12n
0137
(K12n
0137
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 11 4 10 11 1 8 5 9
Solving Sequence
3,5
2
1,9
10 4 12 11 6 7 8
c
2
c
1
c
9
c
4
c
12
c
11
c
5
c
6
c
8
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−9.85228 × 10
76
u
64
6.99704 × 10
77
u
63
+ ··· + 5.10441 × 10
77
b 2.60555 × 10
76
,
1.62634 × 10
77
u
64
1.12461 × 10
78
u
63
+ ··· + 5.10441 × 10
77
a + 2.07206 × 10
79
,
u
65
+ 7u
64
+ ··· 61u + 1i
I
u
2
= h3u
2
a + 4au + u
2
+ b + 2a + u + 1, u
2
a + a
2
+ u
2
+ a u, u
3
+ u
2
1i
I
u
3
= h−4a
2
+ b + a 7, a
3
a
2
+ 2a 1, u 1i
I
u
4
= hb + u + 2, a 2u 3, u
2
+ u 1i
* 4 irreducible components of dim
C
= 0, with total 76 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−9.85 × 10
76
u
64
7.00 × 10
77
u
63
+ · · · + 5.10 × 10
77
b 2.61 ×
10
76
, 1.63 × 10
77
u
64
1.12 × 10
78
u
63
+ · · · + 5.10 × 10
77
a + 2.07 ×
10
79
, u
65
+ 7u
64
+ · · · 61u + 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
1
=
u
2
+ 1
u
2
a
9
=
0.318614u
64
+ 2.20322u
63
+ ··· 187.922u 40.5935
0.193015u
64
+ 1.37078u
63
+ ··· + 31.2483u + 0.0510450
a
10
=
0.181812u
64
1.12398u
63
+ ··· 143.777u 40.7558
0.312557u
64
1.73228u
63
+ ··· + 47.1225u 0.209103
a
4
=
u
u
3
+ u
a
12
=
0.0237982u
64
+ 0.139063u
63
+ ··· 88.0105u 21.6619
1.02089u
64
6.46038u
63
+ ··· + 19.1974u + 0.00528714
a
11
=
0.0237982u
64
+ 0.139063u
63
+ ··· 88.0105u 21.6619
0.793668u
64
5.02777u
63
+ ··· + 0.528898u + 0.310938
a
6
=
0.174107u
64
1.00555u
63
+ ··· 15.0665u 7.78932
0.246119u
64
1.30818u
63
+ ··· + 12.6788u 0.0843556
a
7
=
0.0275853u
64
+ 0.408515u
63
+ ··· 35.1429u 7.46418
0.325138u
64
+ 2.07428u
63
+ ··· 7.33147u + 0.243004
a
8
=
0.404667u
64
+ 2.59172u
63
+ ··· 131.916u 23.4724
0.464640u
64
3.13321u
63
+ ··· + 9.04364u + 0.172982
(ii) Obstruction class = 1
(iii) Cusp Shapes = 1.87765u
64
+ 14.8687u
63
+ ··· + 187.232u + 6.39873
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
65
+ 35u
64
+ ··· + 4379u + 1
c
2
, c
4
u
65
7u
64
+ ··· 61u 1
c
3
, c
6
u
65
4u
64
+ ··· 4u 8
c
5
, c
11
u
65
3u
64
+ ··· + 224u 64
c
7
, c
8
, c
10
u
65
+ 7u
64
+ ··· + 88u 1
c
9
, c
12
u
65
5u
64
+ ··· + 4u 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
65
3y
64
+ ··· + 19078099y 1
c
2
, c
4
y
65
35y
64
+ ··· + 4379y 1
c
3
, c
6
y
65
+ 24y
64
+ ··· + 7056y 64
c
5
, c
11
y
65
47y
64
+ ··· + 283648y 4096
c
7
, c
8
, c
10
y
65
55y
64
+ ··· + 6134y 1
c
9
, c
12
y
65
21y
64
+ ··· + 1448y 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.978199 + 0.188355I
a = 0.097584 0.485198I
b = 0.566996 + 1.279070I
1.000760 0.692383I 6.73751 + 0.I
u = 0.978199 0.188355I
a = 0.097584 + 0.485198I
b = 0.566996 1.279070I
1.000760 + 0.692383I 6.73751 + 0.I
u = 0.989443
a = 0.408778
b = 9.34730
0.561787 200.700
u = 0.792790 + 0.578558I
a = 0.102171 + 0.126924I
b = 1.09401 + 1.32036I
11.02040 + 2.29381I 0
u = 0.792790 0.578558I
a = 0.102171 0.126924I
b = 1.09401 1.32036I
11.02040 2.29381I 0
u = 0.956320 + 0.141139I
a = 0.48690 1.37332I
b = 0.201705 0.989795I
4.60256 2.48429I 2.01382 9.93890I
u = 0.956320 0.141139I
a = 0.48690 + 1.37332I
b = 0.201705 + 0.989795I
4.60256 + 2.48429I 2.01382 + 9.93890I
u = 0.683102 + 0.644381I
a = 1.33162 1.64923I
b = 0.818752 0.323057I
4.65051 + 1.43055I 4.63524 5.15036I
u = 0.683102 0.644381I
a = 1.33162 + 1.64923I
b = 0.818752 + 0.323057I
4.65051 1.43055I 4.63524 + 5.15036I
u = 0.220993 + 0.900580I
a = 1.71144 + 0.11133I
b = 0.241601 + 0.752237I
0.42473 5.58831I 0. + 4.96253I
5
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.220993 0.900580I
a = 1.71144 0.11133I
b = 0.241601 0.752237I
0.42473 + 5.58831I 0. 4.96253I
u = 0.898048 + 0.623866I
a = 1.07198 + 1.80577I
b = 1.82761 + 1.02787I
4.03132 + 3.47720I 0
u = 0.898048 0.623866I
a = 1.07198 1.80577I
b = 1.82761 1.02787I
4.03132 3.47720I 0
u = 0.568826 + 0.935685I
a = 1.252080 + 0.399270I
b = 0.371291 + 1.091860I
1.38895 + 2.95818I 0
u = 0.568826 0.935685I
a = 1.252080 0.399270I
b = 0.371291 1.091860I
1.38895 2.95818I 0
u = 0.124919 + 0.887726I
a = 0.281399 + 0.044752I
b = 0.459233 + 0.326826I
7.57890 3.09040I 6.72860 + 3.02873I
u = 0.124919 0.887726I
a = 0.281399 0.044752I
b = 0.459233 0.326826I
7.57890 + 3.09040I 6.72860 3.02873I
u = 0.307741 + 1.069510I
a = 1.62825 0.54466I
b = 0.412336 1.061990I
4.86618 10.28160I 0
u = 0.307741 1.069510I
a = 1.62825 + 0.54466I
b = 0.412336 + 1.061990I
4.86618 + 10.28160I 0
u = 1.013850 + 0.477455I
a = 0.318794 + 0.444188I
b = 0.224946 0.486741I
0.56978 + 4.38703I 0
6
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.013850 0.477455I
a = 0.318794 0.444188I
b = 0.224946 + 0.486741I
0.56978 4.38703I 0
u = 1.079610 + 0.408263I
a = 0.203024 + 1.130950I
b = 0.42767 + 1.71363I
1.31032 2.58838I 0
u = 1.079610 0.408263I
a = 0.203024 1.130950I
b = 0.42767 1.71363I
1.31032 + 2.58838I 0
u = 0.866464 + 0.780684I
a = 1.00474 + 1.88631I
b = 0.54581 + 2.09233I
3.85230 + 2.93050I 0
u = 0.866464 0.780684I
a = 1.00474 1.88631I
b = 0.54581 2.09233I
3.85230 2.93050I 0
u = 1.143920 + 0.287639I
a = 0.466880 + 0.894239I
b = 2.46022 + 2.14973I
1.58736 + 0.20570I 0
u = 1.143920 0.287639I
a = 0.466880 0.894239I
b = 2.46022 2.14973I
1.58736 0.20570I 0
u = 1.129770 + 0.340293I
a = 0.433099 + 1.075100I
b = 0.135218 + 1.075780I
6.03312 + 3.48808I 0
u = 1.129770 0.340293I
a = 0.433099 1.075100I
b = 0.135218 1.075780I
6.03312 3.48808I 0
u = 0.280055 + 0.763508I
a = 1.74275 + 0.40895I
b = 0.218485 + 0.488467I
2.73256 3.28945I 2.80172 + 2.68321I
7
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.280055 0.763508I
a = 1.74275 0.40895I
b = 0.218485 0.488467I
2.73256 + 3.28945I 2.80172 2.68321I
u = 0.802270
a = 0.0518504
b = 4.68222
7.71518 86.2400
u = 1.100770 + 0.494345I
a = 0.390021 0.786924I
b = 1.74073 1.70018I
0.66498 + 4.63908I 0
u = 1.100770 0.494345I
a = 0.390021 + 0.786924I
b = 1.74073 + 1.70018I
0.66498 4.63908I 0
u = 1.138890 + 0.516830I
a = 0.153293 1.184990I
b = 1.79910 1.91501I
4.82760 4.40824I 0
u = 1.138890 0.516830I
a = 0.153293 + 1.184990I
b = 1.79910 + 1.91501I
4.82760 + 4.40824I 0
u = 0.733644 + 0.132924I
a = 2.44908 2.64522I
b = 0.25925 + 3.39498I
0.646116 0.109642I 45.2047 + 8.8218I
u = 0.733644 0.132924I
a = 2.44908 + 2.64522I
b = 0.25925 3.39498I
0.646116 + 0.109642I 45.2047 8.8218I
u = 0.212963 + 0.702410I
a = 1.69285 0.06283I
b = 0.285610 0.811517I
2.18618 0.21906I 3.41412 + 0.63779I
u = 0.212963 0.702410I
a = 1.69285 + 0.06283I
b = 0.285610 + 0.811517I
2.18618 + 0.21906I 3.41412 0.63779I
8
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.144350 + 0.549506I
a = 0.423148 1.005190I
b = 0.110767 1.274550I
0.19526 + 8.22606I 0
u = 1.144350 0.549506I
a = 0.423148 + 1.005190I
b = 0.110767 + 1.274550I
0.19526 8.22606I 0
u = 1.106960 + 0.690966I
a = 0.095676 + 1.306210I
b = 1.35935 + 1.80926I
0.31021 8.92181I 0
u = 1.106960 0.690966I
a = 0.095676 1.306210I
b = 1.35935 1.80926I
0.31021 + 8.92181I 0
u = 0.481434 + 0.486718I
a = 0.675303 0.696708I
b = 1.023720 0.531357I
2.10912 0.34030I 3.61302 + 0.63149I
u = 0.481434 0.486718I
a = 0.675303 + 0.696708I
b = 1.023720 + 0.531357I
2.10912 + 0.34030I 3.61302 0.63149I
u = 1.291860 + 0.308009I
a = 0.535246 0.916498I
b = 0.460659 1.205350I
5.30684 + 1.54275I 0
u = 1.291860 0.308009I
a = 0.535246 + 0.916498I
b = 0.460659 + 1.205350I
5.30684 1.54275I 0
u = 1.206790 + 0.570547I
a = 0.226214 + 1.184020I
b = 1.59633 + 2.09960I
3.39471 + 10.94230I 0
u = 1.206790 0.570547I
a = 0.226214 1.184020I
b = 1.59633 2.09960I
3.39471 10.94230I 0
9
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.255190 + 0.475295I
a = 0.055501 0.186916I
b = 0.097593 0.921175I
3.47356 1.55230I 0
u = 1.255190 0.475295I
a = 0.055501 + 0.186916I
b = 0.097593 + 0.921175I
3.47356 + 1.55230I 0
u = 0.652150
a = 0.581302
b = 0.626898
1.00335 10.2290
u = 1.228690 + 0.560073I
a = 0.016874 + 0.243267I
b = 0.611981 + 0.962751I
4.31441 + 8.34885I 0
u = 1.228690 0.560073I
a = 0.016874 0.243267I
b = 0.611981 0.962751I
4.31441 8.34885I 0
u = 0.914593 + 1.030720I
a = 0.652075 1.141730I
b = 0.23039 1.45733I
9.17254 + 3.64107I 0
u = 0.914593 1.030720I
a = 0.652075 + 1.141730I
b = 0.23039 + 1.45733I
9.17254 3.64107I 0
u = 0.277804 + 0.539464I
a = 1.21331 + 0.90105I
b = 0.374598 0.465447I
1.65110 0.40415I 3.75505 + 0.76632I
u = 0.277804 0.539464I
a = 1.21331 0.90105I
b = 0.374598 + 0.465447I
1.65110 + 0.40415I 3.75505 0.76632I
u = 1.250110 + 0.654083I
a = 0.032151 1.362680I
b = 1.31022 2.26446I
1.9343 + 16.4466I 0
10
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.250110 0.654083I
a = 0.032151 + 1.362680I
b = 1.31022 + 2.26446I
1.9343 16.4466I 0
u = 1.46199 + 0.21591I
a = 0.775439 + 0.708467I
b = 0.671896 + 0.753763I
1.34521 + 5.69764I 0
u = 1.46199 0.21591I
a = 0.775439 0.708467I
b = 0.671896 0.753763I
1.34521 5.69764I 0
u = 1.64593
a = 0.300783
b = 0.310246
7.15457 0
u = 0.0151310
a = 43.4847
b = 0.562042
1.12640 9.50900
11
II.
I
u
2
= h3u
2
a + 4au + u
2
+ b + 2a + u + 1, u
2
a + a
2
+ u
2
+ a u, u
3
+ u
2
1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
1
=
u
2
+ 1
u
2
a
9
=
a
3u
2
a 4au u
2
2a u 1
a
10
=
u
2
1
2u
2
a 3au 2a u 1
a
4
=
u
u
2
+ u 1
a
12
=
0
2u
2
a + 3au + 2u
2
+ 2a + 2u + 2
a
11
=
0
2u
2
a + 3au + 2u
2
+ 2a + 2u + 2
a
6
=
0
u
a
7
=
u
2
1
u
2
a
8
=
a
u
2
a + 2au + 2u
2
+ 2a + 2u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 21u
2
a + 39au + 11u
2
+ 24a + 19u + 26
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
u
2
+ 2u 1)
2
c
2
(u
3
+ u
2
1)
2
c
4
(u
3
u
2
+ 1)
2
c
5
, c
11
u
6
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
, c
9
(u
2
u 1)
3
c
10
, c
12
(u
2
+ u 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y 1)
2
c
2
, c
4
(y
3
y
2
+ 2y 1)
2
c
5
, c
11
y
6
c
7
, c
8
, c
9
c
10
, c
12
(y
2
3y + 1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
1(vol +
1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.485107 + 0.807858I
b = 0.67924 + 1.71765I
11.90680 + 2.82812I 7.63548 4.05775I
u = 0.877439 + 0.744862I
a = 1.27003 2.11500I
b = 0.55668 2.46251I
4.01109 + 2.82812I 22.3213 + 9.8050I
u = 0.877439 0.744862I
a = 0.485107 0.807858I
b = 0.67924 1.71765I
11.90680 2.82812I 7.63548 + 4.05775I
u = 0.877439 0.744862I
a = 1.27003 + 2.11500I
b = 0.55668 + 2.46251I
4.01109 2.82812I 22.3213 9.8050I
u = 0.754878
a = 0.696013
b = 2.35878
0.126494 1.08690
u = 0.754878
a = 0.265853
b = 4.11365
7.76919 64.0000
15
III. I
u
3
= h−4a
2
+ b + a 7, a
3
a
2
+ 2a 1, u 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
1
a
2
=
1
1
a
1
=
0
1
a
9
=
a
4a
2
a + 7
a
10
=
a
4a
2
2a + 7
a
4
=
1
0
a
12
=
a
2
3a
2
a + 3
a
11
=
a
2
2a
2
a + 3
a
6
=
a
2
+ a 1
0
a
7
=
a
2
+ a 1
0
a
8
=
a
2
3a
2
a + 5
(ii) Obstruction class = 1
(iii) Cusp Shapes = 53a
2
32a + 92
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u 1)
3
c
3
, c
6
u
3
c
4
(u + 1)
3
c
5
u
3
3u
2
+ 2u + 1
c
7
, c
8
u
3
u
2
+ 1
c
9
u
3
+ u
2
+ 2u + 1
c
10
u
3
+ u
2
1
c
11
u
3
+ 3u
2
+ 2u 1
c
12
u
3
u
2
+ 2u 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y 1)
3
c
3
, c
6
y
3
c
5
, c
11
y
3
5y
2
+ 10y 1
c
7
, c
8
, c
10
y
3
y
2
+ 2y 1
c
9
, c
12
y
3
+ 3y
2
+ 2y 1
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.215080 + 1.307140I
b = 0.135484 + 0.941977I
4.66906 + 2.82812I 2.98758 12.02771I
u = 1.00000
a = 0.215080 1.307140I
b = 0.135484 0.941977I
4.66906 2.82812I 2.98758 + 12.02771I
u = 1.00000
a = 0.569840
b = 7.72903
0.531480 90.9750
19
IV. I
u
4
= hb + u + 2, a 2u 3, u
2
+ u 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u 1
a
1
=
u
u 1
a
9
=
2u + 3
u 2
a
10
=
2u + 3
u 2
a
4
=
u
u + 1
a
12
=
u
u 1
a
11
=
u
u
a
6
=
2u + 1
3u 1
a
7
=
u
u
a
8
=
u + 3
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 49
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
u
2
3u + 1
c
2
, c
3
u
2
+ u 1
c
4
, c
6
u
2
u 1
c
5
u
2
+ 3u + 1
c
7
, c
8
(u + 1)
2
c
9
, c
12
u
2
c
10
(u 1)
2
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
11
y
2
7y + 1
c
2
, c
3
, c
4
c
6
y
2
3y + 1
c
7
, c
8
, c
10
(y 1)
2
c
9
, c
12
y
2
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
1(vol +
1CS) Cusp shape
u = 0.618034
a = 4.23607
b = 2.61803
0.657974 49.0000
u = 1.61803
a = 0.236068
b = 0.381966
7.23771 49.0000
23
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u 1)
3
(u
2
3u + 1)(u
3
u
2
+ 2u 1)
2
· (u
65
+ 35u
64
+ ··· + 4379u + 1)
c
2
((u 1)
3
)(u
2
+ u 1)(u
3
+ u
2
1)
2
(u
65
7u
64
+ ··· 61u 1)
c
3
u
3
(u
2
+ u 1)(u
3
u
2
+ 2u 1)
2
(u
65
4u
64
+ ··· 4u 8)
c
4
((u + 1)
3
)(u
2
u 1)(u
3
u
2
+ 1)
2
(u
65
7u
64
+ ··· 61u 1)
c
5
u
6
(u
2
+ 3u + 1)(u
3
3u
2
+ 2u + 1)(u
65
3u
64
+ ··· + 224u 64)
c
6
u
3
(u
2
u 1)(u
3
+ u
2
+ 2u + 1)
2
(u
65
4u
64
+ ··· 4u 8)
c
7
, c
8
((u + 1)
2
)(u
2
u 1)
3
(u
3
u
2
+ 1)(u
65
+ 7u
64
+ ··· + 88u 1)
c
9
u
2
(u
2
u 1)
3
(u
3
+ u
2
+ 2u + 1)(u
65
5u
64
+ ··· + 4u 4)
c
10
((u 1)
2
)(u
2
+ u 1)
3
(u
3
+ u
2
1)(u
65
+ 7u
64
+ ··· + 88u 1)
c
11
u
6
(u
2
3u + 1)(u
3
+ 3u
2
+ 2u 1)(u
65
3u
64
+ ··· + 224u 64)
c
12
u
2
(u
2
+ u 1)
3
(u
3
u
2
+ 2u 1)(u
65
5u
64
+ ··· + 4u 4)
24
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y 1)
3
(y
2
7y + 1)(y
3
+ 3y
2
+ 2y 1)
2
· (y
65
3y
64
+ ··· + 19078099y 1)
c
2
, c
4
(y 1)
3
(y
2
3y + 1)(y
3
y
2
+ 2y 1)
2
· (y
65
35y
64
+ ··· + 4379y 1)
c
3
, c
6
y
3
(y
2
3y + 1)(y
3
+ 3y
2
+ 2y 1)
2
(y
65
+ 24y
64
+ ··· + 7056y 64)
c
5
, c
11
y
6
(y
2
7y + 1)(y
3
5y
2
+ 10y 1)
· (y
65
47y
64
+ ··· + 283648y 4096)
c
7
, c
8
, c
10
(y 1)
2
(y
2
3y + 1)
3
(y
3
y
2
+ 2y 1)
· (y
65
55y
64
+ ··· + 6134y 1)
c
9
, c
12
y
2
(y
2
3y + 1)
3
(y
3
+ 3y
2
+ 2y 1)(y
65
21y
64
+ ··· + 1448y 16)
25