12n
0341
(K12n
0341
)
A knot diagram
1
Linearized knot diagam
3 6 9 8 2 11 3 4 6 12 7 10
Solving Sequence
6,11 3,7
8 12 2 1 5 4 10 9
c
6
c
7
c
11
c
2
c
1
c
5
c
4
c
10
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h7214350820u
41
+ 14261840536u
40
+ ··· + 24614921861b + 13897246666,
− 286630329392u
41
+ 403439047669u
40
+ ··· + 147689531166a − 913761324969,
u
42
− 2u
41
+ ··· + 3u − 3i
I
u
2
= hb − 1, a
2
+ 2au + 3u
2
− 2a − 6u + 3, u
3
− u
2
+ 1i
I
u
3
= hb + 1, a + u + 1, u
3
+ u
2
− 1i
* 3 irreducible components of dim
C
= 0, with total 51 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
= h7.21 × 10
9
u
41
+ 1.43 × 10
10
u
40
+ · · · + 2. 46 × 10
10
b + 1.39 ×10
10
, −2.87 ×
10
11
u
41
+4.03×10
11
u
40
+· · ·+1.48×10
11
a−9.14×10
11
, u
42
−2u
41
+· · ·+3u−3i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
1.94076u
41
− 2.73167u
40
+ ··· − 1.14974u + 6.18704
−0.293089u
41
− 0.579398u
40
+ ··· + 3.28574u − 0.564586
a
7
=
1
u
2
a
8
=
−1.42423u
41
+ 1.18705u
40
+ ··· + 4.86301u + 0.704689
−1.16806u
41
+ 1.78911u
40
+ ··· − 1.38926u − 3.49408
a
12
=
−u
−u
3
+ u
a
2
=
1.64767u
41
− 3.31107u
40
+ ··· + 2.13601u + 5.62246
−0.293089u
41
− 0.579398u
40
+ ··· + 3.28574u − 0.564586
a
1
=
−u
5
− u
−u
7
+ u
5
− 2u
3
+ u
a
5
=
−1.90746u
41
+ 4.07002u
40
+ ··· − 4.13107u − 3.28175
0.164081u
41
+ 0.812895u
40
+ ··· − 2.16751u + 0.836225
a
4
=
1.43869u
41
− 0.929160u
40
+ ··· − 3.24612u + 9.35450
−0.222585u
41
+ 0.354594u
40
+ ··· + 1.79645u + 0.987097
a
10
=
u
3
u
5
− u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
47220782375
24614921861
u
41
−
499793100
24614921861
u
40
+ ··· +
450722117512
24614921861
u −
698025687495
24614921861
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
42
+ 50u
41
+ ··· − 190u + 1
c
2
, c
5
u
42
+ 4u
41
+ ··· + 12u − 1
c
3
, c
4
, c
8
u
42
+ u
41
+ ··· + 16u + 8
c
6
, c
11
u
42
− 2u
41
+ ··· + 3u − 3
c
7
u
42
− u
41
+ ··· + 64u + 8
c
9
u
42
+ 2u
41
+ ··· + 15u − 3
c
10
, c
12
u
42
+ 16u
41
+ ··· + 147u + 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
42
− 106y
41
+ ··· + 9510y + 1
c
2
, c
5
y
42
− 50y
41
+ ··· + 190y + 1
c
3
, c
4
, c
8
y
42
+ 35y
41
+ ··· + 128y + 64
c
6
, c
11
y
42
− 16y
41
+ ··· − 147y + 9
c
7
y
42
− 49y
41
+ ··· + 1536y + 64
c
9
y
42
− 48y
41
+ ··· − 3y + 9
c
10
, c
12
y
42
+ 24y
41
+ ··· − 2367y + 81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.493349 + 0.873287I
a = 0.358491 − 0.000809I
b = 1.62688 + 0.12135I
−7.72387 − 2.08243I −13.63208 + 0.61844I
u = −0.493349 − 0.873287I
a = 0.358491 + 0.000809I
b = 1.62688 − 0.12135I
−7.72387 + 2.08243I −13.63208 − 0.61844I
u = −0.857671 + 0.547734I
a = 2.39738 + 0.56265I
b = 1.297890 + 0.093808I
3.74220 + 2.19658I −11.88073 − 3.02970I
u = −0.857671 − 0.547734I
a = 2.39738 − 0.56265I
b = 1.297890 − 0.093808I
3.74220 − 2.19658I −11.88073 + 3.02970I
u = −1.042580 + 0.004634I
a = 1.183060 + 0.451877I
b = 0.711272 + 0.677400I
−1.44341 − 2.30270I −14.7058 + 3.7232I
u = −1.042580 − 0.004634I
a = 1.183060 − 0.451877I
b = 0.711272 − 0.677400I
−1.44341 + 2.30270I −14.7058 − 3.7232I
u = 0.586636 + 0.736989I
a = −0.267637 + 0.708323I
b = 0.487231 − 0.786141I
3.87323 + 3.15388I −7.00575 − 3.01303I
u = 0.586636 − 0.736989I
a = −0.267637 − 0.708323I
b = 0.487231 + 0.786141I
3.87323 − 3.15388I −7.00575 + 3.01303I
u = 0.587628 + 0.887835I
a = −0.300671 − 0.305913I
b = −1.57989 + 0.27896I
−3.00105 + 7.12424I −10.18276 − 2.97506I
u = 0.587628 − 0.887835I
a = −0.300671 + 0.305913I
b = −1.57989 − 0.27896I
−3.00105 − 7.12424I −10.18276 + 2.97506I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.893636 + 0.236079I
a = −1.66259 + 0.50310I
b = −1.141660 + 0.275387I
−3.22132 − 0.72709I −17.2502 + 5.7568I
u = 0.893636 − 0.236079I
a = −1.66259 − 0.50310I
b = −1.141660 − 0.275387I
−3.22132 + 0.72709I −17.2502 − 5.7568I
u = 0.364728 + 0.832070I
a = −0.385710 + 0.359680I
b = −1.59824 − 0.07981I
−4.34661 − 3.12224I −10.84272 + 2.92675I
u = 0.364728 − 0.832070I
a = −0.385710 − 0.359680I
b = −1.59824 + 0.07981I
−4.34661 + 3.12224I −10.84272 − 2.92675I
u = 0.878925 + 0.694557I
a = 0.310415 − 0.409583I
b = 0.590948 − 0.108791I
2.08568 − 2.67535I −4.92251 + 2.20937I
u = 0.878925 − 0.694557I
a = 0.310415 + 0.409583I
b = 0.590948 + 0.108791I
2.08568 + 2.67535I −4.92251 − 2.20937I
u = 0.949655 + 0.594867I
a = −0.255068 − 0.352987I
b = 0.662783 − 0.635045I
2.08502 − 3.11024I −10.10897 + 3.32337I
u = 0.949655 − 0.594867I
a = −0.255068 + 0.352987I
b = 0.662783 + 0.635045I
2.08502 + 3.11024I −10.10897 − 3.32337I
u = −0.967079 + 0.570637I
a = −1.10245 − 1.08422I
b = −0.758887 + 0.665919I
−1.15338 + 4.42053I −14.9758 − 5.6521I
u = −0.967079 − 0.570637I
a = −1.10245 + 1.08422I
b = −0.758887 − 0.665919I
−1.15338 − 4.42053I −14.9758 + 5.6521I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.739578 + 0.450680I
a = 0.34684 − 1.84376I
b = 0.854538 + 0.412162I
2.96029 − 1.36479I −9.90879 + 4.47886I
u = 0.739578 − 0.450680I
a = 0.34684 + 1.84376I
b = 0.854538 − 0.412162I
2.96029 + 1.36479I −9.90879 − 4.47886I
u = −0.870477 + 0.749362I
a = −0.989419 + 0.125741I
b = 0.0326402 − 0.0700810I
7.96823 + 2.83809I −2.44683 − 2.99950I
u = −0.870477 − 0.749362I
a = −0.989419 − 0.125741I
b = 0.0326402 + 0.0700810I
7.96823 − 2.83809I −2.44683 + 2.99950I
u = −0.667718 + 0.509431I
a = 0.526984 + 0.298925I
b = −0.455324 − 0.527591I
−0.215927 + 0.057739I −12.86905 + 0.99724I
u = −0.667718 − 0.509431I
a = 0.526984 − 0.298925I
b = −0.455324 + 0.527591I
−0.215927 − 0.057739I −12.86905 − 0.99724I
u = −1.192640 + 0.084412I
a = −2.72686 − 0.35998I
b = −1.68766 + 0.19280I
−9.82219 + 5.69889I −16.0238 − 3.4541I
u = −1.192640 − 0.084412I
a = −2.72686 + 0.35998I
b = −1.68766 − 0.19280I
−9.82219 − 5.69889I −16.0238 + 3.4541I
u = 1.20918
a = 2.73290
b = 1.73917
−13.9658 −18.7010
u = −0.893982 + 0.823604I
a = 0.017481 − 0.836922I
b = −1.322530 + 0.004918I
3.39487 + 3.06804I −10.44365 − 2.87354I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.893982 − 0.823604I
a = 0.017481 + 0.836922I
b = −1.322530 − 0.004918I
3.39487 − 3.06804I −10.44365 + 2.87354I
u = 1.029980 + 0.651441I
a = 1.19267 − 0.82522I
b = 0.548001 + 0.888471I
2.56002 − 8.46686I −9.68679 + 7.74880I
u = 1.029980 − 0.651441I
a = 1.19267 + 0.82522I
b = 0.548001 − 0.888471I
2.56002 + 8.46686I −9.68679 − 7.74880I
u = 1.106730 + 0.589792I
a = −1.63532 + 1.44131I
b = −1.67176 − 0.00321I
−6.57334 − 2.08658I −13.90990 + 1.57056I
u = 1.106730 − 0.589792I
a = −1.63532 − 1.44131I
b = −1.67176 + 0.00321I
−6.57334 + 2.08658I −13.90990 − 1.57056I
u = −1.107090 + 0.662634I
a = 1.51861 + 1.67496I
b = 1.68068 − 0.19305I
−9.58882 + 7.76603I −15.5418 − 5.0561I
u = −1.107090 − 0.662634I
a = 1.51861 − 1.67496I
b = 1.68068 + 0.19305I
−9.58882 − 7.76603I −15.5418 + 5.0561I
u = 1.084910 + 0.710010I
a = −1.42076 + 1.88196I
b = −1.60580 − 0.32883I
−4.52539 − 13.04420I −12.00000 + 7.21592I
u = 1.084910 − 0.710010I
a = −1.42076 − 1.88196I
b = −1.60580 + 0.32883I
−4.52539 + 13.04420I −12.00000 − 7.21592I
u = 0.451064 + 0.481038I
a = −0.346528 − 1.234360I
b = 0.610635 + 0.462564I
3.08466 − 1.37137I −7.10497 + 4.42267I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.451064 − 0.481038I
a = −0.346528 + 1.234360I
b = 0.610635 − 0.462564I
3.08466 + 1.37137I −7.10497 − 4.42267I
u = −0.370951
a = 0.749235
b = −0.302638
−0.594790 −16.5260
9
II. I
u
2
= hb − 1, a
2
+ 2au + 3u
2
− 2a − 6u + 3, u
3
− u
2
+ 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
a
1
a
7
=
1
u
2
a
8
=
−a − 3u + 4
−u
2
a + u
2
+ 1
a
12
=
−u
−u
2
+ u + 1
a
2
=
a + 1
1
a
1
=
1
0
a
5
=
−a
−1
a
4
=
u
2
a + a − 1
u
2
a − au − u
2
− a + 1
a
10
=
u
2
− 1
−u
2
a
9
=
−1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u − 12
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
6
c
2
(u + 1)
6
c
3
, c
4
, c
7
c
8
(u
2
+ 2)
3
c
6
(u
3
− u
2
+ 1)
2
c
9
, c
10
(u
3
− u
2
+ 2u − 1)
2
c
11
(u
3
+ u
2
− 1)
2
c
12
(u
3
+ u
2
+ 2u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
7
c
8
(y + 2)
6
c
6
, c
11
(y
3
− y
2
+ 2y −1)
2
c
9
, c
10
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 1.175960 − 0.571534I
b = 1.00000
6.31400 − 2.82812I −8.49024 + 2.97945I
u = 0.877439 + 0.744862I
a = −0.930832 − 0.918189I
b = 1.00000
6.31400 − 2.82812I −8.49024 + 2.97945I
u = 0.877439 − 0.744862I
a = 1.175960 + 0.571534I
b = 1.00000
6.31400 + 2.82812I −8.49024 − 2.97945I
u = 0.877439 − 0.744862I
a = −0.930832 + 0.918189I
b = 1.00000
6.31400 + 2.82812I −8.49024 − 2.97945I
u = −0.754878
a = 1.75488 + 2.48177I
b = 1.00000
2.17641 −15.0200
u = −0.754878
a = 1.75488 − 2.48177I
b = 1.00000
2.17641 −15.0200
13
III. I
u
3
= hb + 1, a + u + 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
−u − 1
−1
a
7
=
1
u
2
a
8
=
1
u
2
a
12
=
−u
u
2
+ u − 1
a
2
=
−u − 2
−1
a
1
=
−1
0
a
5
=
−u − 1
−1
a
4
=
−u − 1
−1
a
10
=
−u
2
+ 1
u
2
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
+ 2u − 16
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
7
c
8
u
3
c
5
(u + 1)
3
c
6
u
3
+ u
2
− 1
c
9
, c
12
u
3
+ u
2
+ 2u + 1
c
10
u
3
− u
2
+ 2u − 1
c
11
u
3
− u
2
+ 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
3
c
3
, c
4
, c
7
c
8
y
3
c
6
, c
11
y
3
− y
2
+ 2y −1
c
9
, c
10
, c
12
y
3
+ 3y
2
+ 2y −1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.122561 − 0.744862I
b = −1.00000
1.37919 + 2.82812I −16.8946 − 3.7388I
u = −0.877439 − 0.744862I
a = −0.122561 + 0.744862I
b = −1.00000
1.37919 − 2.82812I −16.8946 + 3.7388I
u = 0.754878
a = −1.75488
b = −1.00000
−2.75839 −12.2110
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
42
+ 50u
41
+ ··· − 190u + 1)
c
2
((u − 1)
3
)(u + 1)
6
(u
42
+ 4u
41
+ ··· + 12u − 1)
c
3
, c
4
, c
8
u
3
(u
2
+ 2)
3
(u
42
+ u
41
+ ··· + 16u + 8)
c
5
((u − 1)
6
)(u + 1)
3
(u
42
+ 4u
41
+ ··· + 12u − 1)
c
6
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
42
− 2u
41
+ ··· + 3u − 3)
c
7
u
3
(u
2
+ 2)
3
(u
42
− u
41
+ ··· + 64u + 8)
c
9
((u
3
− u
2
+ 2u − 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
42
+ 2u
41
+ ··· + 15u − 3)
c
10
((u
3
− u
2
+ 2u − 1)
3
)(u
42
+ 16u
41
+ ··· + 147u + 9)
c
11
(u
3
− u
2
+ 1)(u
3
+ u
2
− 1)
2
(u
42
− 2u
41
+ ··· + 3u − 3)
c
12
((u
3
+ u
2
+ 2u + 1)
3
)(u
42
+ 16u
41
+ ··· + 147u + 9)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
42
− 106y
41
+ ··· + 9510y + 1)
c
2
, c
5
((y −1)
9
)(y
42
− 50y
41
+ ··· + 190y + 1)
c
3
, c
4
, c
8
y
3
(y + 2)
6
(y
42
+ 35y
41
+ ··· + 128y + 64)
c
6
, c
11
((y
3
− y
2
+ 2y −1)
3
)(y
42
− 16y
41
+ ··· − 147y + 9)
c
7
y
3
(y + 2)
6
(y
42
− 49y
41
+ ··· + 1536y + 64)
c
9
((y
3
+ 3y
2
+ 2y −1)
3
)(y
42
− 48y
41
+ ··· − 3y + 9)
c
10
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
42
+ 24y
41
+ ··· − 2367y + 81)
19
