12a
0221
(K12a
0221
)
A knot diagram
1
Linearized knot diagam
3 6 7 9 10 2 1 12 11 5 4 8
Solving Sequence
2,6
3 7 4 1 8 12 9 11 10 5
c
2
c
6
c
3
c
1
c
7
c
12
c
8
c
11
c
9
c
5
c
4
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
84
+ u
83
+ ··· + 2u + 1i
* 1 irreducible components of dim
C
= 0, with total 84 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
84
+ u
83
+ · · · + 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
−u
2
a
7
=
u
u
a
4
=
u
4
+ u
2
+ 1
u
4
a
1
=
u
2
+ 1
−u
4
a
8
=
−u
7
− 2u
5
− 2u
3
u
9
+ u
7
+ u
5
+ u
a
12
=
u
12
+ 3u
10
+ 5u
8
+ 4u
6
+ 2u
4
+ u
2
+ 1
−u
14
− 2u
12
− 3u
10
− 2u
8
− 2u
6
− 2u
4
− u
2
a
9
=
−u
17
− 4u
15
− 9u
13
− 12u
11
− 11u
9
− 8u
7
− 6u
5
− 4u
3
− u
u
19
+ 3u
17
+ 6u
15
+ 7u
13
+ 7u
11
+ 7u
9
+ 6u
7
+ 4u
5
+ u
3
+ u
a
11
=
u
22
+ 5u
20
+ ··· + 2u
2
+ 1
u
22
+ 4u
20
+ 9u
18
+ 12u
16
+ 10u
14
+ 6u
12
+ 3u
10
+ 2u
8
− u
6
− 2u
4
− u
2
a
10
=
u
63
+ 14u
61
+ ··· + 56u
7
+ 12u
5
u
63
+ 13u
61
+ ··· − 6u
7
+ u
a
5
=
u
40
+ 9u
38
+ ··· + 2u
2
+ 1
−u
42
− 8u
40
+ ··· − 2u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
83
+ 4u
82
+ ··· + 12u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
84
+ 37u
83
+ ··· + 2u + 1
c
2
, c
6
u
84
− u
83
+ ··· − 2u + 1
c
3
u
84
+ u
83
+ ··· − 2u + 1
c
4
u
84
+ u
83
+ ··· − 1110u + 1237
c
5
, c
10
u
84
− u
83
+ ··· + u
2
+ 1
c
7
, c
8
, c
12
u
84
− 5u
83
+ ··· − 150u + 13
c
9
u
84
− 41u
83
+ ··· − 2u + 1
c
11
u
84
− 5u
83
+ ··· − 20u + 133
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
84
+ 21y
83
+ ··· + 10y + 1
c
2
, c
6
y
84
+ 37y
83
+ ··· + 2y + 1
c
3
y
84
+ 5y
83
+ ··· + 66y + 1
c
4
y
84
− 31y
83
+ ··· − 27364962y + 1530169
c
5
, c
10
y
84
+ 41y
83
+ ··· + 2y + 1
c
7
, c
8
, c
12
y
84
+ 89y
83
+ ··· + 11742y + 169
c
9
y
84
+ 5y
83
+ ··· + 10y + 1
c
11
y
84
− 11y
83
+ ··· + 78070y + 17689
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.223462 + 0.933764I
−0.07570 − 1.69164I 0
u = −0.223462 − 0.933764I
−0.07570 + 1.69164I 0
u = 0.791209 + 0.516161I
10.32730 + 7.27403I 5.32581 − 5.73192I
u = 0.791209 − 0.516161I
10.32730 − 7.27403I 5.32581 + 5.73192I
u = 0.037106 + 1.056750I
−0.19944 − 2.21631I 0
u = 0.037106 − 1.056750I
−0.19944 + 2.21631I 0
u = 0.796662 + 0.503525I
12.03280 − 0.97322I 7.54914 + 0.I
u = 0.796662 − 0.503525I
12.03280 + 0.97322I 7.54914 + 0.I
u = −0.787400 + 0.509956I
7.67805 − 2.34302I 0
u = −0.787400 − 0.509956I
7.67805 + 2.34302I 0
u = 0.543765 + 0.916385I
2.63357 − 1.71163I 0
u = 0.543765 − 0.916385I
2.63357 + 1.71163I 0
u = −0.805009 + 0.474254I
11.86550 + 2.32230I 7.32986 + 0.I
u = −0.805009 − 0.474254I
11.86550 − 2.32230I 7.32986 + 0.I
u = −0.807000 + 0.462064I
10.0185 + 10.5475I 4.83061 − 6.04400I
u = −0.807000 − 0.462064I
10.0185 − 10.5475I 4.83061 + 6.04400I
u = −0.508321 + 0.945367I
0.05765 − 2.39517I 0
u = −0.508321 − 0.945367I
0.05765 + 2.39517I 0
u = 0.801408 + 0.464683I
7.42012 − 5.56366I 0. + 2.51060I
u = 0.801408 − 0.464683I
7.42012 + 5.56366I 0. − 2.51060I
u = 0.308336 + 1.036850I
−3.73977 − 0.08549I 0
u = 0.308336 − 1.036850I
−3.73977 + 0.08549I 0
u = −0.285037 + 1.047880I
−1.75442 + 4.74800I 0
u = −0.285037 − 1.047880I
−1.75442 − 4.74800I 0
u = −0.769820 + 0.487123I
5.19035 − 0.86260I 1.23878 + 3.04762I
u = −0.769820 − 0.487123I
5.19035 + 0.86260I 1.23878 − 3.04762I
u = 0.778145 + 0.470305I
5.09099 − 3.85034I 0.89323 + 3.65261I
u = 0.778145 − 0.470305I
5.09099 + 3.85034I 0.89323 − 3.65261I
u = −0.339809 + 0.830779I
−0.13330 − 1.63331I −0.57375 + 3.86192I
u = −0.339809 − 0.830779I
−0.13330 + 1.63331I −0.57375 − 3.86192I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.358186 + 1.043080I
−4.14324 + 1.55939I 0
u = 0.358186 − 1.043080I
−4.14324 − 1.55939I 0
u = 0.043589 + 1.113230I
1.93267 − 3.70749I 0
u = 0.043589 − 1.113230I
1.93267 + 3.70749I 0
u = 0.551965 + 0.969258I
3.24663 + 5.73101I 0
u = 0.551965 − 0.969258I
3.24663 − 5.73101I 0
u = −0.026149 + 1.121100I
6.27570 + 0.52297I 0
u = −0.026149 − 1.121100I
6.27570 − 0.52297I 0
u = −0.381786 + 1.056500I
−2.57890 − 6.08174I 0
u = −0.381786 − 1.056500I
−2.57890 + 6.08174I 0
u = −0.047926 + 1.123680I
4.49933 + 8.62941I 0
u = −0.047926 − 1.123680I
4.49933 − 8.62941I 0
u = 0.564858 + 0.667954I
3.34666 + 6.18409I 4.75304 − 7.47482I
u = 0.564858 − 0.667954I
3.34666 − 6.18409I 4.75304 + 7.47482I
u = −0.458432 + 1.061660I
−2.07874 − 0.79068I 0
u = −0.458432 − 1.061660I
−2.07874 + 0.79068I 0
u = 0.483376 + 1.064070I
−3.30588 + 5.23729I 0
u = 0.483376 − 1.064070I
−3.30588 − 5.23729I 0
u = −0.510868 + 0.650088I
0.91179 − 1.80281I 1.30059 + 3.91326I
u = −0.510868 − 0.650088I
0.91179 + 1.80281I 1.30059 − 3.91326I
u = −0.534113 + 1.054150I
1.75841 − 4.59094I 0
u = −0.534113 − 1.054150I
1.75841 + 4.59094I 0
u = 0.575651 + 0.581028I
4.35803 − 1.20708I 6.98007 + 0.38131I
u = 0.575651 − 0.581028I
4.35803 + 1.20708I 6.98007 − 0.38131I
u = 0.513175 + 1.074520I
−2.37945 + 6.93925I 0
u = 0.513175 − 1.074520I
−2.37945 − 6.93925I 0
u = −0.521503 + 1.082880I
−0.20890 − 11.73350I 0
u = −0.521503 − 1.082880I
−0.20890 + 11.73350I 0
u = −0.616198 + 1.072140I
3.44700 − 4.37796I 0
u = −0.616198 − 1.072140I
3.44700 + 4.37796I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.632761 + 1.065390I
6.01874 − 3.00258I 0
u = −0.632761 − 1.065390I
6.01874 + 3.00258I 0
u = 0.637028 + 1.063380I
8.69167 − 1.90319I 0
u = 0.637028 − 1.063380I
8.69167 + 1.90319I 0
u = 0.616009 + 1.082200I
3.26859 + 9.11086I 0
u = 0.616009 − 1.082200I
3.26859 − 9.11086I 0
u = 0.635377 + 1.071920I
10.33310 + 6.35256I 0
u = 0.635377 − 1.071920I
10.33310 − 6.35256I 0
u = −0.628985 + 1.088730I
10.02760 − 7.69935I 0
u = −0.628985 − 1.088730I
10.02760 + 7.69935I 0
u = 0.624160 + 1.091740I
5.54652 + 10.91310I 0
u = 0.624160 − 1.091740I
5.54652 − 10.91310I 0
u = −0.625644 + 1.094730I
8.1266 − 15.9167I 0
u = −0.625644 − 1.094730I
8.1266 + 15.9167I 0
u = −0.607812 + 0.381445I
3.64708 + 0.08854I 5.69951 − 0.71462I
u = −0.607812 − 0.381445I
3.64708 − 0.08854I 5.69951 + 0.71462I
u = −0.632909 + 0.297597I
1.98999 + 7.25331I 1.90672 − 7.55824I
u = −0.632909 − 0.297597I
1.98999 − 7.25331I 1.90672 + 7.55824I
u = 0.594633 + 0.293043I
−0.22402 − 2.57780I −1.72559 + 3.89319I
u = 0.594633 − 0.293043I
−0.22402 + 2.57780I −1.72559 − 3.89319I
u = 0.508540 + 0.204173I
−1.08367 − 1.23781I −4.17334 + 4.04965I
u = 0.508540 − 0.204173I
−1.08367 + 1.23781I −4.17334 − 4.04965I
u = −0.512235 + 0.097566I
0.33885 − 2.95765I −1.40754 + 2.86155I
u = −0.512235 − 0.097566I
0.33885 + 2.95765I −1.40754 − 2.86155I
7
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
84
+ 37u
83
+ ··· + 2u + 1
c
2
, c
6
u
84
− u
83
+ ··· − 2u + 1
c
3
u
84
+ u
83
+ ··· − 2u + 1
c
4
u
84
+ u
83
+ ··· − 1110u + 1237
c
5
, c
10
u
84
− u
83
+ ··· + u
2
+ 1
c
7
, c
8
, c
12
u
84
− 5u
83
+ ··· − 150u + 13
c
9
u
84
− 41u
83
+ ··· − 2u + 1
c
11
u
84
− 5u
83
+ ··· − 20u + 133
8
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
84
+ 21y
83
+ ··· + 10y + 1
c
2
, c
6
y
84
+ 37y
83
+ ··· + 2y + 1
c
3
y
84
+ 5y
83
+ ··· + 66y + 1
c
4
y
84
− 31y
83
+ ··· − 27364962y + 1530169
c
5
, c
10
y
84
+ 41y
83
+ ··· + 2y + 1
c
7
, c
8
, c
12
y
84
+ 89y
83
+ ··· + 11742y + 169
c
9
y
84
+ 5y
83
+ ··· + 10y + 1
c
11
y
84
− 11y
83
+ ··· + 78070y + 17689
9
