12n
0142
(K12n
0142
)
A knot diagram
1
Linearized knot diagam
3 4 10 2 9 3 12 4 6 7 8 11
Solving Sequence
3,10 4,6
7 11 2 1 9 5 8 12
c
3
c
6
c
10
c
2
c
1
c
9
c
5
c
8
c
12
c
4
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h2.35580 × 10
24
u
34
+ 5.23556 × 10
24
u
33
+ ··· + 1.45032 × 10
25
b − 1.44449 × 10
25
,
2.40552 × 10
24
u
34
+ 8.47444 × 10
24
u
33
+ ··· + 7.25161 × 10
24
a + 1.23374 × 10
25
, u
35
+ 2u
34
+ ··· + 2u − 1i
I
u
2
= hb
4
+ 4b
3
u − 4b
3
− 4b
2
u + u − 1, a − u + 1, u
2
− u + 1i
I
u
3
= hb
3
− 3b
2
u − 3b
2
+ 3bu + 1, a + u + 1, u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 49 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
= h2.36 × 10
24
u
34
+ 5.24 × 10
24
u
33
+ · · · + 1.45 × 10
25
b − 1.44 × 10
25
, 2.41 ×
10
24
u
34
+8.47×10
24
u
33
+· · ·+7.25×10
24
a+1.23×10
25
, u
35
+2u
34
+· · ·+2u−1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
6
=
−0.331722u
34
− 1.16863u
33
+ ··· − 6.30773u − 1.70134
−0.162433u
34
− 0.360993u
33
+ ··· + 0.130097u + 0.995980
a
7
=
−0.494155u
34
− 1.52962u
33
+ ··· − 6.17763u − 0.705356
−0.162433u
34
− 0.360993u
33
+ ··· + 0.130097u + 0.995980
a
11
=
0.593442u
34
+ 0.968496u
33
+ ··· − 3.97361u + 2.63388
−0.474317u
34
− 1.15537u
33
+ ··· + 1.27477u + 0.0691574
a
2
=
u
2
+ 1
u
4
a
1
=
u
4
+ u
2
+ 1
u
4
a
9
=
1.46457u
34
+ 3.18756u
33
+ ··· − 3.14002u + 2.39592
−0.396812u
34
− 1.06370u
33
+ ··· − 0.108352u + 0.168805
a
5
=
u
4
+ u
2
+ 1
u
6
+ u
2
a
8
=
1.03514u
34
+ 2.01243u
33
+ ··· − 4.19611u + 2.30630
−0.466022u
34
− 1.17415u
33
+ ··· + 0.0947758u − 0.147474
a
12
=
−0.214267u
34
− 0.843549u
33
+ ··· − 2.91396u + 3.23369
−0.447509u
34
− 0.903401u
33
+ ··· + 3.98711u − 0.173391
(ii) Obstruction class = −1
(iii) Cusp Shapes =
293178633293666810040513
1812902632604430856362551
u
34
−
1485220871768859334081099
7251610530417723425450204
u
33
+ ··· −
22519370902193600570526111
1812902632604430856362551
u −
6630145935705280334996315
3625805265208861712725102
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
35
+ 54u
34
+ ··· + 132u − 1
c
2
, c
4
u
35
− 6u
34
+ ··· − 4u + 1
c
3
u
35
+ 2u
34
+ ··· + 2u − 1
c
5
, c
9
u
35
+ 3u
34
+ ··· − 83u + 13
c
6
u
35
+ 2u
34
+ ··· + 178664u + 28669
c
7
, c
11
u
35
+ u
34
+ ··· + 4u + 4
c
8
u
35
− 2u
34
+ ··· + 342120u + 112661
c
10
u
35
− u
34
+ ··· − 1020u + 404
c
12
u
35
− 15u
34
+ ··· − 80u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
35
− 138y
34
+ ··· + 4900y −1
c
2
, c
4
y
35
+ 54y
34
+ ··· + 132y −1
c
3
y
35
+ 6y
34
+ ··· − 4y −1
c
5
, c
9
y
35
− 55y
34
+ ··· − 1795y −169
c
6
y
35
+ 42y
34
+ ··· − 5590382760y −821911561
c
7
, c
11
y
35
+ 15y
34
+ ··· − 80y −16
c
8
y
35
− 90y
34
+ ··· − 117701124860y −12692500921
c
10
y
35
+ 15y
34
+ ··· − 4334416y −163216
c
12
y
35
+ 15y
34
+ ··· + 768y −256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.604506 + 0.837594I
a = −0.448438 − 0.357846I
b = 0.689783 + 0.188000I
−0.48340 + 2.36916I 1.57611 − 4.50521I
u = −0.604506 − 0.837594I
a = −0.448438 + 0.357846I
b = 0.689783 − 0.188000I
−0.48340 − 2.36916I 1.57611 + 4.50521I
u = 0.390242 + 0.879154I
a = 0.310445 − 0.133927I
b = −0.23755 + 1.43913I
2.56414 − 5.90559I 4.16775 + 8.40016I
u = 0.390242 − 0.879154I
a = 0.310445 + 0.133927I
b = −0.23755 − 1.43913I
2.56414 + 5.90559I 4.16775 − 8.40016I
u = 0.291312 + 0.893061I
a = −0.518425 + 1.133840I
b = 1.01576 − 1.21326I
−0.669990 − 1.198850I 0.589488 − 0.204869I
u = 0.291312 − 0.893061I
a = −0.518425 − 1.133840I
b = 1.01576 + 1.21326I
−0.669990 + 1.198850I 0.589488 + 0.204869I
u = 0.201059 + 0.882276I
a = 0.570013 − 0.061429I
b = −1.146370 + 0.781662I
3.35367 + 0.99605I 6.08668 + 0.22244I
u = 0.201059 − 0.882276I
a = 0.570013 + 0.061429I
b = −1.146370 − 0.781662I
3.35367 − 0.99605I 6.08668 − 0.22244I
u = −0.396959 + 0.761909I
a = 0.353852 − 0.098553I
b = −0.252788 − 0.797693I
0.24170 + 1.75473I −0.43635 − 4.44927I
u = −0.396959 − 0.761909I
a = 0.353852 + 0.098553I
b = −0.252788 + 0.797693I
0.24170 − 1.75473I −0.43635 + 4.44927I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.985569 + 0.598689I
a = −1.338090 + 0.075762I
b = 0.543514 − 1.234170I
−5.47439 − 0.97053I −6.95717 + 0.21297I
u = 0.985569 − 0.598689I
a = −1.338090 − 0.075762I
b = 0.543514 + 1.234170I
−5.47439 + 0.97053I −6.95717 − 0.21297I
u = −0.975167 + 0.760910I
a = −1.202040 + 0.021054I
b = 1.02372 + 1.38759I
−4.82985 + 6.52855I −5.30987 − 5.93880I
u = −0.975167 − 0.760910I
a = −1.202040 − 0.021054I
b = 1.02372 − 1.38759I
−4.82985 − 6.52855I −5.30987 + 5.93880I
u = −0.704227 + 1.101200I
a = −0.164791 − 1.088000I
b = −0.89159 + 1.41508I
−3.55483 − 0.18354I −5.27168 + 1.44891I
u = −0.704227 − 1.101200I
a = −0.164791 + 1.088000I
b = −0.89159 − 1.41508I
−3.55483 + 0.18354I −5.27168 − 1.44891I
u = 0.581677 + 1.185630I
a = −0.272346 + 1.132270I
b = −0.42884 − 2.03294I
−3.30526 − 4.99735I −4.49983 + 4.97284I
u = 0.581677 − 1.185630I
a = −0.272346 − 1.132270I
b = −0.42884 + 2.03294I
−3.30526 + 4.99735I −4.49983 − 4.97284I
u = −0.574590 + 0.262572I
a = 0.828257 − 0.616246I
b = 0.014775 − 0.612549I
−0.98087 + 1.09493I −5.80678 − 3.92220I
u = −0.574590 − 0.262572I
a = 0.828257 + 0.616246I
b = 0.014775 + 0.612549I
−0.98087 − 1.09493I −5.80678 + 3.92220I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.978178 + 0.990672I
a = 0.991907 + 0.901079I
b = −0.57751 − 1.46799I
−10.30070 + 3.59687I −2.12620 − 2.06254I
u = −0.978178 − 0.990672I
a = 0.991907 − 0.901079I
b = −0.57751 + 1.46799I
−10.30070 − 3.59687I −2.12620 + 2.06254I
u = −1.12687 + 0.87702I
a = 1.01739 + 0.99553I
b = 0.581005 − 1.085740I
−15.7264 − 5.1315I −4.99224 + 2.26947I
u = −1.12687 − 0.87702I
a = 1.01739 − 0.99553I
b = 0.581005 + 1.085740I
−15.7264 + 5.1315I −4.99224 − 2.26947I
u = 0.553000 + 0.079189I
a = 1.19162 − 0.95512I
b = 0.541286 − 0.641792I
0.51946 − 3.30014I −3.55846 + 2.28251I
u = 0.553000 − 0.079189I
a = 1.19162 + 0.95512I
b = 0.541286 + 0.641792I
0.51946 + 3.30014I −3.55846 − 2.28251I
u = 1.11285 + 0.95383I
a = 1.03608 − 0.96278I
b = 0.35648 + 1.52175I
−17.5409 − 1.1078I −6.65030 + 1.84113I
u = 1.11285 − 0.95383I
a = 1.03608 + 0.96278I
b = 0.35648 − 1.52175I
−17.5409 + 1.1078I −6.65030 − 1.84113I
u = −0.93478 + 1.13030I
a = 1.030020 + 0.817450I
b = −1.15562 − 2.30328I
−14.8325 + 12.6268I −3.98580 − 6.37716I
u = −0.93478 − 1.13030I
a = 1.030020 − 0.817450I
b = −1.15562 + 2.30328I
−14.8325 − 12.6268I −3.98580 + 6.37716I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.98846 + 1.10575I
a = 1.042800 − 0.852917I
b = −0.75564 + 2.24982I
−16.9917 − 6.5550I −6.22803 + 2.34101I
u = 0.98846 − 1.10575I
a = 1.042800 + 0.852917I
b = −0.75564 − 2.24982I
−16.9917 + 6.5550I −6.22803 − 2.34101I
u = −0.002637 + 0.401944I
a = −0.40921 − 2.57831I
b = 1.287540 + 0.119160I
0.99468 + 3.59879I −0.61353 − 4.53406I
u = −0.002637 − 0.401944I
a = −0.40921 + 2.57831I
b = 1.287540 − 0.119160I
0.99468 − 3.59879I −0.61353 + 4.53406I
u = 0.387505
a = −3.03811
b = 0.784090
−1.97384 −5.96760
8
II. I
u
2
= hb
4
+ 4b
3
u − 4b
3
− 4b
2
u + u − 1, a − u + 1, u
2
− u + 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
u − 1
a
6
=
u − 1
b
a
7
=
b + u − 1
b
a
11
=
b
2
u − 2b − u + 1
b
2
u − b + u
a
2
=
u
−u
a
1
=
0
−u
a
9
=
−u + 1
−b + u
a
5
=
0
u
a
8
=
−b − u + 1
−bu + u
a
12
=
2b
2
u − 4b
b
3
u − b
3
− b
2
u − b
2
− 2bu + b + u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4b
2
u + 4b
2
+ 8bu + 4u − 4
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
5
(u + 1)
8
c
6
u
8
+ 4u
7
+ 12u
6
+ 16u
5
+ 15u
4
− 8u
3
− 4u
2
+ 1
c
7
, c
11
(u
4
+ 2u
2
+ 2)
2
c
8
u
8
− 4u
7
+ 12u
6
− 16u
5
+ 15u
4
+ 8u
3
− 4u
2
+ 1
c
9
(u − 1)
8
c
10
(u
4
− 2u
2
+ 2)
2
c
12
(u
2
− 2u + 2)
4
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
(y
2
+ y + 1)
4
c
5
, c
9
(y −1)
8
c
6
, c
8
y
8
+ 8y
7
+ 46y
6
+ 160y
5
+ 387y
4
− 160y
3
+ 46y
2
− 8y + 1
c
7
, c
11
(y
2
+ 2y + 2)
4
c
10
(y
2
− 2y + 2)
4
c
12
(y
2
+ 4)
4
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = 0.344777 + 0.313008I
0.82247 − 5.69375I −2.00000 + 7.46410I
u = 0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = −0.443461 − 0.142082I
0.82247 + 1.63398I −2.00000 − 0.53590I
u = 0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = 1.44346 − 1.58997I
0.82247 + 1.63398I −2.00000 − 0.53590I
u = 0.500000 + 0.866025I
a = −0.500000 + 0.866025I
b = 0.65522 − 2.04506I
0.82247 − 5.69375I −2.00000 + 7.46410I
u = 0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = −0.443461 + 0.142082I
0.82247 + 5.69375I −2.00000 − 7.46410I
u = 0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = 0.344777 − 0.313008I
0.82247 − 1.63398I −2.00000 + 0.53590I
u = 0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = 1.44346 + 1.58997I
0.82247 − 1.63398I −2.00000 + 0.53590I
u = 0.500000 − 0.866025I
a = −0.500000 − 0.866025I
b = 0.65522 + 2.04506I
0.82247 + 5.69375I −2.00000 − 7.46410I
12
III. I
u
3
= hb
3
− 3b
2
u − 3b
2
+ 3bu + 1, a + u + 1, u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u − 1
a
6
=
−u − 1
b
a
7
=
b − u − 1
b
a
11
=
b
2
u + 2b − u − 1
b
2
u + b + u
a
2
=
−u
u
a
1
=
0
u
a
9
=
−u − 1
b + u
a
5
=
0
−u
a
8
=
b − u − 1
−bu + u
a
12
=
0
−b
2
+ 2bu + 2b
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2b
2
u − 2b
2
+ 4bu − 4u − 6
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
2
− u + 1)
3
c
2
, c
3
, c
6
c
8
(u
2
+ u + 1)
3
c
5
(u − 1)
6
c
7
, c
10
, c
11
c
12
u
6
c
9
(u + 1)
6
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
(y
2
+ y + 1)
3
c
5
, c
9
(y −1)
6
c
7
, c
10
, c
11
c
12
y
6
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = −0.500000 − 0.866025I
b = 0.500000 + 0.866025I
−1.64493 + 2.02988I −6.00000 − 3.46410I
u = −0.500000 + 0.866025I
a = −0.500000 − 0.866025I
b = 0.500000 + 0.866025I
−1.64493 + 2.02988I −6.00000 − 3.46410I
u = −0.500000 + 0.866025I
a = −0.500000 − 0.866025I
b = 0.500000 + 0.866025I
−1.64493 + 2.02988I −6.00000 − 3.46410I
u = −0.500000 − 0.866025I
a = −0.500000 + 0.866025I
b = 0.500000 − 0.866025I
−1.64493 − 2.02988I −6.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = −0.500000 + 0.866025I
b = 0.500000 − 0.866025I
−1.64493 − 2.02988I −6.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = −0.500000 + 0.866025I
b = 0.500000 − 0.866025I
−1.64493 − 2.02988I −6.00000 + 3.46410I
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
7
)(u
35
+ 54u
34
+ ··· + 132u − 1)
c
2
((u
2
+ u + 1)
7
)(u
35
− 6u
34
+ ··· − 4u + 1)
c
3
((u
2
− u + 1)
4
)(u
2
+ u + 1)
3
(u
35
+ 2u
34
+ ··· + 2u − 1)
c
4
((u
2
− u + 1)
7
)(u
35
− 6u
34
+ ··· − 4u + 1)
c
5
((u − 1)
6
)(u + 1)
8
(u
35
+ 3u
34
+ ··· − 83u + 13)
c
6
(u
2
+ u + 1)
3
(u
8
+ 4u
7
+ 12u
6
+ 16u
5
+ 15u
4
− 8u
3
− 4u
2
+ 1)
· (u
35
+ 2u
34
+ ··· + 178664u + 28669)
c
7
, c
11
u
6
(u
4
+ 2u
2
+ 2)
2
(u
35
+ u
34
+ ··· + 4u + 4)
c
8
(u
2
+ u + 1)
3
(u
8
− 4u
7
+ 12u
6
− 16u
5
+ 15u
4
+ 8u
3
− 4u
2
+ 1)
· (u
35
− 2u
34
+ ··· + 342120u + 112661)
c
9
((u − 1)
8
)(u + 1)
6
(u
35
+ 3u
34
+ ··· − 83u + 13)
c
10
u
6
(u
4
− 2u
2
+ 2)
2
(u
35
− u
34
+ ··· − 1020u + 404)
c
12
u
6
(u
2
− 2u + 2)
4
(u
35
− 15u
34
+ ··· − 80u + 16)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
7
)(y
35
− 138y
34
+ ··· + 4900y −1)
c
2
, c
4
((y
2
+ y + 1)
7
)(y
35
+ 54y
34
+ ··· + 132y −1)
c
3
((y
2
+ y + 1)
7
)(y
35
+ 6y
34
+ ··· − 4y −1)
c
5
, c
9
((y −1)
14
)(y
35
− 55y
34
+ ··· − 1795y −169)
c
6
(y
2
+ y + 1)
3
· (y
8
+ 8y
7
+ 46y
6
+ 160y
5
+ 387y
4
− 160y
3
+ 46y
2
− 8y + 1)
· (y
35
+ 42y
34
+ ··· − 5590382760y −821911561)
c
7
, c
11
y
6
(y
2
+ 2y + 2)
4
(y
35
+ 15y
34
+ ··· − 80y −16)
c
8
(y
2
+ y + 1)
3
· (y
8
+ 8y
7
+ 46y
6
+ 160y
5
+ 387y
4
− 160y
3
+ 46y
2
− 8y + 1)
· (y
35
− 90y
34
+ ··· − 117701124860y −12692500921)
c
10
y
6
(y
2
− 2y + 2)
4
(y
35
+ 15y
34
+ ··· − 4334416y −163216)
c
12
y
6
(y
2
+ 4)
4
(y
35
+ 15y
34
+ ··· + 768y −256)
18
