12a
0375
(K12a
0375
)
A knot diagram
1
Linearized knot diagam
3 6 9 10 2 12 1 11 4 5 8 7
Solving Sequence
4,10 2,5
6 11 9 3 1 8 12 7
c
4
c
5
c
10
c
9
c
3
c
1
c
8
c
11
c
7
c
2
, c
6
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−4u
42
+ u
41
+ ··· + 4b − 4, 2u
42
− u
41
+ ··· + 4a − 2, u
43
− 2u
42
+ ··· + 2u
2
− 2i
I
u
2
= h2u
5
a − 2u
5
− a
2
u
2
− 8u
3
a + 2u
2
a + 8u
3
+ 2a
2
+ 8au − 2u
2
+ b − 4a − 8u + 4,
2u
5
a
2
− u
5
a − 6u
3
a
2
+ u
5
+ 2a
2
u
2
+ 6u
3
a + a
3
+ 4a
2
u − u
2
a − 4u
3
− 4a
2
− 8au + u
2
+ 5a + 4u − 3,
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1i
I
u
3
= hb + u − 1, 2a − u, u
2
− 2i
I
v
1
= ha, b + 1, v −1i
* 4 irreducible components of dim
C
= 0, with total 64 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−4u
42
+u
41
+· · ·+4b−4, 2u
42
−u
41
+· · ·+4a−2, u
43
−2u
42
+· · ·+2u
2
−2i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
2
=
−
1
2
u
42
+
1
4
u
41
+ ··· −
1
2
u +
1
2
u
42
−
1
4
u
41
+ ··· +
1
2
u + 1
a
5
=
1
−u
2
a
6
=
1
2
u
42
−
23
2
u
40
+ ··· + u
2
+
3
2
−u
42
+
1
4
u
41
+ ··· −
1
2
u − 1
a
11
=
u
−u
3
+ u
a
9
=
−u
u
a
3
=
−u
2
+ 1
u
2
a
1
=
−
1
2
u
42
+
21
2
u
40
+ ··· − u −
1
2
u
42
− 22u
40
+ ··· +
1
2
u + 1
a
8
=
u
5
− 2u
3
− u
−u
7
+ 3u
5
− 2u
3
+ u
a
12
=
u
9
− 4u
7
+ 3u
5
+ 2u
3
+ u
−u
11
+ 5u
9
− 8u
7
+ 5u
5
− 3u
3
+ u
a
7
=
1
4
u
32
−
17
4
u
30
+ ··· −
3
2
u +
1
2
−
1
4
u
32
+ 4u
30
+ ··· −
1
2
u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
42
− 46u
40
+ ··· − 2u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
43
+ 22u
42
+ ··· + 9u + 1
c
2
, c
5
u
43
+ 2u
42
+ ··· + u − 1
c
3
, c
4
, c
9
c
10
u
43
+ 2u
42
+ ··· − 2u
2
+ 2
c
6
, c
7
, c
12
u
43
− 2u
42
+ ··· − 11u − 1
c
8
, c
11
u
43
+ 6u
42
+ ··· + 160u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
43
+ 2y
42
+ ··· + 53y −1
c
2
, c
5
y
43
− 22y
42
+ ··· + 9y −1
c
3
, c
4
, c
9
c
10
y
43
− 46y
42
+ ··· + 8y −4
c
6
, c
7
, c
12
y
43
− 38y
42
+ ··· + 89y −1
c
8
, c
11
y
43
+ 30y
42
+ ··· + 26112y −256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.591746 + 0.636717I
a = −1.23572 − 1.63525I
b = −0.100756 + 0.318643I
−1.84282 + 10.94570I 6.30092 − 8.93673I
u = 0.591746 − 0.636717I
a = −1.23572 + 1.63525I
b = −0.100756 − 0.318643I
−1.84282 − 10.94570I 6.30092 + 8.93673I
u = −0.765729 + 0.373369I
a = −0.32757 + 1.59857I
b = 0.134514 − 0.340307I
5.12970 − 5.35425I 12.2003 + 7.7214I
u = −0.765729 − 0.373369I
a = −0.32757 − 1.59857I
b = 0.134514 + 0.340307I
5.12970 + 5.35425I 12.2003 − 7.7214I
u = 0.826606 + 0.202990I
a = 0.511476 + 0.581636I
b = 0.430064 + 0.050748I
6.06160 + 0.79364I 14.9061 − 0.8864I
u = 0.826606 − 0.202990I
a = 0.511476 − 0.581636I
b = 0.430064 − 0.050748I
6.06160 − 0.79364I 14.9061 + 0.8864I
u = −0.546701 + 0.616087I
a = −1.33752 + 1.76286I
b = −0.098208 − 0.285427I
−6.30176 − 6.51240I 1.84350 + 6.82731I
u = −0.546701 − 0.616087I
a = −1.33752 − 1.76286I
b = −0.098208 + 0.285427I
−6.30176 + 6.51240I 1.84350 − 6.82731I
u = −0.582396 + 0.579810I
a = 0.659858 − 0.192915I
b = 0.279952 − 0.384491I
1.26998 − 5.99398I 9.54778 + 6.06507I
u = −0.582396 − 0.579810I
a = 0.659858 + 0.192915I
b = 0.279952 + 0.384491I
1.26998 + 5.99398I 9.54778 − 6.06507I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.403178 + 0.680325I
a = 0.873527 + 0.218379I
b = −0.795082 − 0.902473I
−2.40444 − 6.51462I 5.07632 + 3.55431I
u = 0.403178 − 0.680325I
a = 0.873527 − 0.218379I
b = −0.795082 + 0.902473I
−2.40444 + 6.51462I 5.07632 − 3.55431I
u = −0.443849 + 0.633747I
a = 0.902439 − 0.228432I
b = −0.888609 + 0.891580I
−6.60622 + 2.27578I 0.698054 − 0.385736I
u = −0.443849 − 0.633747I
a = 0.902439 + 0.228432I
b = −0.888609 − 0.891580I
−6.60622 − 2.27578I 0.698054 + 0.385736I
u = −0.377654 + 0.599705I
a = 0.738904 − 0.108280I
b = 0.103319 − 0.421977I
0.67597 + 1.96643I 8.14582 + 0.08681I
u = −0.377654 − 0.599705I
a = 0.738904 + 0.108280I
b = 0.103319 + 0.421977I
0.67597 − 1.96643I 8.14582 − 0.08681I
u = 1.34352
a = −0.457746
b = 1.25945
6.42503 14.7720
u = 0.596066 + 0.273723I
a = 0.15234 − 2.25587I
b = 0.108473 + 0.195176I
−0.17994 + 3.15116I 7.13825 − 9.28828I
u = 0.596066 − 0.273723I
a = 0.15234 + 2.25587I
b = 0.108473 − 0.195176I
−0.17994 − 3.15116I 7.13825 + 9.28828I
u = −0.084134 + 0.604122I
a = 0.845349 − 0.074679I
b = −0.465947 + 0.534008I
2.98218 + 1.98828I 8.33137 − 3.20557I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.084134 − 0.604122I
a = 0.845349 + 0.074679I
b = −0.465947 − 0.534008I
2.98218 − 1.98828I 8.33137 + 3.20557I
u = −1.43197
a = 0.0229961
b = 1.00111
3.32572 0
u = −1.43137 + 0.20558I
a = 0.132330 − 0.142499I
b = 0.955154 − 0.077955I
3.46742 + 3.34369I 0
u = −1.43137 − 0.20558I
a = 0.132330 + 0.142499I
b = 0.955154 + 0.077955I
3.46742 − 3.34369I 0
u = 1.47158 + 0.11058I
a = −0.881376 − 0.770659I
b = 1.53453 + 1.36753I
6.53226 + 0.41130I 0
u = 1.47158 − 0.11058I
a = −0.881376 + 0.770659I
b = 1.53453 − 1.36753I
6.53226 − 0.41130I 0
u = 1.47924 + 0.17979I
a = 0.131246 + 0.108669I
b = 0.986913 + 0.072464I
−0.365210 + 0.609893I 0
u = 1.47924 − 0.17979I
a = 0.131246 − 0.108669I
b = 0.986913 − 0.072464I
−0.365210 − 0.609893I 0
u = −0.463968
a = 1.94534
b = 0.138255
0.736315 13.7490
u = 1.53818 + 0.18818I
a = 0.21264 + 2.03609I
b = −0.50972 − 4.35591I
0.59349 + 9.42918I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.53818 − 0.18818I
a = 0.21264 − 2.03609I
b = −0.50972 + 4.35591I
0.59349 − 9.42918I 0
u = −1.55142 + 0.05723I
a = −0.80889 − 2.01985I
b = 1.44195 + 4.18487I
7.05051 − 4.24670I 0
u = −1.55142 − 0.05723I
a = −0.80889 + 2.01985I
b = 1.44195 − 4.18487I
7.05051 + 4.24670I 0
u = 1.55532 + 0.17621I
a = −0.711365 − 0.950566I
b = 0.99205 + 1.76418I
8.38956 + 8.75469I 0
u = 1.55532 − 0.17621I
a = −0.711365 + 0.950566I
b = 0.99205 − 1.76418I
8.38956 − 8.75469I 0
u = −1.55776 + 0.19984I
a = 0.19109 − 1.91237I
b = −0.48048 + 4.13035I
5.2865 − 14.0160I 0
u = −1.55776 − 0.19984I
a = 0.19109 + 1.91237I
b = −0.48048 − 4.13035I
5.2865 + 14.0160I 0
u = 0.155892 + 0.389253I
a = 0.939590 + 0.061625I
b = −0.782027 − 0.295786I
−1.51738 − 0.78597I −1.81615 + 1.01522I
u = 0.155892 − 0.389253I
a = 0.939590 − 0.061625I
b = −0.782027 + 0.295786I
−1.51738 + 0.78597I −1.81615 − 1.01522I
u = −1.60026 + 0.04435I
a = −0.79147 + 1.29002I
b = 1.26150 − 2.64623I
14.2395 − 1.6305I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.60026 − 0.04435I
a = −0.79147 − 1.29002I
b = 1.26150 + 2.64623I
14.2395 + 1.6305I 0
u = 1.59968 + 0.08511I
a = −0.45217 + 1.80198I
b = 0.69300 − 3.81498I
13.1581 + 6.9538I 0
u = 1.59968 − 0.08511I
a = −0.45217 − 1.80198I
b = 0.69300 + 3.81498I
13.1581 − 6.9538I 0
9
II. I
u
2
= h2u
5
a − 2u
5
+ · · · − 4a + 4, 2u
5
a
2
− u
5
a + · · · + 5a − 3, u
6
+ u
5
−
3u
4
− 2u
3
+ 2u
2
− u − 1i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
2
=
a
−2u
5
a + 2u
5
+ ··· + 4a − 4
a
5
=
1
−u
2
a
6
=
−2u
5
a + 2u
5
+ ··· + 5a − 4
2u
5
a − 2u
5
+ ··· − 4a + 4
a
11
=
u
−u
3
+ u
a
9
=
−u
u
a
3
=
−u
2
+ 1
u
2
a
1
=
−u
5
a
2
− 2u
5
a + ··· − a
2
+ a
u
5
a
2
− 2u
5
a + ··· + 6a − 6
a
8
=
u
5
− 2u
3
− u
−u
5
+ u
4
+ 2u
3
− 3u
2
+ u + 1
a
12
=
u
2
− 1
−u
2
a
7
=
u
4
a
2
+ u
4
a − 2a
2
u
2
− 3u
2
a + 2u
2
+ 3a − 2
2u
5
a − 2u
5
+ ··· − 6a + 6
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
3
− 8u + 10
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
18
+ 12u
17
+ ··· + 5u + 1
c
2
, c
5
, c
6
c
7
, c
12
u
18
− 6u
16
+ ··· + u − 1
c
3
, c
4
, c
9
c
10
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
3
c
8
, c
11
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
3
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
18
− 12y
17
+ ··· − 17y + 1
c
2
, c
5
, c
6
c
7
, c
12
y
18
− 12y
17
+ ··· − 5y + 1
c
3
, c
4
, c
9
c
10
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
3
c
8
, c
11
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
3
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.493180 + 0.575288I
a = 0.941013 + 0.239784I
b = −1.009960 − 0.876429I
−2.96024 + 1.97241I 4.57572 − 3.68478I
u = 0.493180 + 0.575288I
a = 0.703854 + 0.163676I
b = 0.204229 + 0.389849I
−2.96024 + 1.97241I 4.57572 − 3.68478I
u = 0.493180 + 0.575288I
a = −1.46493 − 2.00338I
b = −0.086351 + 0.244114I
−2.96024 + 1.97241I 4.57572 − 3.68478I
u = 0.493180 − 0.575288I
a = 0.941013 − 0.239784I
b = −1.009960 + 0.876429I
−2.96024 − 1.97241I 4.57572 + 3.68478I
u = 0.493180 − 0.575288I
a = 0.703854 − 0.163676I
b = 0.204229 − 0.389849I
−2.96024 − 1.97241I 4.57572 + 3.68478I
u = 0.493180 − 0.575288I
a = −1.46493 + 2.00338I
b = −0.086351 − 0.244114I
−2.96024 − 1.97241I 4.57572 + 3.68478I
u = −0.483672
a = 1.12121
b = −1.42631
0.738851 13.4170
u = −0.483672
a = 1.85982 + 0.59462I
b = 0.146924 − 0.011821I
0.738851 13.4170
u = −0.483672
a = 1.85982 − 0.59462I
b = 0.146924 + 0.011821I
0.738851 13.4170
u = −1.52087 + 0.16310I
a = −0.751848 + 0.903227I
b = 1.13095 − 1.63417I
3.69558 − 4.59213I 8.58114 + 3.20482I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.52087 + 0.16310I
a = 0.137996 − 0.084846I
b = 1.011050 − 0.070016I
3.69558 − 4.59213I 8.58114 + 3.20482I
u = −1.52087 + 0.16310I
a = 0.15868 − 2.23953I
b = −0.39625 + 4.72775I
3.69558 − 4.59213I 8.58114 + 3.20482I
u = −1.52087 − 0.16310I
a = −0.751848 − 0.903227I
b = 1.13095 + 1.63417I
3.69558 + 4.59213I 8.58114 − 3.20482I
u = −1.52087 − 0.16310I
a = 0.137996 + 0.084846I
b = 1.011050 + 0.070016I
3.69558 + 4.59213I 8.58114 − 3.20482I
u = −1.52087 − 0.16310I
a = 0.15868 + 2.23953I
b = −0.39625 − 4.72775I
3.69558 + 4.59213I 8.58114 − 3.20482I
u = 1.53904
a = 0.110457
b = 1.03249
7.66009 12.2690
u = 1.53904
a = −1.20042 + 1.54308I
b = 2.19632 − 3.14900I
7.66009 12.2690
u = 1.53904
a = −1.20042 − 1.54308I
b = 2.19632 + 3.14900I
7.66009 12.2690
14
III. I
u
3
= hb + u − 1, 2a − u, u
2
− 2i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
2
=
1
2
u
−u + 1
a
5
=
1
−2
a
6
=
1
2
u + 1
−u − 1
a
11
=
u
−u
a
9
=
−u
u
a
3
=
−1
2
a
1
=
1
2
u + 1
−u − 1
a
8
=
−u
u
a
12
=
u
−u
a
7
=
−
1
2
u + 1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
(u − 1)
2
c
2
, c
12
(u + 1)
2
c
3
, c
4
, c
9
c
10
u
2
− 2
c
8
, c
11
u
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
12
(y − 1)
2
c
3
, c
4
, c
9
c
10
(y − 2)
2
c
8
, c
11
y
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = 0.707107
b = −0.414214
4.93480 8.00000
u = −1.41421
a = −0.707107
b = 2.41421
4.93480 8.00000
18
IV. I
v
1
= ha, b + 1, v − 1i
(i) Arc colorings
a
4
=
1
0
a
10
=
1
0
a
2
=
0
−1
a
5
=
1
0
a
6
=
1
1
a
11
=
1
0
a
9
=
1
0
a
3
=
1
0
a
1
=
−1
−1
a
8
=
1
0
a
12
=
1
0
a
7
=
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
u − 1
c
3
, c
4
, c
8
c
9
, c
10
, c
11
u
c
5
, c
6
, c
7
u + 1
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
12
y − 1
c
3
, c
4
, c
8
c
9
, c
10
, c
11
y
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
0 0
22
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
3
)(u
18
+ 12u
17
+ ··· + 5u + 1)(u
43
+ 22u
42
+ ··· + 9u + 1)
c
2
(u − 1)(u + 1)
2
(u
18
− 6u
16
+ ··· + u − 1)(u
43
+ 2u
42
+ ··· + u − 1)
c
3
, c
4
, c
9
c
10
u(u
2
− 2)(u
6
− u
5
+ ··· + u − 1)
3
(u
43
+ 2u
42
+ ··· − 2u
2
+ 2)
c
5
((u − 1)
2
)(u + 1)(u
18
− 6u
16
+ ··· + u − 1)(u
43
+ 2u
42
+ ··· + u − 1)
c
6
, c
7
((u − 1)
2
)(u + 1)(u
18
− 6u
16
+ ··· + u − 1)(u
43
− 2u
42
+ ··· − 11u − 1)
c
8
, c
11
u
3
(u
6
+ u
5
+ ··· + u − 1)
3
(u
43
+ 6u
42
+ ··· + 160u + 16)
c
12
(u − 1)(u + 1)
2
(u
18
− 6u
16
+ ··· + u − 1)(u
43
− 2u
42
+ ··· − 11u − 1)
23
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
3
)(y
18
− 12y
17
+ ··· − 17y + 1)(y
43
+ 2y
42
+ ··· + 53y − 1)
c
2
, c
5
((y − 1)
3
)(y
18
− 12y
17
+ ··· − 5y + 1)(y
43
− 22y
42
+ ··· + 9y − 1)
c
3
, c
4
, c
9
c
10
y(y − 2)
2
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
3
· (y
43
− 46y
42
+ ··· + 8y − 4)
c
6
, c
7
, c
12
((y − 1)
3
)(y
18
− 12y
17
+ ··· − 5y + 1)(y
43
− 38y
42
+ ··· + 89y − 1)
c
8
, c
11
y
3
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
3
· (y
43
+ 30y
42
+ ··· + 26112y − 256)
24
