12a
0031
(K12a
0031
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 12 11 4 1 8 7 10
Solving Sequence
7,11 4,8
9 5 12 6 3 2 10 1
c
7
c
8
c
4
c
11
c
6
c
3
c
2
c
10
c
12
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
76
− 8u
75
+ ··· + 2b − 2, u
76
+ 3u
75
+ ··· + 2a + 4, u
77
+ 3u
76
+ ··· − 5u − 1i
I
u
2
= h−u
2
a + b, −u
3
a + u
3
+ a
2
− 2au − u
2
+ 3u − 2, u
4
− u
3
+ 3u
2
− 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 85 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−2u
76
−8u
75
+· · ·+2b−2, u
76
+3u
75
+· · ·+2a+4, u
77
+3u
76
+· · ·−5u−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
−
1
2
u
76
−
3
2
u
75
+ ··· +
1
2
u − 2
u
76
+ 4u
75
+ ··· +
5
2
u + 1
a
8
=
1
u
2
a
9
=
u
9
+ 4u
7
+ 3u
5
− 2u
3
+ u
u
11
+ 5u
9
+ 8u
7
+ 5u
5
+ 3u
3
+ u
a
5
=
−
1
2
u
76
−
3
2
u
75
+ ··· +
23
2
u + 1
−2u
76
− 4u
75
+ ··· +
9
2
u + 1
a
12
=
−u
u
a
6
=
u
2
+ 1
−u
2
a
3
=
−u
76
− 3u
75
+ ··· +
19
2
u +
1
2
3
2
u
75
+ 3u
74
+ ··· +
3
2
u +
1
2
a
2
=
1
2
u
73
+ u
72
+ ··· +
5
2
u −
3
2
1
2
u
75
+ u
74
+ ··· +
1
2
u −
1
2
a
10
=
u
u
3
+ u
a
1
=
u
5
+ 2u
3
− u
u
7
+ 3u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
5
2
u
76
− 5u
75
+ ··· +
37
2
u +
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
77
+ 39u
76
+ ··· − 5u − 1
c
2
, c
5
u
77
+ 5u
76
+ ··· + 5u + 1
c
3
u
77
− 5u
76
+ ··· + 1267u + 593
c
4
, c
8
u
77
− u
76
+ ··· + 128u + 256
c
6
, c
7
, c
10
c
11
u
77
− 3u
76
+ ··· − 5u + 1
c
9
, c
12
u
77
+ 13u
76
+ ··· + 6101u + 563
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
77
+ 3y
76
+ ··· + 31y −1
c
2
, c
5
y
77
+ 39y
76
+ ··· − 5y −1
c
3
y
77
− 33y
76
+ ··· − 96621y −351649
c
4
, c
8
y
77
− 45y
76
+ ··· + 770048y −65536
c
6
, c
7
, c
10
c
11
y
77
+ 85y
76
+ ··· − y −1
c
9
, c
12
y
77
+ 53y
76
+ ··· − 29451637y −316969
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.612026 + 0.595452I
a = −0.89352 − 1.98650I
b = −0.1277660 − 0.0256461I
−7.2236 + 12.2987I 0. − 9.53747I
u = −0.612026 − 0.595452I
a = −0.89352 + 1.98650I
b = −0.1277660 + 0.0256461I
−7.2236 − 12.2987I 0. + 9.53747I
u = 0.260354 + 0.812312I
a = 1.02817 + 1.51523I
b = −0.114745 − 0.093276I
−1.53971 − 6.73327I 0. + 7.96395I
u = 0.260354 − 0.812312I
a = 1.02817 − 1.51523I
b = −0.114745 + 0.093276I
−1.53971 + 6.73327I 0. − 7.96395I
u = −0.624875 + 0.555027I
a = −0.83280 − 1.63607I
b = 0.212245 + 0.051955I
−9.25981 + 3.52917I −10.42870 − 3.53484I
u = −0.624875 − 0.555027I
a = −0.83280 + 1.63607I
b = 0.212245 − 0.051955I
−9.25981 − 3.52917I −10.42870 + 3.53484I
u = −0.601147 + 0.578458I
a = 1.00763 + 1.86576I
b = 0.0492372 − 0.1177350I
−4.47407 + 7.07756I −4.00000 − 6.24950I
u = −0.601147 − 0.578458I
a = 1.00763 − 1.86576I
b = 0.0492372 + 0.1177350I
−4.47407 − 7.07756I −4.00000 + 6.24950I
u = 0.368902 + 0.726526I
a = 1.08568 + 1.00548I
b = 0.098041 + 0.137813I
−2.26420 + 0.74681I −6.71284 + 0.I
u = 0.368902 − 0.726526I
a = 1.08568 − 1.00548I
b = 0.098041 − 0.137813I
−2.26420 − 0.74681I −6.71284 + 0.I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.581932 + 0.528527I
a = 1.076950 − 0.383806I
b = 0.549148 − 0.521361I
−3.73609 − 5.85675I −7.46944 + 7.02890I
u = 0.581932 − 0.528527I
a = 1.076950 + 0.383806I
b = 0.549148 + 0.521361I
−3.73609 + 5.85675I −7.46944 − 7.02890I
u = −0.647536 + 0.435417I
a = 0.595895 + 0.580977I
b = −0.967049 − 0.517583I
−9.61367 + 0.77228I −11.33626 − 2.74096I
u = −0.647536 − 0.435417I
a = 0.595895 − 0.580977I
b = −0.967049 + 0.517583I
−9.61367 − 0.77228I −11.33626 + 2.74096I
u = 0.216325 + 0.728357I
a = −0.73792 − 1.46005I
b = −0.085170 + 0.124094I
0.94993 − 2.34524I 0.16398 + 5.01561I
u = 0.216325 − 0.728357I
a = −0.73792 + 1.46005I
b = −0.085170 − 0.124094I
0.94993 + 2.34524I 0.16398 − 5.01561I
u = −0.651890 + 0.384365I
a = 0.484760 + 0.187734I
b = −1.126890 − 0.771784I
−7.84727 − 8.02560I −9.37662 + 3.63845I
u = −0.651890 − 0.384365I
a = 0.484760 − 0.187734I
b = −1.126890 + 0.771784I
−7.84727 + 8.02560I −9.37662 − 3.63845I
u = −0.549898 + 0.515511I
a = 1.63904 + 1.32471I
b = −0.176291 − 0.694995I
−1.46167 + 4.56452I −6.90154 − 7.67596I
u = −0.549898 − 0.515511I
a = 1.63904 − 1.32471I
b = −0.176291 + 0.694995I
−1.46167 − 4.56452I −6.90154 + 7.67596I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.630013 + 0.399648I
a = −0.678355 − 0.262081I
b = 0.988730 + 0.743780I
−5.00108 − 2.89884I −6.71451 + 0.09803I
u = −0.630013 − 0.399648I
a = −0.678355 + 0.262081I
b = 0.988730 − 0.743780I
−5.00108 + 2.89884I −6.71451 − 0.09803I
u = 0.587897 + 0.453548I
a = 0.867008 − 0.545729I
b = 0.265415 − 0.763966I
−3.95708 + 1.85496I −8.41123 − 0.20091I
u = 0.587897 − 0.453548I
a = 0.867008 + 0.545729I
b = 0.265415 + 0.763966I
−3.95708 − 1.85496I −8.41123 + 0.20091I
u = 0.442510 + 0.584913I
a = −0.802226 − 0.268287I
b = −0.272720 + 0.049507I
−0.32404 − 2.14218I −4.74496 + 2.48807I
u = 0.442510 − 0.584913I
a = −0.802226 + 0.268287I
b = −0.272720 − 0.049507I
−0.32404 + 2.14218I −4.74496 − 2.48807I
u = 0.528497 + 0.498843I
a = −0.855254 + 0.305815I
b = −0.290235 + 0.435521I
−0.95957 − 1.82879I −3.28595 + 3.84265I
u = 0.528497 − 0.498843I
a = −0.855254 − 0.305815I
b = −0.290235 − 0.435521I
−0.95957 + 1.82879I −3.28595 − 3.84265I
u = −0.546595 + 0.470302I
a = −1.66270 − 0.74796I
b = 0.446393 + 0.836617I
−1.59803 − 0.77982I −8.08630 − 0.44983I
u = −0.546595 − 0.470302I
a = −1.66270 + 0.74796I
b = 0.446393 − 0.836617I
−1.59803 + 0.77982I −8.08630 + 0.44983I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.012863 + 0.625748I
a = −0.24266 − 1.97148I
b = −0.448324 + 0.216980I
1.96808 − 1.39534I 3.79892 + 3.93642I
u = 0.012863 − 0.625748I
a = −0.24266 + 1.97148I
b = −0.448324 − 0.216980I
1.96808 + 1.39534I 3.79892 − 3.93642I
u = 0.565989 + 0.059555I
a = 0.457201 − 0.152438I
b = −1.024150 − 0.209368I
−4.32714 − 3.90789I −11.06800 + 4.23782I
u = 0.565989 − 0.059555I
a = 0.457201 + 0.152438I
b = −1.024150 + 0.209368I
−4.32714 + 3.90789I −11.06800 − 4.23782I
u = −0.16646 + 1.42571I
a = 0.735679 + 0.265341I
b = 0.034524 − 0.389916I
−2.08471 − 5.12370I 0
u = −0.16646 − 1.42571I
a = 0.735679 − 0.265341I
b = 0.034524 + 0.389916I
−2.08471 + 5.12370I 0
u = −0.114032 + 0.552592I
a = −0.06709 + 2.41386I
b = 0.722830 − 0.128559I
0.83609 + 2.95728I 2.05488 − 2.48382I
u = −0.114032 − 0.552592I
a = −0.06709 − 2.41386I
b = 0.722830 + 0.128559I
0.83609 − 2.95728I 2.05488 + 2.48382I
u = −0.15820 + 1.45065I
a = −0.450304 − 0.080083I
b = −0.404070 + 0.012297I
0.930752 − 0.132997I 0
u = −0.15820 − 1.45065I
a = −0.450304 + 0.080083I
b = −0.404070 − 0.012297I
0.930752 + 0.132997I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.428156 + 0.327496I
a = −0.612085 + 0.560726I
b = 0.295741 + 0.382777I
−1.06079 − 0.97162I −8.61633 + 4.50314I
u = 0.428156 − 0.327496I
a = −0.612085 − 0.560726I
b = 0.295741 − 0.382777I
−1.06079 + 0.97162I −8.61633 − 4.50314I
u = −0.18507 + 1.46627I
a = 0.655388 − 0.345780I
b = −0.118447 + 0.592915I
−3.46897 + 3.73526I 0
u = −0.18507 − 1.46627I
a = 0.655388 + 0.345780I
b = −0.118447 − 0.592915I
−3.46897 − 3.73526I 0
u = 0.05370 + 1.49908I
a = −0.14350 − 1.41174I
b = −0.00296 + 2.42243I
4.96940 − 2.28253I 0
u = 0.05370 − 1.49908I
a = −0.14350 + 1.41174I
b = −0.00296 − 2.42243I
4.96940 + 2.28253I 0
u = 0.15725 + 1.49987I
a = −0.65467 + 1.57198I
b = 0.74161 − 2.63774I
2.44139 − 0.75267I 0
u = 0.15725 − 1.49987I
a = −0.65467 − 1.57198I
b = 0.74161 + 2.63774I
2.44139 + 0.75267I 0
u = −0.14789 + 1.51967I
a = 0.656348 + 0.939409I
b = −2.08556 − 2.34113I
4.99954 + 1.64526I 0
u = −0.14789 − 1.51967I
a = 0.656348 − 0.939409I
b = −2.08556 + 2.34113I
4.99954 − 1.64526I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.469622
a = −0.697268
b = 0.775181
−1.35612 −8.05750
u = 0.14962 + 1.53175I
a = 0.87869 − 1.19452I
b = −1.26558 + 2.21528I
5.81104 − 4.23431I 0
u = 0.14962 − 1.53175I
a = 0.87869 + 1.19452I
b = −1.26558 − 2.21528I
5.81104 + 4.23431I 0
u = −0.15800 + 1.53365I
a = −0.51423 − 1.48179I
b = 1.52086 + 3.41056I
5.35249 + 7.09171I 0
u = −0.15800 − 1.53365I
a = −0.51423 + 1.48179I
b = 1.52086 − 3.41056I
5.35249 − 7.09171I 0
u = 0.17126 + 1.53380I
a = −1.11733 + 1.42706I
b = 1.50885 − 2.72068I
3.09805 − 8.56408I 0
u = 0.17126 − 1.53380I
a = −1.11733 − 1.42706I
b = 1.50885 + 2.72068I
3.09805 + 8.56408I 0
u = −0.01975 + 1.55002I
a = −0.20694 − 2.05311I
b = −0.26388 + 4.26464I
7.98485 + 3.36438I 0
u = −0.01975 − 1.55002I
a = −0.20694 + 2.05311I
b = −0.26388 − 4.26464I
7.98485 − 3.36438I 0
u = −0.19294 + 1.54033I
a = −0.36571 + 1.79258I
b = 0.52093 − 3.50683I
−2.33258 + 6.50393I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.19294 − 1.54033I
a = −0.36571 − 1.79258I
b = 0.52093 + 3.50683I
−2.33258 − 6.50393I 0
u = 0.00388 + 1.56191I
a = 0.52275 + 1.90663I
b = −0.60338 − 3.96485I
9.37638 − 1.46035I 0
u = 0.00388 − 1.56191I
a = 0.52275 − 1.90663I
b = −0.60338 + 3.96485I
9.37638 + 1.46035I 0
u = −0.18383 + 1.55341I
a = 0.13863 − 2.08558I
b = −0.25324 + 4.29249I
2.61159 + 9.94268I 0
u = −0.18383 − 1.55341I
a = 0.13863 + 2.08558I
b = −0.25324 − 4.29249I
2.61159 − 9.94268I 0
u = 0.12365 + 1.56577I
a = 1.021460 − 0.318953I
b = −1.83865 + 0.74033I
6.94313 − 4.18276I 0
u = 0.12365 − 1.56577I
a = 1.021460 + 0.318953I
b = −1.83865 − 0.74033I
6.94313 + 4.18276I 0
u = −0.18940 + 1.56024I
a = −0.27066 + 2.24261I
b = 0.65587 − 4.53549I
−0.0544 + 15.2383I 0
u = −0.18940 − 1.56024I
a = −0.27066 − 2.24261I
b = 0.65587 + 4.53549I
−0.0544 − 15.2383I 0
u = 0.04522 + 1.58797I
a = 1.07752 + 1.49008I
b = −2.06927 − 3.07631I
8.80796 − 3.21781I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.04522 − 1.58797I
a = 1.07752 − 1.49008I
b = −2.06927 + 3.07631I
8.80796 + 3.21781I 0
u = 0.09898 + 1.59218I
a = −1.39381 − 0.49922I
b = 2.77842 + 0.83797I
5.57971 − 0.94650I 0
u = 0.09898 − 1.59218I
a = −1.39381 + 0.49922I
b = 2.77842 − 0.83797I
5.57971 + 0.94650I 0
u = 0.05588 + 1.60746I
a = −1.45261 − 1.41156I
b = 2.97258 + 2.91807I
6.68062 − 7.81052I 0
u = 0.05588 − 1.60746I
a = −1.45261 + 1.41156I
b = 2.97258 − 2.91807I
6.68062 + 7.81052I 0
u = −0.208116 + 0.135285I
a = −0.62579 + 2.30791I
b = 0.289369 + 0.569805I
−0.31712 − 1.73087I −2.80212 + 4.71364I
u = −0.208116 − 0.135285I
a = −0.62579 − 2.30791I
b = 0.289369 − 0.569805I
−0.31712 + 1.73087I −2.80212 − 4.71364I
12
II.
I
u
2
= h−u
2
a + b, −u
3
a + u
3
+ a
2
− 2au − u
2
+ 3u − 2, u
4
− u
3
+ 3u
2
− 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
a
u
2
a
a
8
=
1
u
2
a
9
=
1
u
2
a
5
=
a
u
2
a
a
12
=
−u
u
a
6
=
u
2
+ 1
−u
2
a
3
=
au
u
3
a + au
a
2
=
−u
3
+ au − 2u
u
3
a + au + u
2
− u + 1
a
10
=
u
u
3
+ u
a
1
=
−u
2
− 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
3
a + 5u
2
a + 3u
3
− 7au − 3u
2
+ 5a + 9u − 8
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
4
, c
8
u
8
c
6
, c
7
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
9
(u
4
− u
3
+ u
2
+ 1)
2
c
10
, c
11
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
12
(u
4
+ u
3
+ u
2
+ 1)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
4
c
4
, c
8
y
8
c
6
, c
7
, c
10
c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
9
, c
12
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.395123 + 0.506844I
a = −0.696993 + 1.034520I
b = −0.344123 − 0.383415I
−0.21101 − 3.44499I −4.65255 + 7.52635I
u = 0.395123 + 0.506844I
a = 1.244420 + 0.086354I
b = −0.159985 + 0.489727I
−0.211005 + 0.614778I −1.64912 + 1.57080I
u = 0.395123 − 0.506844I
a = −0.696993 − 1.034520I
b = −0.344123 + 0.383415I
−0.21101 + 3.44499I −4.65255 − 7.52635I
u = 0.395123 − 0.506844I
a = 1.244420 − 0.086354I
b = −0.159985 − 0.489727I
−0.211005 − 0.614778I −1.64912 − 1.57080I
u = 0.10488 + 1.55249I
a = −0.780901 + 0.181257I
b = 1.81454 − 0.68917I
6.79074 − 1.13408I 1.80063 − 0.49697I
u = 0.10488 + 1.55249I
a = 0.233478 − 0.766909I
b = −0.31043 + 1.91602I
6.79074 − 5.19385I −1.99896 + 6.53786I
u = 0.10488 − 1.55249I
a = −0.780901 − 0.181257I
b = 1.81454 + 0.68917I
6.79074 + 1.13408I 1.80063 + 0.49697I
u = 0.10488 − 1.55249I
a = 0.233478 + 0.766909I
b = −0.31043 − 1.91602I
6.79074 + 5.19385I −1.99896 − 6.53786I
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
77
+ 39u
76
+ ··· − 5u − 1)
c
2
((u
2
+ u + 1)
4
)(u
77
+ 5u
76
+ ··· + 5u + 1)
c
3
((u
2
− u + 1)
4
)(u
77
− 5u
76
+ ··· + 1267u + 593)
c
4
, c
8
u
8
(u
77
− u
76
+ ··· + 128u + 256)
c
5
((u
2
− u + 1)
4
)(u
77
+ 5u
76
+ ··· + 5u + 1)
c
6
, c
7
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
77
− 3u
76
+ ··· − 5u + 1)
c
9
((u
4
− u
3
+ u
2
+ 1)
2
)(u
77
+ 13u
76
+ ··· + 6101u + 563)
c
10
, c
11
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
77
− 3u
76
+ ··· − 5u + 1)
c
12
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
77
+ 13u
76
+ ··· + 6101u + 563)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
77
+ 3y
76
+ ··· + 31y −1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
77
+ 39y
76
+ ··· − 5y −1)
c
3
((y
2
+ y + 1)
4
)(y
77
− 33y
76
+ ··· − 96621y −351649)
c
4
, c
8
y
8
(y
77
− 45y
76
+ ··· + 770048y −65536)
c
6
, c
7
, c
10
c
11
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
77
+ 85y
76
+ ··· − y −1)
c
9
, c
12
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
77
+ 53y
76
+ ··· − 2.94516 × 10
7
y −316969)
18
