12a
0579
(K12a
0579
)
A knot diagram
1
Linearized knot diagam
3 7 9 10 11 2 1 6 12 5 4 8
Solving Sequence
3,7
2 1 8 6 9 4 12 10 11 5
c
2
c
1
c
7
c
6
c
8
c
3
c
12
c
9
c
11
c
5
c
4
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
87
+ 2u
86
+ ··· − 3u − 1i
I
u
2
= hu − 1i
* 2 irreducible components of dim
C
= 0, with total 88 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
87
+ 2u
86
+ · · · − 3u − 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
8
=
u
5
− 2u
3
+ u
u
5
− u
3
+ u
a
6
=
u
−u
3
+ u
a
9
=
−u
9
+ 2u
7
− u
5
− 2u
3
+ u
u
11
− 3u
9
+ 4u
7
− u
5
− u
3
+ u
a
4
=
−u
20
+ 5u
18
− 11u
16
+ 10u
14
+ 2u
12
− 13u
10
+ 9u
8
− 3u
4
+ u
2
+ 1
u
22
− 6u
20
+ 17u
18
− 26u
16
+ 20u
14
− 13u
10
+ 10u
8
− u
6
− 2u
4
+ u
2
a
12
=
−u
8
+ 3u
6
− 3u
4
+ 1
−u
8
+ 2u
6
− 2u
4
a
10
=
−u
27
+ 8u
25
+ ··· + 4u
5
− u
3
−u
27
+ 7u
25
+ ··· − u
3
+ u
a
11
=
u
50
− 13u
48
+ ··· + u
2
+ 1
−u
52
+ 14u
50
+ ··· − 6u
8
− u
4
a
5
=
u
76
− 21u
74
+ ··· + u
2
+ 1
u
76
− 20u
74
+ ··· + 10u
8
− u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
86
− 92u
84
+ ··· − 4u
2
+ 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
87
+ 46u
86
+ ··· − u + 1
c
2
, c
6
u
87
− 2u
86
+ ··· − 3u + 1
c
3
u
87
− 2u
86
+ ··· − 15553u + 1789
c
4
, c
5
, c
10
u
87
− 39u
85
+ ··· − u + 1
c
7
, c
12
u
87
− 3u
86
+ ··· + 59u + 11
c
8
u
87
− 12u
86
+ ··· + 3u + 1
c
9
u
87
− 18u
86
+ ··· + 65191u − 4073
c
11
u
87
− 3u
86
+ ··· − 3u + 11
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
87
− 10y
86
+ ··· + 3y −1
c
2
, c
6
y
87
− 46y
86
+ ··· − y −1
c
3
y
87
− 22y
86
+ ··· + 148162943y −3200521
c
4
, c
5
, c
10
y
87
− 78y
86
+ ··· − y −1
c
7
, c
12
y
87
+ 69y
86
+ ··· − 1579y −121
c
8
y
87
+ 2y
86
+ ··· + 187y −1
c
9
y
87
+ 26y
86
+ ··· − 406378973y −16589329
c
11
y
87
− 3y
86
+ ··· + 581y −121
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.966141 + 0.257989I
3.05247 − 1.13677I 0
u = −0.966141 − 0.257989I
3.05247 + 1.13677I 0
u = −0.884260 + 0.473666I
1.43118 + 4.23576I 0
u = −0.884260 − 0.473666I
1.43118 − 4.23576I 0
u = −0.840538 + 0.517807I
1.69738 + 3.45223I 0
u = −0.840538 − 0.517807I
1.69738 − 3.45223I 0
u = 0.862004 + 0.537117I
0.53831 − 7.10255I 0
u = 0.862004 − 0.537117I
0.53831 + 7.10255I 0
u = 0.818200 + 0.544961I
7.81764 − 1.63453I 7.64996 + 3.75286I
u = 0.818200 − 0.544961I
7.81764 + 1.63453I 7.64996 − 3.75286I
u = −1.014630 + 0.076379I
−3.56476 + 3.14041I 0
u = −1.014630 − 0.076379I
−3.56476 − 3.14041I 0
u = −0.862976 + 0.550546I
6.01709 + 10.55840I 0
u = −0.862976 − 0.550546I
6.01709 − 10.55840I 0
u = 1.034630 + 0.100691I
1.61270 − 6.52842I 0
u = 1.034630 − 0.100691I
1.61270 + 6.52842I 0
u = 0.877745 + 0.381441I
−1.57258 − 1.45395I −5.27659 + 2.81251I
u = 0.877745 − 0.381441I
−1.57258 + 1.45395I −5.27659 − 2.81251I
u = 0.949322
−1.75071 −4.59820
u = 0.707240 + 0.548889I
8.13382 − 2.76721I 8.70750 + 3.54192I
u = 0.707240 − 0.548889I
8.13382 + 2.76721I 8.70750 − 3.54192I
u = −0.644779 + 0.563536I
6.63286 − 6.10066I 6.67875 + 3.11647I
u = −0.644779 − 0.563536I
6.63286 + 6.10066I 6.67875 − 3.11647I
u = −0.682539 + 0.509279I
2.15178 + 0.76326I 5.46969 − 3.60542I
u = −0.682539 − 0.509279I
2.15178 − 0.76326I 5.46969 + 3.60542I
u = 0.641757 + 0.540416I
1.15794 + 2.74428I 2.34838 − 3.28740I
u = 0.641757 − 0.540416I
1.15794 − 2.74428I 2.34838 + 3.28740I
u = −0.147114 + 0.807933I
2.59735 − 11.10170I 2.85354 + 7.04638I
u = −0.147114 − 0.807933I
2.59735 + 11.10170I 2.85354 − 7.04638I
u = 0.139710 + 0.803211I
−2.82602 + 7.50372I −1.71194 − 6.83043I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.139710 − 0.803211I
−2.82602 − 7.50372I −1.71194 + 6.83043I
u = −1.093090 + 0.461867I
3.40048 − 0.74196I 0
u = −1.093090 − 0.461867I
3.40048 + 0.74196I 0
u = −0.104841 + 0.792610I
−1.84325 − 3.95466I −0.78666 + 4.04344I
u = −0.104841 − 0.792610I
−1.84325 + 3.95466I −0.78666 − 4.04344I
u = −0.131118 + 0.787561I
−1.39783 − 3.69601I 1.00853 + 1.78136I
u = −0.131118 − 0.787561I
−1.39783 + 3.69601I 1.00853 − 1.78136I
u = 1.121560 + 0.452590I
−2.20477 − 2.15024I 0
u = 1.121560 − 0.452590I
−2.20477 + 2.15024I 0
u = −0.055872 + 0.787918I
0.12538 + 2.84570I 0.09778 − 2.42036I
u = −0.055872 − 0.787918I
0.12538 − 2.84570I 0.09778 + 2.42036I
u = 0.078455 + 0.785741I
−4.55325 + 0.55742I −5.08529 + 0.67714I
u = 0.078455 − 0.785741I
−4.55325 − 0.55742I −5.08529 − 0.67714I
u = 0.157374 + 0.772504I
4.93345 + 2.24336I 5.56036 − 2.08431I
u = 0.157374 − 0.772504I
4.93345 − 2.24336I 5.56036 + 2.08431I
u = −1.138470 + 0.476384I
−1.96887 + 5.60074I 0
u = −1.138470 − 0.476384I
−1.96887 − 5.60074I 0
u = 1.131420 + 0.493450I
3.87168 − 8.16081I 0
u = 1.131420 − 0.493450I
3.87168 + 8.16081I 0
u = −1.184400 + 0.373392I
0.98186 + 1.54403I 0
u = −1.184400 − 0.373392I
0.98186 − 1.54403I 0
u = 1.202200 + 0.386893I
−5.34474 − 0.27499I 0
u = 1.202200 − 0.386893I
−5.34474 + 0.27499I 0
u = −1.210270 + 0.378326I
−6.87411 − 3.52324I 0
u = −1.210270 − 0.378326I
−6.87411 + 3.52324I 0
u = 1.212320 + 0.372596I
−1.49892 + 7.13914I 0
u = 1.212320 − 0.372596I
−1.49892 − 7.13914I 0
u = 1.207670 + 0.400251I
−5.72997 − 0.13864I 0
u = 1.207670 − 0.400251I
−5.72997 + 0.13864I 0
u = −1.206850 + 0.414420I
−8.33467 + 3.61849I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.206850 − 0.414420I
−8.33467 − 3.61849I 0
u = 1.208380 + 0.424632I
−3.59584 − 7.10442I 0
u = 1.208380 − 0.424632I
−3.59584 + 7.10442I 0
u = 1.179390 + 0.510697I
1.94326 − 7.00326I 0
u = 1.179390 − 0.510697I
1.94326 + 7.00326I 0
u = −1.199560 + 0.477330I
−3.22033 + 1.74513I 0
u = −1.199560 − 0.477330I
−3.22033 − 1.74513I 0
u = 1.196800 + 0.486367I
−7.82292 − 5.20062I 0
u = 1.196800 − 0.486367I
−7.82292 + 5.20062I 0
u = −1.189780 + 0.506005I
−4.50194 + 8.46175I 0
u = −1.189780 − 0.506005I
−4.50194 − 8.46175I 0
u = −1.195880 + 0.496933I
−5.04500 + 8.67949I 0
u = −1.195880 − 0.496933I
−5.04500 − 8.67949I 0
u = 1.193320 + 0.511980I
−5.93104 − 12.33810I 0
u = 1.193320 − 0.511980I
−5.93104 + 12.33810I 0
u = −1.193390 + 0.515647I
−0.4909 + 15.9669I 0
u = −1.193390 − 0.515647I
−0.4909 − 15.9669I 0
u = 0.232114 + 0.655329I
6.46120 + 3.72588I 7.39278 − 3.37040I
u = 0.232114 − 0.655329I
6.46120 − 3.72588I 7.39278 + 3.37040I
u = −0.527497 + 0.441971I
2.35577 − 0.33907I 3.68741 + 0.05432I
u = −0.527497 − 0.441971I
2.35577 + 0.33907I 3.68741 − 0.05432I
u = −0.329405 + 0.589621I
5.58601 + 4.90010I 6.38719 − 3.76074I
u = −0.329405 − 0.589621I
5.58601 − 4.90010I 6.38719 + 3.76074I
u = −0.181861 + 0.602722I
0.74652 − 1.35095I 3.80548 + 4.35021I
u = −0.181861 − 0.602722I
0.74652 + 1.35095I 3.80548 − 4.35021I
u = 0.308308 + 0.533007I
0.19366 − 1.78298I 1.92568 + 4.24061I
u = 0.308308 − 0.533007I
0.19366 + 1.78298I 1.92568 − 4.24061I
7
II. I
u
2
= hu − 1i
(i) Arc colorings
a
3
=
1
0
a
7
=
0
1
a
2
=
1
−1
a
1
=
0
−1
a
8
=
0
1
a
6
=
1
0
a
9
=
−1
1
a
4
=
0
1
a
12
=
0
−1
a
10
=
−1
0
a
11
=
0
−1
a
5
=
1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
10
u + 1
c
7
, c
11
, c
12
u
c
9
u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
8
, c
9
, c
10
y −1
c
7
, c
11
, c
12
y
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
−1.64493 −6.00000
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u + 1)(u
87
+ 46u
86
+ ··· − u + 1)
c
2
, c
6
(u + 1)(u
87
− 2u
86
+ ··· − 3u + 1)
c
3
(u + 1)(u
87
− 2u
86
+ ··· − 15553u + 1789)
c
4
, c
5
, c
10
(u + 1)(u
87
− 39u
85
+ ··· − u + 1)
c
7
, c
12
u(u
87
− 3u
86
+ ··· + 59u + 11)
c
8
(u + 1)(u
87
− 12u
86
+ ··· + 3u + 1)
c
9
(u − 1)(u
87
− 18u
86
+ ··· + 65191u − 4073)
c
11
u(u
87
− 3u
86
+ ··· − 3u + 11)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
87
− 10y
86
+ ··· + 3y −1)
c
2
, c
6
(y −1)(y
87
− 46y
86
+ ··· − y −1)
c
3
(y −1)(y
87
− 22y
86
+ ··· + 1.48163 × 10
8
y −3200521)
c
4
, c
5
, c
10
(y −1)(y
87
− 78y
86
+ ··· − y −1)
c
7
, c
12
y(y
87
+ 69y
86
+ ··· − 1579y −121)
c
8
(y −1)(y
87
+ 2y
86
+ ··· + 187y −1)
c
9
(y −1)(y
87
+ 26y
86
+ ··· − 4.06379 × 10
8
y −1.65893 × 10
7
)
c
11
y(y
87
− 3y
86
+ ··· + 581y −121)
13
