12n
0150
(K12n
0150
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 12 11 3 5 6 7 10 9
Solving Sequence
6,11
7 10 12
3,5
2 4 9 1 8
c
6
c
10
c
11
c
5
c
2
c
4
c
9
c
12
c
8
c
1
, c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
36
− u
35
+ ··· + b − 2u, −u
36
+ u
35
+ ··· + a + 1, u
38
− 2u
37
+ ··· − 5u + 1i
I
u
2
= hb + u, −u
3
+ a − u + 1, u
4
+ u
2
− u + 1i
I
u
3
= h−u
5
− u
4
− 2u
3
− 2u
2
+ b − 2u − 1, u
4
+ u
2
+ a, u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1i
* 3 irreducible components of dim
C
= 0, with total 48 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
36
−u
35
+· · ·+b−2u, −u
36
+u
35
+· · ·+a+1, u
38
−2u
37
+· · ·−5u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
10
=
u
u
3
+ u
a
12
=
u
3
u
5
+ u
3
+ u
a
3
=
u
36
− u
35
+ ··· − 2u
3
− 1
−u
36
+ u
35
+ ··· − 3u
2
+ 2u
a
5
=
u
8
+ u
6
+ u
4
+ 1
u
10
+ 2u
8
+ 3u
6
+ 2u
4
+ u
2
a
2
=
−u
34
+ u
33
+ ··· + 3u − 1
−u
36
+ u
35
+ ··· − 2u
2
+ 2u
a
4
=
u
36
− u
35
+ ··· − 4u + 2
−u
36
− u
35
+ ··· + u
2
− u
a
9
=
−u
3
u
3
+ u
a
1
=
−u
11
− 2u
9
− 2u
7
+ u
3
u
11
+ 3u
9
+ 4u
7
+ 3u
5
+ u
3
+ u
a
8
=
u
21
+ 4u
19
+ 9u
17
+ 12u
15
+ 12u
13
+ 10u
11
+ 9u
9
+ 6u
7
+ 3u
5
+ u
u
23
+ 5u
21
+ ··· + 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
37
−12u
36
+46u
35
−104u
34
+244u
33
−468u
32
+837u
31
−1403u
30
+2076u
29
−3097u
28
+
3974u
27
− 5326u
26
+ 6111u
25
− 7416u
24
+ 7798u
23
− 8645u
22
+ 8473u
21
− 8673u
20
+
8000u
19
−7644u
18
+6639u
17
−5962u
16
+4886u
15
−4146u
14
+3236u
13
−2602u
12
+1976u
11
−
1512u
10
+ 1108u
9
− 780u
8
+ 540u
7
− 353u
6
+ 237u
5
− 147u
4
+ 100u
3
− 62u
2
+ 37u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
38
+ 57u
37
+ ··· + 4u + 1
c
2
, c
4
u
38
− 11u
37
+ ··· + 10u − 1
c
3
, c
7
u
38
− u
37
+ ··· − 1024u − 1024
c
5
u
38
+ 10u
37
+ ··· + 313u + 43
c
6
, c
10
u
38
+ 2u
37
+ ··· + 5u + 1
c
8
u
38
+ 2u
37
+ ··· + 3u + 1
c
9
u
38
− 2u
37
+ ··· − 24u + 8
c
11
u
38
− 18u
37
+ ··· + 5u + 1
c
12
u
38
− 6u
37
+ ··· − 2663u + 61
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
38
− 141y
37
+ ··· − 12y + 1
c
2
, c
4
y
38
− 57y
37
+ ··· − 4y + 1
c
3
, c
7
y
38
− 63y
37
+ ··· + 8912896y + 1048576
c
5
y
38
+ 18y
37
+ ··· + 243y + 1849
c
6
, c
10
y
38
+ 18y
37
+ ··· − 5y + 1
c
8
y
38
− 78y
37
+ ··· − 5y + 1
c
9
y
38
− 6y
37
+ ··· − 1680y + 64
c
11
y
38
+ 6y
37
+ ··· − 77y + 1
c
12
y
38
− 18y
37
+ ··· − 5705405y + 3721
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.190184 + 1.006430I
a = 0.666672 + 0.650134I
b = 0.171965 − 0.057486I
−0.043597 + 0.941262I −5.60914 − 1.60302I
u = 0.190184 − 1.006430I
a = 0.666672 − 0.650134I
b = 0.171965 + 0.057486I
−0.043597 − 0.941262I −5.60914 + 1.60302I
u = −0.330777 + 0.907409I
a = 0.06706 + 2.30485I
b = 1.66441 − 0.88267I
−0.97161 + 1.42227I −5.26135 − 0.25979I
u = −0.330777 − 0.907409I
a = 0.06706 − 2.30485I
b = 1.66441 + 0.88267I
−0.97161 − 1.42227I −5.26135 + 0.25979I
u = −0.732889 + 0.615448I
a = −0.77820 − 1.38388I
b = 0.885934 + 0.086371I
−16.4068 + 4.3334I −12.33818 − 2.98248I
u = −0.732889 − 0.615448I
a = −0.77820 + 1.38388I
b = 0.885934 − 0.086371I
−16.4068 − 4.3334I −12.33818 + 2.98248I
u = 0.392112 + 1.025700I
a = −0.398310 − 0.990025I
b = 0.344483 + 0.379769I
1.28562 − 2.93709I −3.65871 + 5.49454I
u = 0.392112 − 1.025700I
a = −0.398310 + 0.990025I
b = 0.344483 − 0.379769I
1.28562 + 2.93709I −3.65871 − 5.49454I
u = 0.808282 + 0.376060I
a = −0.308296 − 1.239560I
b = 2.75261 − 1.36307I
−15.0919 + 7.2035I −11.42537 − 2.83717I
u = 0.808282 − 0.376060I
a = −0.308296 + 1.239560I
b = 2.75261 + 1.36307I
−15.0919 − 7.2035I −11.42537 + 2.83717I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.283537 + 1.096170I
a = −0.116887 − 1.308640I
b = −0.749932 + 0.972888I
3.86766 + 0.18519I 3.63146 − 0.12416I
u = −0.283537 − 1.096170I
a = −0.116887 + 1.308640I
b = −0.749932 − 0.972888I
3.86766 − 0.18519I 3.63146 + 0.12416I
u = −0.694068 + 0.485559I
a = 1.073670 − 0.121248I
b = −0.86069 − 1.59817I
−4.83970 + 0.46505I −12.83571 − 0.64239I
u = −0.694068 − 0.485559I
a = 1.073670 + 0.121248I
b = −0.86069 + 1.59817I
−4.83970 − 0.46505I −12.83571 + 0.64239I
u = 0.725358 + 0.427516I
a = 0.910150 + 0.189413I
b = −1.062290 + 0.018396I
−4.53414 + 2.77322I −12.23678 − 1.99066I
u = 0.725358 − 0.427516I
a = 0.910150 − 0.189413I
b = −1.062290 − 0.018396I
−4.53414 − 2.77322I −12.23678 + 1.99066I
u = 0.520749 + 1.037540I
a = −0.591726 − 0.036334I
b = −0.0031098 − 0.1233600I
0.43968 − 3.37790I −4.33141 + 2.37402I
u = 0.520749 − 1.037540I
a = −0.591726 + 0.036334I
b = −0.0031098 + 0.1233600I
0.43968 + 3.37790I −4.33141 − 2.37402I
u = −0.635679 + 0.975329I
a = −0.60721 + 1.51515I
b = −0.083609 − 0.380918I
−15.3384 + 0.8717I −10.72381 − 2.45601I
u = −0.635679 − 0.975329I
a = −0.60721 − 1.51515I
b = −0.083609 + 0.380918I
−15.3384 − 0.8717I −10.72381 + 2.45601I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.169305 + 1.157370I
a = −2.96351 + 1.28333I
b = 1.31226 − 2.38449I
−10.03480 + 4.57409I −5.57316 − 1.67266I
u = 0.169305 − 1.157370I
a = −2.96351 − 1.28333I
b = 1.31226 + 2.38449I
−10.03480 − 4.57409I −5.57316 + 1.67266I
u = −0.578647 + 1.054140I
a = −1.47418 − 2.16069I
b = −0.53165 + 2.48877I
−3.15884 + 4.44653I −9.57300 − 4.70180I
u = −0.578647 − 1.054140I
a = −1.47418 + 2.16069I
b = −0.53165 − 2.48877I
−3.15884 − 4.44653I −9.57300 + 4.70180I
u = −0.698518 + 0.329444I
a = −0.420323 + 0.439871I
b = 0.191136 + 0.950537I
−0.20642 − 2.47480I −3.77716 + 2.78494I
u = −0.698518 − 0.329444I
a = −0.420323 − 0.439871I
b = 0.191136 − 0.950537I
−0.20642 + 2.47480I −3.77716 − 2.78494I
u = 0.581924 + 1.085040I
a = 0.77887 + 1.33818I
b = −0.985828 − 0.678181I
−2.59793 − 7.77172I −8.83321 + 6.58917I
u = 0.581924 − 1.085040I
a = 0.77887 − 1.33818I
b = −0.985828 + 0.678181I
−2.59793 + 7.77172I −8.83321 − 6.58917I
u = 0.561548 + 0.523699I
a = −0.514245 + 0.265142I
b = 0.143026 + 0.191094I
−1.11182 − 0.98490I −6.72845 + 3.76025I
u = 0.561548 − 0.523699I
a = −0.514245 − 0.265142I
b = 0.143026 − 0.191094I
−1.11182 + 0.98490I −6.72845 − 3.76025I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.424583 + 1.162280I
a = 3.07997 + 1.73117I
b = −3.42587 + 1.10441I
−6.91800 − 4.12327I −5.26243 + 3.48548I
u = 0.424583 − 1.162280I
a = 3.07997 − 1.73117I
b = −3.42587 − 1.10441I
−6.91800 + 4.12327I −5.26243 − 3.48548I
u = −0.549992 + 1.111450I
a = 0.916076 + 1.068210I
b = 0.33127 − 1.48940I
2.05645 + 7.27341I −0.29909 − 6.10879I
u = −0.549992 − 1.111450I
a = 0.916076 − 1.068210I
b = 0.33127 + 1.48940I
2.05645 − 7.27341I −0.29909 + 6.10879I
u = 0.738732
a = −1.26309
b = −2.66464
−10.3080 −9.27210
u = 0.596983 + 1.127320I
a = −0.65101 − 3.73015I
b = 3.48928 + 1.94339I
−12.8566 − 12.4568I −8.47504 + 6.79450I
u = 0.596983 − 1.127320I
a = −0.65101 + 3.73015I
b = 3.48928 − 1.94339I
−12.8566 + 12.4568I −8.47504 − 6.79450I
u = 0.327423
a = −1.07406
b = 0.497882
−1.00232 −10.1070
8
II. I
u
2
= hb + u, −u
3
+ a − u + 1, u
4
+ u
2
− u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
10
=
u
u
3
+ u
a
12
=
u
3
u
2
a
3
=
u
3
+ u − 1
−u
a
5
=
−u
3
+ u
2
− u + 1
−u
2
+ u − 1
a
2
=
2u
3
− u
2
+ 2u − 2
u
2
− 2u + 1
a
4
=
u
3
+ u − 1
−u
a
9
=
−u
3
u
3
+ u
a
1
=
u
3
− u
2
+ u − 1
u
2
− u + 1
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
3
+ 4u
2
− u − 10
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
7
u
4
c
4
(u + 1)
4
c
5
u
4
+ 2u
3
+ 3u
2
+ u + 1
c
6
u
4
+ u
2
− u + 1
c
8
, c
10
, c
12
u
4
+ u
2
+ u + 1
c
9
u
4
+ 3u
3
+ 4u
2
+ 3u + 2
c
11
u
4
− 2u
3
+ 3u
2
− u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
7
y
4
c
5
, c
11
y
4
+ 2y
3
+ 7y
2
+ 5y + 1
c
6
, c
8
, c
10
c
12
y
4
+ 2y
3
+ 3y
2
+ y + 1
c
9
y
4
− y
3
+ 2y
2
+ 7y + 4
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.547424 + 0.585652I
a = −0.851808 + 0.911292I
b = −0.547424 − 0.585652I
−2.62503 − 1.39709I −11.91838 + 2.95607I
u = 0.547424 − 0.585652I
a = −0.851808 − 0.911292I
b = −0.547424 + 0.585652I
−2.62503 + 1.39709I −11.91838 − 2.95607I
u = −0.547424 + 1.120870I
a = 0.351808 + 0.720342I
b = 0.547424 − 1.120870I
0.98010 + 7.64338I −7.58162 − 7.23121I
u = −0.547424 − 1.120870I
a = 0.351808 − 0.720342I
b = 0.547424 + 1.120870I
0.98010 − 7.64338I −7.58162 + 7.23121I
12
III. I
u
3
=
h−u
5
−u
4
−2u
3
−2u
2
+b−2u−1, u
4
+u
2
+a, u
6
+u
5
+2u
4
+2u
3
+2u
2
+2u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u
2
a
10
=
u
u
3
+ u
a
12
=
u
3
u
5
+ u
3
+ u
a
3
=
−u
4
− u
2
u
5
+ u
4
+ 2u
3
+ 2u
2
+ 2u + 1
a
5
=
u
4
+ u
2
+ u + 1
−2u
5
− u
4
− 3u
3
− 2u
2
− 3u − 2
a
2
=
−2u
4
− 2u
2
− u − 1
3u
5
+ 2u
4
+ 5u
3
+ 4u
2
+ 5u + 3
a
4
=
−u
4
− u
2
u
5
+ u
4
+ 2u
3
+ 2u
2
+ 2u + 1
a
9
=
−u
3
u
3
+ u
a
1
=
−u
4
− u
2
− u − 1
2u
5
+ u
4
+ 3u
3
+ 2u
2
+ 3u + 2
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
4
+ 3u
3
− u
2
+ 4u − 7
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
7
u
6
c
4
(u + 1)
6
c
5
u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1
c
6
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
8
, c
10
, c
12
u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1
c
9
(u
3
− u
2
+ 1)
2
c
11
u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
7
y
6
c
5
, c
11
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
c
6
, c
8
, c
10
c
12
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
9
(y
3
− y
2
+ 2y −1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.498832 + 1.001300I
a = 1.183530 + 0.507021I
b = −1.39861 + 0.80012I
−1.37919 − 2.82812I −7.94996 + 3.74291I
u = 0.498832 − 1.001300I
a = 1.183530 − 0.507021I
b = −1.39861 − 0.80012I
−1.37919 + 2.82812I −7.94996 − 3.74291I
u = −0.284920 + 1.115140I
a = 0.215080 − 0.841795I
b = −0.784920 + 0.841795I
2.75839 −4.80521 + 0.27335I
u = −0.284920 − 1.115140I
a = 0.215080 + 0.841795I
b = −0.784920 − 0.841795I
2.75839 −4.80521 − 0.27335I
u = −0.713912 + 0.305839I
a = −0.398606 + 0.800120I
b = 0.183526 + 0.507021I
−1.37919 − 2.82812I −10.74483 + 3.34054I
u = −0.713912 − 0.305839I
a = −0.398606 − 0.800120I
b = 0.183526 − 0.507021I
−1.37919 + 2.82812I −10.74483 − 3.34054I
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
10
)(u
38
+ 57u
37
+ ··· + 4u + 1)
c
2
((u − 1)
10
)(u
38
− 11u
37
+ ··· + 10u − 1)
c
3
, c
7
u
10
(u
38
− u
37
+ ··· − 1024u − 1024)
c
4
((u + 1)
10
)(u
38
− 11u
37
+ ··· + 10u − 1)
c
5
(u
4
+ 2u
3
+ 3u
2
+ u + 1)(u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
· (u
38
+ 10u
37
+ ··· + 313u + 43)
c
6
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
38
+ 2u
37
+ ··· + 5u + 1)
c
8
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
38
+ 2u
37
+ ··· + 3u + 1)
c
9
((u
3
− u
2
+ 1)
2
)(u
4
+ 3u
3
+ ··· + 3u + 2)(u
38
− 2u
37
+ ··· − 24u + 8)
c
10
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
38
+ 2u
37
+ ··· + 5u + 1)
c
11
(u
4
− 2u
3
+ 3u
2
− u + 1)(u
6
− 3u
5
+ 4u
4
− 2u
3
+ 1)
· (u
38
− 18u
37
+ ··· + 5u + 1)
c
12
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
38
− 6u
37
+ ··· − 2663u + 61)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
10
)(y
38
− 141y
37
+ ··· − 12y + 1)
c
2
, c
4
((y −1)
10
)(y
38
− 57y
37
+ ··· − 4y + 1)
c
3
, c
7
y
10
(y
38
− 63y
37
+ ··· + 8912896y + 1048576)
c
5
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
38
+ 18y
37
+ ··· + 243y + 1849)
c
6
, c
10
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
38
+ 18y
37
+ ··· − 5y + 1)
c
8
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
38
− 78y
37
+ ··· − 5y + 1)
c
9
(y
3
− y
2
+ 2y −1)
2
(y
4
− y
3
+ 2y
2
+ 7y + 4)
· (y
38
− 6y
37
+ ··· − 1680y + 64)
c
11
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
38
+ 6y
37
+ ··· − 77y + 1)
c
12
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
38
− 18y
37
+ ··· − 5705405y + 3721)
18
