.511985 + 0.231496I
1.26195 − 4.79212I 0
u = −1.121590 − 0.183537I
a = −0.313878 + 0.392217I
b = −0.511985 − 0.231496I
1.26195 + 4.79212I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.204880 + 0.082857I
a = 0.377988 − 0.117095I
b = 0.653932 + 0.222912I
2.68827 + 0.56424I 0
u = 1.204880 − 0.082857I
a = 0.377988 + 0.117095I
b = 0.653932 − 0.222912I
2.68827 − 0.56424I 0
u = −0.304717 + 0.718199I
a = −3.02965 + 2.28213I
b = −2.85170 − 0.19298I
−4.47476 − 12.25720I −8.16453 + 9.33775I
u = −0.304717 − 0.718199I
a = −3.02965 − 2.28213I
b = −2.85170 + 0.19298I
−4.47476 + 12.25720I −8.16453 − 9.33775I
u = −0.644967 + 0.406996I
a = 0.81693 − 1.29819I
b = 2.40453 − 0.03748I
−3.18368 + 8.27801I −5.74606 − 4.15811I
u = −0.644967 − 0.406996I
a = 0.81693 + 1.29819I
b = 2.40453 + 0.03748I
−3.18368 − 8.27801I −5.74606 + 4.15811I
u = −0.302147 + 0.698378I
a = 1.69695 − 0.39760I
b = 1.200740 + 0.251003I
0.59979 − 7.98524I −4.28596 + 8.59628I
u = −0.302147 − 0.698378I
a = 1.69695 + 0.39760I
b = 1.200740 − 0.251003I
0.59979 + 7.98524I −4.28596 − 8.59628I
u = 0.290094 + 0.690020I
a = 3.62527 + 2.53458I
b = 3.21306 − 0.37602I
−2.54869 + 6.00860I −7.19871 − 6.21639I
u = 0.290094 − 0.690020I
a = 3.62527 − 2.53458I
b = 3.21306 + 0.37602I
−2.54869 − 6.00860I −7.19871 + 6.21639I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.213699 + 0.712661I
a = −3.75536 + 1.00291I
b = −2.50732 − 1.01411I
−10.08270 − 3.69819I −13.22796 + 4.35181I
u = −0.213699 − 0.712661I
a = −3.75536 − 1.00291I
b = −2.50732 + 1.01411I
−10.08270 + 3.69819I −13.22796 − 4.35181I
u = 0.370897 + 0.644380I
a = −0.213962 − 0.831526I
b = 0.070557 + 0.135650I
−1.75429 − 0.31789I −7.24883 + 0.38454I
u = 0.370897 − 0.644380I
a = −0.213962 + 0.831526I
b = 0.070557 − 0.135650I
−1.75429 + 0.31789I −7.24883 − 0.38454I
u = 0.317056 + 0.659169I
a = −1.69293 − 0.31547I
b = −1.113990 + 0.441879I
1.84865 + 2.71798I −1.15601 − 3.64002I
u = 0.317056 − 0.659169I
a = −1.69293 + 0.31547I
b = −1.113990 − 0.441879I
1.84865 − 2.71798I −1.15601 + 3.64002I
u = −0.285082 + 0.670702I
a = 0.291900 − 0.687149I
b = −0.001367 + 0.187746I
−1.84610 − 3.37653I −8.02440 + 5.55346I
u = −0.285082 − 0.670702I
a = 0.291900 + 0.687149I
b = −0.001367 − 0.187746I
−1.84610 + 3.37653I −8.02440 − 5.55346I
u = 0.492588 + 0.515342I
a = 0.002431 − 0.995227I
b = 0.206724 + 0.064894I
−1.23426 + 4.17652I −5.38905 − 6.75676I
u = 0.492588 − 0.515342I
a = 0.002431 + 0.995227I
b = 0.206724 − 0.064894I
−1.23426 − 4.17652I −5.38905 + 6.75676I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.594464 + 0.384370I
a = −0.364909 + 1.124990I
b = −0.818330 + 0.327095I
1.81169 + 4.17335I −1.36388 − 3.22734I
u = −0.594464 − 0.384370I
a = −0.364909 − 1.124990I
b = −0.818330 − 0.327095I
1.81169 − 4.17335I −1.36388 + 3.22734I
u = −0.088996 + 0.698267I
a = −3.20560 − 0.37370I
b = −1.60239 − 1.14493I
−7.07287 + 5.12939I −12.10062 − 3.05911I
u = −0.088996 − 0.698267I
a = −3.20560 + 0.37370I
b = −1.60239 + 1.14493I
−7.07287 − 5.12939I −12.10062 + 3.05911I
u = 1.296840 + 0.254244I
a = 1.74259 + 1.37380I
b = 1.00578 − 1.42157I
−2.76937 − 1.68516I 0
u = 1.296840 − 0.254244I
a = 1.74259 − 1.37380I
b = 1.00578 + 1.42157I
−2.76937 + 1.68516I 0
u = 0.573829 + 0.340250I
a = −0.46762 − 1.57866I
b = −2.49170 − 0.11337I
−1.28488 − 2.33578I −4.10416 + 0.70719I
u = 0.573829 − 0.340250I
a = −0.46762 + 1.57866I
b = −2.49170 + 0.11337I
−1.28488 + 2.33578I −4.10416 − 0.70719I
u = −0.195487 + 0.637733I
a = 1.57545 − 0.73490I
b = 0.760351 − 0.133230I
−2.97228 − 2.37551I −11.65767 + 5.29927I
u = −0.195487 − 0.637733I
a = 1.57545 + 0.73490I
b = 0.760351 + 0.133230I
−2.97228 + 2.37551I −11.65767 − 5.29927I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.498839 + 0.429548I
a = 0.095972 + 1.205710I
b = 0.747967 + 0.564715I
2.71612 + 0.92129I 1.22364 − 3.37489I
u = 0.498839 − 0.429548I
a = 0.095972 − 1.205710I
b = 0.747967 − 0.564715I
2.71612 − 0.92129I 1.22364 + 3.37489I
u = 0.149386 + 0.639664I
a = 4.28046 − 0.54291I
b = 1.84399 − 1.81743I
−4.29764 + 0.19506I −10.69913 − 1.71756I
u = 0.149386 − 0.639664I
a = 4.28046 + 0.54291I
b = 1.84399 + 1.81743I
−4.29764 − 0.19506I −10.69913 + 1.71756I
u = −0.102193 + 0.639114I
a = 0.170959 − 0.334698I
b = 0.022360 + 0.303744I
−1.74600 + 1.64794I −8.67678 − 2.62841I
u = −0.102193 − 0.639114I
a = 0.170959 + 0.334698I
b = 0.022360 − 0.303744I
−1.74600 − 1.64794I −8.67678 + 2.62841I
u = 1.342630 + 0.217708I
a = −0.181733 − 0.286949I
b = 0.358856 + 0.469214I
2.77719 + 1.39279I 0
u = 1.342630 − 0.217708I
a = −0.181733 + 0.286949I
b = 0.358856 − 0.469214I
2.77719 − 1.39279I 0
u = −1.358500 + 0.238616I
a = −2.58446 + 2.02139I
b = −1.43667 − 2.58596I
0.48014 − 3.36112I 0
u = −1.358500 − 0.238616I
a = −2.58446 − 2.02139I
b = −1.43667 + 2.58596I
0.48014 + 3.36112I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.39118
a = 1.49038
b = −1.25287
−1.58799 0
u = 1.376020 + 0.246751I
a = −0.511564 − 0.904596I
b = −0.925433 − 0.030697I
2.02789 + 5.59288I 0
u = 1.376020 − 0.246751I
a = −0.511564 + 0.904596I
b = −0.925433 + 0.030697I
2.02789 − 5.59288I 0
u = 0.258444 + 0.537961I
a = −0.824637 + 0.076619I
b = −0.204825 + 0.532118I
0.029281 + 1.335260I 0.16290 − 4.75932I
u = 0.258444 − 0.537961I
a = −0.824637 − 0.076619I
b = −0.204825 − 0.532118I
0.029281 − 1.335260I 0.16290 + 4.75932I
u = 1.379690 + 0.282039I
a = 1.40623 + 2.55472I
b = 2.38274 − 1.47835I
−5.02912 + 7.30246I 0
u = 1.379690 − 0.282039I
a = 1.40623 − 2.55472I
b = 2.38274 + 1.47835I
−5.02912 − 7.30246I 0
u = −0.484810 + 0.332092I
a = −0.282807 − 1.155640I
b = −0.315394 − 0.023660I
−0.688337 − 0.084123I −4.53467 − 0.27129I
u = −0.484810 − 0.332092I
a = −0.282807 + 1.155640I
b = −0.315394 + 0.023660I
−0.688337 + 0.084123I −4.53467 + 0.27129I
u = −1.40291 + 0.21886I
a = 0.629712 − 0.516662I
b = 0.184210 + 0.726781I
5.35900 − 4.16935I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.40291 − 0.21886I
a = 0.629712 + 0.516662I
b = 0.184210 − 0.726781I
5.35900 + 4.16935I 0
u = 1.42125 + 0.14963I
a = −0.171108 − 0.870282I
b = 0.134991 − 0.469562I
5.21320 + 1.97379I 0
u = 1.42125 − 0.14963I
a = −0.171108 + 0.870282I
b = 0.134991 + 0.469562I
5.21320 − 1.97379I 0
u = −1.42562 + 0.13409I
a = −1.76380 − 0.89594I
b = 2.09227 − 0.95884I
4.84148 + 0.63389I 0
u = −1.42562 − 0.13409I
a = −1.76380 + 0.89594I
b = 2.09227 + 0.95884I
4.84148 − 0.63389I 0
u = 1.41411 + 0.26293I
a = −0.194651 − 0.604284I
b = 0.211606 + 0.184132I
3.58503 + 6.78684I 0
u = 1.41411 − 0.26293I
a = −0.194651 + 0.604284I
b = 0.211606 − 0.184132I
3.58503 − 6.78684I 0
u = −1.41705 + 0.27010I
a = −0.48533 + 3.31578I
b = −3.42567 − 0.72914I
2.90698 − 9.50849I 0
u = −1.41705 − 0.27010I
a = −0.48533 − 3.31578I
b = −3.42567 + 0.72914I
2.90698 + 9.50849I 0
u = 1.43860 + 0.13075I
a = −0.429349 + 0.494991I
b = 0.793565 + 0.831374I
8.13195 − 2.40072I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.43860 − 0.13075I
a = −0.429349 − 0.494991I
b = 0.793565 − 0.831374I
8.13195 + 2.40072I 0
u = −1.43698 + 0.15851I
a = 0.623047 + 0.425490I
b = −0.754946 + 0.982953I
8.81344 − 3.06716I 0
u = −1.43698 − 0.15851I
a = 0.623047 − 0.425490I
b = −0.754946 − 0.982953I
8.81344 + 3.06716I 0
u = −1.42446 + 0.25593I
a = 0.593762 − 1.006570I
b = 1.32370 + 0.54326I
7.42079 − 6.06552I 0
u = −1.42446 − 0.25593I
a = 0.593762 + 1.006570I
b = 1.32370 − 0.54326I
7.42079 + 6.06552I 0
u = 1.42267 + 0.27266I
a = −0.527480 − 0.978161I
b = −1.40879 + 0.34846I
6.11525 + 11.52370I 0
u = 1.42267 − 0.27266I
a = −0.527480 + 0.978161I
b = −1.40879 − 0.34846I
6.11525 − 11.52370I 0
u = 1.42568 + 0.28091I
a = 0.29899 + 2.87831I
b = 3.01900 − 0.42804I
1.0585 + 15.8929I 0
u = 1.42568 − 0.28091I
a = 0.29899 − 2.87831I
b = 3.01900 + 0.42804I
1.0585 − 15.8929I 0
u = 1.45026 + 0.11671I
a = 1.37770 − 0.77158I
b = −2.03725 − 0.56570I
3.38763 − 6.58591I 0
12
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.45026 − 0.11671I
a = 1.37770 + 0.77158I
b = −2.03725 + 0.56570I
3.38763 + 6.58591I 0
u = −1.44132 + 0.24187I
a = 0.194375 − 0.685579I
b = −0.194750 + 0.044574I
4.05862 − 2.91828I 0
u = −1.44132 − 0.24187I
a = 0.194375 + 0.685579I
b = −0.194750 − 0.044574I
4.05862 + 2.91828I 0
u = −1.45150 + 0.17791I
a = 0.175028 − 0.800545I
b = −0.203825 − 0.257398I
4.96998 − 6.66919I 0
u = −1.45150 − 0.17791I
a = 0.175028 + 0.800545I
b = −0.203825 + 0.257398I
4.96998 + 6.66919I 0
u = −0.320908
a = −1.40782
b = −0.462488
−1.08070 −8.42340
13
II. I
u
2
= hb, −u
4
− u
3
+ u
2
+ a + u, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
5
=
1
−u
2
a
12
=
u
−u
3
+ u
a
6
=
−u
2
+ 1
u
4
− 2u
2
a
9
=
u
4
+ u
3
− u
2
− u
0
a
7
=
u
4
+ u
3
− 2u
2
− u + 1
u
4
− 2u
2
a
1
=
−u
3
+ 2u
−u
3
+ u
a
8
=
u
4
+ u
3
− 2u
2
− u + 1
u
4
− 2u
2
a
10
=
u
2
− 1
−u
4
+ 2u
2
a
3
=
−u
3
+ 2u
−u
3
+ u
a
2
=
−u
3
+ 2u
−u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
3
+ u
2
− 8u − 3
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u
5
c
3
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
4
, c
5
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
6
(u − 1)
5
c
8
, c
9
(u + 1)
5
c
10
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
11
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
12
u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
y
5
c
3
, c
10
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
4
, c
5
, c
11
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
6
, c
8
, c
9
(y −1)
5
c
12
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.21774
a = 1.30408
b = 0
0.756147 −2.23020
u = 0.309916 + 0.549911I
a = −0.428550 − 1.039280I
b = 0
−1.31583 + 1.53058I −6.94263 − 4.09764I
u = 0.309916 − 0.549911I
a = −0.428550 + 1.039280I
b = 0
−1.31583 − 1.53058I −6.94263 + 4.09764I
u = −1.41878 + 0.21917I
a = 0.276511 − 0.728237I
b = 0
4.22763 − 4.40083I −2.94226 + 4.18967I
u = −1.41878 − 0.21917I
a = 0.276511 + 0.728237I
b = 0
4.22763 + 4.40083I −2.94226 − 4.18967I
17
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
5
(u
86
+ 33u
85
+ ··· + 7680u + 1024)
c
2
, c
7
u
5
(u
86
+ u
85
+ ··· + 64u + 32)
c
3
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
86
+ 2u
85
+ ··· − 3956u − 757)
c
4
, c
5
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
86
− 2u
85
+ ··· − 4u − 1)
c
6
((u − 1)
5
)(u
86
− 6u
85
+ ··· + 6u − 1)
c
8
, c
9
((u + 1)
5
)(u
86
− 6u
85
+ ··· + 6u − 1)
c
10
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
86
− 18u
85
+ ··· − 31954u + 1153)
c
11
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
86
− 2u
85
+ ··· − 4u − 1)
c
12
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)(u
86
+ 6u
85
+ ··· + 48u + 5)
18
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
5
(y
86
+ 31y
85
+ ··· − 2.89669 × 10
7
y + 1048576)
c
2
, c
7
y
5
(y
86
− 33y
85
+ ··· − 7680y + 1024)
c
3
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
86
− 6y
85
+ ··· − 1.34062 × 10
7
y + 573049)
c
4
, c
5
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
86
− 78y
85
+ ··· − 8y + 1)
c
6
, c
8
, c
9
((y −1)
5
)(y
86
− 74y
85
+ ··· − 18y + 1)
c
10
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
· (y
86
+ 30y
85
+ ··· − 95840184y + 1329409)
c
12
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
86
− 6y
85
+ ··· − 324y + 25)
19
