12a
0823
(K12a
0823
)
A knot diagram
1
Linearized knot diagam
4 5 8 2 9 11 12 3 1 7 6 10
Solving Sequence
7,10
11 6 12 8
1,4
2 3 9 5
c
10
c
6
c
11
c
7
c
12
c
1
c
3
c
9
c
5
c
2
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
76
+ 4u
75
+ ··· + b − 2, −2u
75
− 2u
74
+ ··· + a − 5u, u
77
+ 2u
76
+ ··· − 3u − 1i
I
u
2
= hb + u, a − u + 1, u
3
+ 2u + 1i
I
u
3
= h−u
3
+ b − 2u + 1, a − 1, u
4
− u
3
+ 2u
2
− 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 84 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
=
h2u
76
+4u
75
+· · ·+b −2, −2u
75
−2u
74
+· · ·+a −5u, u
77
+2u
76
+· · ·−3u −1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
u
u
3
+ u
a
12
=
u
2
+ 1
u
4
+ 2u
2
a
8
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
1
=
−u
4
− u
2
+ 1
u
4
+ 2u
2
a
4
=
2u
75
+ 2u
74
+ ··· + 10u
2
+ 5u
−2u
76
− 4u
75
+ ··· + 4u + 2
a
2
=
−u
75
− u
74
+ ··· − 9u
2
− 5u
u
76
+ 2u
75
+ ··· − u − 1
a
3
=
u
71
+ u
70
+ ··· + 7u + 1
−u
71
− 33u
69
+ ··· − 6u
2
− u
a
9
=
u
8
+ 3u
6
+ u
4
− 2u
2
+ 1
−u
8
− 4u
6
− 4u
4
a
5
=
u
19
+ 8u
17
+ 24u
15
+ 30u
13
+ 7u
11
− 10u
9
+ 4u
7
+ 6u
5
− 3u
3
+ 2u
−u
19
− 9u
17
− 32u
15
− 55u
13
− 43u
11
− 9u
9
− 4u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
76
+ 8u
75
+ ··· − 7u − 17
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
77
− 8u
76
+ ··· + 8u − 1
c
3
, c
8
u
77
+ u
76
+ ··· − 64u − 128
c
5
u
77
+ 2u
76
+ ··· − 29557u − 8221
c
6
, c
10
, c
11
u
77
+ 2u
76
+ ··· − 3u − 1
c
7
u
77
− 2u
76
+ ··· − 5736u − 1480
c
9
, c
12
u
77
− 12u
76
+ ··· − 1271u + 131
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
77
− 74y
76
+ ··· + 40y −1
c
3
, c
8
y
77
+ 45y
76
+ ··· + 12288y −16384
c
5
y
77
− 24y
76
+ ··· + 1924884845y −67584841
c
6
, c
10
, c
11
y
77
+ 72y
76
+ ··· + 17y −1
c
7
y
77
+ 24y
76
+ ··· − 8997104y −2190400
c
9
, c
12
y
77
+ 60y
76
+ ··· + 256185y −17161
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.191592 + 1.136970I
a = 1.25258 − 0.95288I
b = 0.250746 + 0.488141I
−7.29823 + 2.50311I 0
u = 0.191592 − 1.136970I
a = 1.25258 + 0.95288I
b = 0.250746 − 0.488141I
−7.29823 − 2.50311I 0
u = 0.648234 + 0.470762I
a = 1.50700 − 0.30976I
b = 0.10743 + 2.01764I
0.35199 − 2.14980I −12.14186 + 3.38026I
u = 0.648234 − 0.470762I
a = 1.50700 + 0.30976I
b = 0.10743 − 2.01764I
0.35199 + 2.14980I −12.14186 − 3.38026I
u = −0.698472 + 0.381933I
a = 1.38596 − 1.79815I
b = −2.15973 − 1.56032I
−5.29571 + 11.59140I −11.4531 − 8.4016I
u = −0.698472 − 0.381933I
a = 1.38596 + 1.79815I
b = −2.15973 + 1.56032I
−5.29571 − 11.59140I −11.4531 + 8.4016I
u = 0.168145 + 1.208940I
a = −1.04755 + 0.98371I
b = 0.544196 − 0.621463I
−0.375128 − 0.568928I 0
u = 0.168145 − 1.208940I
a = −1.04755 − 0.98371I
b = 0.544196 + 0.621463I
−0.375128 + 0.568928I 0
u = −0.543692 + 0.555330I
a = 0.05223 + 1.78509I
b = 2.07554 − 1.21742I
−4.61972 − 7.40639I −9.96869 + 2.60400I
u = −0.543692 − 0.555330I
a = 0.05223 − 1.78509I
b = 2.07554 + 1.21742I
−4.61972 + 7.40639I −9.96869 − 2.60400I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.674418 + 0.378910I
a = −0.51345 + 1.54986I
b = 2.19270 + 0.48087I
0.66148 + 7.43215I −8.44278 − 8.42340I
u = −0.674418 − 0.378910I
a = −0.51345 − 1.54986I
b = 2.19270 − 0.48087I
0.66148 − 7.43215I −8.44278 + 8.42340I
u = 0.663405 + 0.365765I
a = −0.41627 − 2.27327I
b = 1.86641 − 0.34084I
−2.04586 − 5.11569I −10.51933 + 5.70011I
u = 0.663405 − 0.365765I
a = −0.41627 + 2.27327I
b = 1.86641 + 0.34084I
−2.04586 + 5.11569I −10.51933 − 5.70011I
u = −0.193391 + 1.227550I
a = −0.055721 − 1.084680I
b = −1.309470 − 0.174136I
−2.23828 + 3.03724I 0
u = −0.193391 − 1.227550I
a = −0.055721 + 1.084680I
b = −1.309470 + 0.174136I
−2.23828 − 3.03724I 0
u = 0.636507 + 0.409924I
a = −0.048390 + 1.303560I
b = −1.261700 − 0.283097I
3.19162 − 3.00945I −2.81731 + 4.39327I
u = 0.636507 − 0.409924I
a = −0.048390 − 1.303560I
b = −1.261700 + 0.283097I
3.19162 + 3.00945I −2.81731 − 4.39327I
u = 0.581959 + 0.456608I
a = −0.857892 − 0.656816I
b = 1.210810 − 0.588589I
3.40919 − 0.96956I −2.16570 + 2.84354I
u = 0.581959 − 0.456608I
a = −0.857892 + 0.656816I
b = 1.210810 + 0.588589I
3.40919 + 0.96956I −2.16570 − 2.84354I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.530488 + 0.512932I
a = 0.53897 − 1.77641I
b = −1.88073 + 0.09967I
1.24122 − 3.41237I −6.71316 + 2.36836I
u = −0.530488 − 0.512932I
a = 0.53897 + 1.77641I
b = −1.88073 − 0.09967I
1.24122 + 3.41237I −6.71316 − 2.36836I
u = 0.205270 + 1.249720I
a = 0.558990 − 0.770110I
b = −0.751957 + 1.042090I
0.05560 − 5.43305I 0
u = 0.205270 − 1.249720I
a = 0.558990 + 0.770110I
b = −0.751957 − 1.042090I
0.05560 + 5.43305I 0
u = −0.639232 + 0.359723I
a = −0.341288 − 0.577957I
b = −1.35130 + 0.64186I
−0.67724 + 2.50744I −11.34570 − 3.59018I
u = −0.639232 − 0.359723I
a = −0.341288 + 0.577957I
b = −1.35130 − 0.64186I
−0.67724 − 2.50744I −11.34570 + 3.59018I
u = −0.665128 + 0.278108I
a = −0.319540 − 0.999371I
b = −0.008826 − 0.427772I
−8.17542 + 0.47611I −14.7158 − 2.7445I
u = −0.665128 − 0.278108I
a = −0.319540 + 0.999371I
b = −0.008826 + 0.427772I
−8.17542 − 0.47611I −14.7158 + 2.7445I
u = 0.244033 + 1.257390I
a = −0.490522 + 0.142450I
b = 0.92817 − 1.27652I
−6.42273 − 9.14754I 0
u = 0.244033 − 1.257390I
a = −0.490522 − 0.142450I
b = 0.92817 + 1.27652I
−6.42273 + 9.14754I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.140039 + 1.287520I
a = 0.024976 + 0.385611I
b = 0.504759 − 0.003724I
3.01135 + 2.29522I 0
u = −0.140039 − 1.287520I
a = 0.024976 − 0.385611I
b = 0.504759 + 0.003724I
3.01135 − 2.29522I 0
u = 0.501793 + 0.490701I
a = 0.860902 + 0.742398I
b = −1.82723 − 0.06586I
−1.44185 + 1.23982I −8.66150 + 0.75675I
u = 0.501793 − 0.490701I
a = 0.860902 − 0.742398I
b = −1.82723 + 0.06586I
−1.44185 − 1.23982I −8.66150 − 0.75675I
u = 0.030612 + 1.300700I
a = −1.43632 − 0.09108I
b = 0.806088 + 1.056020I
1.81589 − 0.96118I 0
u = 0.030612 − 1.300700I
a = −1.43632 + 0.09108I
b = 0.806088 − 1.056020I
1.81589 + 0.96118I 0
u = −0.317735 + 0.605450I
a = 0.780897 − 0.702370I
b = 0.427338 − 0.189042I
−6.84784 + 3.07044I −11.28075 − 3.47607I
u = −0.317735 − 0.605450I
a = 0.780897 + 0.702370I
b = 0.427338 + 0.189042I
−6.84784 − 3.07044I −11.28075 + 3.47607I
u = 0.677966 + 0.062750I
a = −1.315350 + 0.451550I
b = −0.615406 + 0.909035I
−10.48850 − 5.79061I −17.0367 + 4.4441I
u = 0.677966 − 0.062750I
a = −1.315350 − 0.451550I
b = −0.615406 − 0.909035I
−10.48850 + 5.79061I −17.0367 − 4.4441I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.508655 + 0.424322I
a = −1.11823 + 1.18156I
b = 0.738302 + 1.005860I
−0.212543 + 1.167780I −10.21646 − 3.31814I
u = −0.508655 − 0.424322I
a = −1.11823 − 1.18156I
b = 0.738302 − 1.005860I
−0.212543 − 1.167780I −10.21646 + 3.31814I
u = −0.046345 + 1.356970I
a = 0.429786 + 0.365022I
b = 0.054673 − 0.996286I
4.57102 + 1.91273I 0
u = −0.046345 − 1.356970I
a = 0.429786 − 0.365022I
b = 0.054673 + 0.996286I
4.57102 − 1.91273I 0
u = −0.634684
a = 2.01220
b = 1.27931
−5.94766 −16.5240
u = 0.630519 + 0.033395I
a = 0.592203 + 0.279484I
b = 0.148921 − 0.802727I
−3.86408 − 2.37823I −15.7846 + 4.2598I
u = 0.630519 − 0.033395I
a = 0.592203 − 0.279484I
b = 0.148921 + 0.802727I
−3.86408 + 2.37823I −15.7846 − 4.2598I
u = −0.25087 + 1.40762I
a = 0.0478365 + 0.1135190I
b = −0.127222 + 0.654262I
−2.79475 + 3.80266I 0
u = −0.25087 − 1.40762I
a = 0.0478365 − 0.1135190I
b = −0.127222 − 0.654262I
−2.79475 − 3.80266I 0
u = −0.07700 + 1.42792I
a = 0.476070 − 0.097068I
b = −1.006700 + 0.604162I
−0.55532 + 4.30754I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.07700 − 1.42792I
a = 0.476070 + 0.097068I
b = −1.006700 − 0.604162I
−0.55532 − 4.30754I 0
u = −0.20019 + 1.44614I
a = 2.10795 + 0.53113I
b = −0.92627 − 1.55636I
5.75987 + 3.83600I 0
u = −0.20019 − 1.44614I
a = 2.10795 − 0.53113I
b = −0.92627 + 1.55636I
5.75987 − 3.83600I 0
u = −0.24286 + 1.44306I
a = −1.29039 + 1.70660I
b = 1.72235 − 0.79745I
5.11778 + 5.74279I 0
u = −0.24286 − 1.44306I
a = −1.29039 − 1.70660I
b = 1.72235 + 0.79745I
5.11778 − 5.74279I 0
u = 0.18621 + 1.45304I
a = −2.92205 − 0.73186I
b = 2.10609 − 0.10486I
4.73087 − 1.28332I 0
u = 0.18621 − 1.45304I
a = −2.92205 + 0.73186I
b = 2.10609 + 0.10486I
4.73087 + 1.28332I 0
u = 0.25081 + 1.44678I
a = 2.48049 + 1.77440I
b = −1.99607 + 0.46680I
3.78033 − 8.45891I 0
u = 0.25081 − 1.44678I
a = 2.48049 − 1.77440I
b = −1.99607 − 0.46680I
3.78033 + 8.45891I 0
u = −0.25374 + 1.45283I
a = 2.99683 − 1.30974I
b = −2.52211 − 0.52460I
6.55336 + 10.82390I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.25374 − 1.45283I
a = 2.99683 + 1.30974I
b = −2.52211 + 0.52460I
6.55336 − 10.82390I 0
u = −0.18475 + 1.46518I
a = −2.66257 + 1.18028I
b = 2.11134 + 0.37230I
7.57007 − 0.82609I 0
u = −0.18475 − 1.46518I
a = −2.66257 − 1.18028I
b = 2.11134 − 0.37230I
7.57007 + 0.82609I 0
u = 0.23586 + 1.45906I
a = −1.42551 − 1.47383I
b = 1.45011 + 0.16039I
9.20898 − 6.20501I 0
u = 0.23586 − 1.45906I
a = −1.42551 + 1.47383I
b = 1.45011 − 0.16039I
9.20898 + 6.20501I 0
u = 0.21070 + 1.46383I
a = 2.24094 − 0.01125I
b = −1.43080 + 0.77753I
9.58297 − 3.87222I 0
u = 0.21070 − 1.46383I
a = 2.24094 + 0.01125I
b = −1.43080 − 0.77753I
9.58297 + 3.87222I 0
u = −0.26298 + 1.45689I
a = −3.77652 + 0.35669I
b = 2.35692 + 1.70683I
0.6203 + 15.0997I 0
u = −0.26298 − 1.45689I
a = −3.77652 − 0.35669I
b = 2.35692 − 1.70683I
0.6203 − 15.0997I 0
u = −0.17434 + 1.47906I
a = 2.32909 − 2.44851I
b = −2.38054 + 0.94253I
1.93159 − 4.85744I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.17434 − 1.47906I
a = 2.32909 + 2.44851I
b = −2.38054 − 0.94253I
1.93159 + 4.85744I 0
u = −0.500819
a = −0.741044
b = −0.370645
−0.969871 −9.81070
u = 0.22825 + 1.48172I
a = −1.26848 + 2.30843I
b = −0.22862 − 2.16336I
6.66843 − 5.34198I 0
u = 0.22825 − 1.48172I
a = −1.26848 − 2.30843I
b = −0.22862 + 2.16336I
6.66843 + 5.34198I 0
u = −0.258129 + 0.312221I
a = −0.91822 + 1.08073I
b = 0.070880 + 0.493977I
−0.483753 + 0.975967I −7.83160 − 6.63357I
u = −0.258129 − 0.312221I
a = −0.91822 − 1.08073I
b = 0.070880 − 0.493977I
−0.483753 − 0.975967I −7.83160 + 6.63357I
u = 0.276652
a = 2.84998
b = −0.686858
−2.04721 0.166320
12
II. I
u
2
= hb + u, a − u + 1, u
3
+ 2u + 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
u
−u − 1
a
12
=
u
2
+ 1
−u
a
8
=
u
2
− u
−u
2
a
1
=
u
2
+ u + 1
−u
a
4
=
u − 1
−u
a
2
=
u
2
+ 2u
−2u
a
3
=
u − 1
−u
a
9
=
u
2
− u
−u
2
a
5
=
−u
2
− u − 1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7u
2
+ 5u − 18
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
9
u
3
+ 2u − 1
c
7
u
3
− 3u
2
+ 5u − 2
c
10
, c
11
, c
12
u
3
+ 2u + 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
3
c
3
, c
8
y
3
c
5
, c
6
, c
9
c
10
, c
11
, c
12
y
3
+ 4y
2
+ 4y − 1
c
7
y
3
+ y
2
+ 13y − 4
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.22670 + 1.46771I
a = −0.77330 + 1.46771I
b = −0.22670 − 1.46771I
7.79580 − 5.13794I −2.14701 + 2.68036I
u = 0.22670 − 1.46771I
a = −0.77330 − 1.46771I
b = −0.22670 + 1.46771I
7.79580 + 5.13794I −2.14701 − 2.68036I
u = −0.453398
a = −1.45340
b = 0.453398
−2.43213 −21.7060
16
III. I
u
3
= h−u
3
+ b − 2u + 1, a − 1, u
4
− u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
u
u
3
+ u
a
12
=
u
2
+ 1
u
3
+ 2u − 1
a
8
=
−u
3
− 2u + 1
−u
3
+ u
2
− u + 2
a
1
=
−u
3
+ u
2
− 2u + 2
u
3
+ 2u − 1
a
4
=
1
u
3
+ 2u − 1
a
2
=
−u
3
+ u
2
− 2u + 3
2u
3
+ 4u − 2
a
3
=
1
u
3
+ 2u − 1
a
9
=
−u
3
− 2u + 1
−u
3
+ u
2
− u + 2
a
5
=
u
3
− u
2
+ 2u − 2
−u
3
− 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
3
+ 2u
2
+ 2u − 7
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
8
u
4
c
4
(u + 1)
4
c
5
, c
6
, c
9
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
7
(u
2
+ u + 1)
2
c
10
, c
11
, c
12
u
4
− u
3
+ 2u
2
− 2u + 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
4
c
3
, c
8
y
4
c
5
, c
6
, c
9
c
10
, c
11
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
c
7
(y
2
+ y + 1)
2
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.621744 + 0.440597I
a = 1.00000
b = 0.121744 + 1.306620I
1.64493 − 2.02988I −5.73686 + 3.25323I
u = 0.621744 − 0.440597I
a = 1.00000
b = 0.121744 − 1.306620I
1.64493 + 2.02988I −5.73686 − 3.25323I
u = −0.121744 + 1.306620I
a = 1.00000
b = −0.621744 + 0.440597I
1.64493 + 2.02988I −8.76314 − 4.54099I
u = −0.121744 − 1.306620I
a = 1.00000
b = −0.621744 − 0.440597I
1.64493 − 2.02988I −8.76314 + 4.54099I
20
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
7
)(u
77
− 8u
76
+ ··· + 8u − 1)
c
3
, c
8
u
7
(u
77
+ u
76
+ ··· − 64u − 128)
c
4
((u + 1)
7
)(u
77
− 8u
76
+ ··· + 8u − 1)
c
5
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
77
+ 2u
76
+ ··· − 29557u − 8221)
c
6
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
77
+ 2u
76
+ ··· − 3u − 1)
c
7
((u
2
+ u + 1)
2
)(u
3
− 3u
2
+ 5u − 2)(u
77
− 2u
76
+ ··· − 5736u − 1480)
c
9
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
77
− 12u
76
+ ··· − 1271u + 131)
c
10
, c
11
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
77
+ 2u
76
+ ··· − 3u − 1)
c
12
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
77
− 12u
76
+ ··· − 1271u + 131)
21
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y − 1)
7
)(y
77
− 74y
76
+ ··· + 40y − 1)
c
3
, c
8
y
7
(y
77
+ 45y
76
+ ··· + 12288y − 16384)
c
5
(y
3
+ 4y
2
+ 4y − 1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
77
− 24y
76
+ ··· + 1924884845y − 67584841)
c
6
, c
10
, c
11
(y
3
+ 4y
2
+ 4y − 1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
77
+ 72y
76
+ ··· + 17y − 1)
c
7
(y
2
+ y + 1)
2
(y
3
+ y
2
+ 13y − 4)
· (y
77
+ 24y
76
+ ··· − 8997104y − 2190400)
c
9
, c
12
(y
3
+ 4y
2
+ 4y − 1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
77
+ 60y
76
+ ··· + 256185y − 17161)
22
