12a
0238
(K12a
0238
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 5 12 1 11 4 9 8
Solving Sequence
1,9
8 12
4,7
3 2 11 10 5 6
c
8
c
12
c
7
c
3
c
1
c
11
c
9
c
4
c
6
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
76
− u
75
+ ··· + 2b − 1, 97u
76
+ 269u
75
+ ··· + 4a + 83, u
77
+ 4u
76
+ ··· + 2u + 1i
I
u
2
= hb, a
3
+ a
2
+ 2a + 1, u − 1i
* 2 irreducible components of dim
C
= 0, with total 80 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
76
−u
75
+· · ·+2b−1, 97u
76
+269u
75
+· · ·+4a+83, u
77
+4u
76
+· · ·+2u+1i
(i) Arc colorings
a
1
=
0
u
a
9
=
1
0
a
8
=
1
−u
2
a
12
=
u
−u
3
+ u
a
4
=
−24.2500u
76
− 67.2500u
75
+ ··· − 17.2500u − 20.7500
−
1
2
u
76
+
1
2
u
75
+ ··· +
1
2
u +
1
2
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
−16.2500u
76
− 45.2500u
75
+ ··· − 9.25000u − 14.7500
43
4
u
76
+
123
4
u
75
+ ··· +
43
4
u +
37
4
a
2
=
1
4
u
76
+
3
4
u
75
+ ··· −
23
4
u +
5
4
−
1
4
u
76
−
3
4
u
75
+ ··· −
1
4
u −
1
4
a
11
=
−u
3
+ 2u
−u
3
+ u
a
10
=
u
6
− 3u
4
+ 2u
2
+ 1
u
6
− 2u
4
+ u
2
a
5
=
−12.7500u
76
− 35.7500u
75
+ ··· − 4.75000u − 11.2500
33
2
u
76
+
89
2
u
75
+ ··· +
33
2
u +
25
2
a
6
=
−
1
4
u
76
−
3
4
u
75
+ ··· +
23
4
u −
1
4
u
19
− 7u
17
+ ··· − 6u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 42u
76
+
231
2
u
75
+ ··· + 43u +
53
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
77
+ 24u
76
+ ··· + 13u + 1
c
2
, c
5
u
77
+ 2u
76
+ ··· − 3u − 1
c
3
u
77
− 2u
76
+ ··· + 15757u − 4753
c
4
, c
10
u
77
+ u
76
+ ··· − 36u − 8
c
7
, c
8
, c
12
u
77
− 4u
76
+ ··· + 2u − 1
c
9
, c
11
u
77
+ 21u
76
+ ··· − 432u − 64
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
77
+ 60y
76
+ ··· + 61y −1
c
2
, c
5
y
77
− 24y
76
+ ··· + 13y −1
c
3
y
77
+ 24y
76
+ ··· + 306820997y −22591009
c
4
, c
10
y
77
+ 21y
76
+ ··· − 432y −64
c
7
, c
8
, c
12
y
77
− 62y
76
+ ··· + 26y −1
c
9
, c
11
y
77
+ 65y
76
+ ··· + 232704y −4096
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.10650
a = 0.143141
b = −0.621888
−2.08407 0
u = 0.097508 + 0.886073I
a = −0.93635 − 2.39335I
b = −0.98803 − 1.89695I
7.58861 − 11.05880I 0
u = 0.097508 − 0.886073I
a = −0.93635 + 2.39335I
b = −0.98803 + 1.89695I
7.58861 + 11.05880I 0
u = 0.084767 + 0.881704I
a = 0.82424 + 2.45291I
b = 0.91399 + 1.93588I
8.41612 − 5.12553I 0
u = 0.084767 − 0.881704I
a = 0.82424 − 2.45291I
b = 0.91399 − 1.93588I
8.41612 + 5.12553I 0
u = 0.808057 + 0.354660I
a = 0.398327 + 0.653038I
b = 0.154983 − 0.382042I
−0.06970 − 2.22982I 0
u = 0.808057 − 0.354660I
a = 0.398327 − 0.653038I
b = 0.154983 + 0.382042I
−0.06970 + 2.22982I 0
u = 0.105277 + 0.833083I
a = −0.63268 − 2.02197I
b = −0.79075 − 1.65686I
1.48971 − 5.79921I −8.00000 + 6.61448I
u = 0.105277 − 0.833083I
a = −0.63268 + 2.02197I
b = −0.79075 + 1.65686I
1.48971 + 5.79921I −8.00000 − 6.61448I
u = −0.017153 + 0.834554I
a = −0.16213 + 2.72359I
b = 0.26575 + 2.08207I
8.84895 − 1.05261I −1.57002 + 2.20450I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.017153 − 0.834554I
a = −0.16213 − 2.72359I
b = 0.26575 − 2.08207I
8.84895 + 1.05261I −1.57002 − 2.20450I
u = 0.055315 + 0.827179I
a = 0.29493 + 2.28516I
b = 0.57429 + 1.82055I
4.87219 − 2.94616I −1.74736 + 3.05981I
u = 0.055315 − 0.827179I
a = 0.29493 − 2.28516I
b = 0.57429 − 1.82055I
4.87219 + 2.94616I −1.74736 − 3.05981I
u = −0.032046 + 0.825605I
a = 0.31484 − 2.72986I
b = −0.16740 − 2.07689I
8.13865 + 4.89439I −2.77959 − 2.93491I
u = −0.032046 − 0.825605I
a = 0.31484 + 2.72986I
b = −0.16740 + 2.07689I
8.13865 − 4.89439I −2.77959 + 2.93491I
u = 0.708066 + 0.417899I
a = −0.248020 − 0.776113I
b = −0.095443 + 0.246952I
−0.36212 + 2.86972I −12.09253 − 2.69400I
u = 0.708066 − 0.417899I
a = −0.248020 + 0.776113I
b = −0.095443 − 0.246952I
−0.36212 − 2.86972I −12.09253 + 2.69400I
u = −1.191750 + 0.014551I
a = −0.15908 + 1.89115I
b = 0.016839 − 0.355506I
0.50247 + 2.96603I 0
u = −1.191750 − 0.014551I
a = −0.15908 − 1.89115I
b = 0.016839 + 0.355506I
0.50247 − 2.96603I 0
u = 1.143870 + 0.381225I
a = 1.095240 − 0.097934I
b = 0.245777 − 1.386210I
−1.69396 + 1.41527I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.143870 − 0.381225I
a = 1.095240 + 0.097934I
b = 0.245777 + 1.386210I
−1.69396 − 1.41527I 0
u = 0.027991 + 0.770737I
a = 0.17046 − 2.07494I
b = −0.29207 − 1.67629I
1.86989 + 0.11087I −7.62813 − 0.98718I
u = 0.027991 − 0.770737I
a = 0.17046 + 2.07494I
b = −0.29207 + 1.67629I
1.86989 − 0.11087I −7.62813 + 0.98718I
u = 1.252950 + 0.126715I
a = 0.063260 − 0.352922I
b = −1.28347 − 0.85286I
−1.123090 + 0.200004I 0
u = 1.252950 − 0.126715I
a = 0.063260 + 0.352922I
b = −1.28347 + 0.85286I
−1.123090 − 0.200004I 0
u = 1.181210 + 0.451223I
a = 1.52706 − 0.14826I
b = 0.65607 − 1.67635I
4.26251 + 6.28683I 0
u = 1.181210 − 0.451223I
a = 1.52706 + 0.14826I
b = 0.65607 + 1.67635I
4.26251 − 6.28683I 0
u = 0.384694 + 0.626756I
a = 0.401391 − 0.155030I
b = 0.489003 + 0.393854I
0.62637 − 6.82755I −8.78463 + 9.37129I
u = 0.384694 − 0.626756I
a = 0.401391 + 0.155030I
b = 0.489003 − 0.393854I
0.62637 + 6.82755I −8.78463 − 9.37129I
u = 1.194370 + 0.441307I
a = −1.50076 + 0.24788I
b = −0.57839 + 1.76571I
5.00644 + 0.39923I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.194370 − 0.441307I
a = −1.50076 − 0.24788I
b = −0.57839 − 1.76571I
5.00644 − 0.39923I 0
u = 1.218780 + 0.371520I
a = −1.131810 + 0.490210I
b = −0.05039 + 1.82641I
1.29259 − 1.36850I 0
u = 1.218780 − 0.371520I
a = −1.131810 − 0.490210I
b = −0.05039 − 1.82641I
1.29259 + 1.36850I 0
u = 1.27596
a = 0.162112
b = 1.61262
−5.58576 0
u = 1.289500 + 0.093349I
a = 0.118640 + 0.332675I
b = 1.61535 + 0.70924I
−1.70313 − 4.94998I 0
u = 1.289500 − 0.093349I
a = 0.118640 − 0.332675I
b = 1.61535 − 0.70924I
−1.70313 + 4.94998I 0
u = −1.242140 + 0.370004I
a = −1.83348 − 0.78636I
b = 0.47585 − 1.69036I
4.40113 − 0.59472I 0
u = −1.242140 − 0.370004I
a = −1.83348 + 0.78636I
b = 0.47585 + 1.69036I
4.40113 + 0.59472I 0
u = 0.486786 + 0.502721I
a = 0.054759 − 0.547642I
b = 0.197801 + 0.269349I
−3.72778 − 1.85875I −16.4618 + 5.2268I
u = 0.486786 − 0.502721I
a = 0.054759 + 0.547642I
b = 0.197801 − 0.269349I
−3.72778 + 1.85875I −16.4618 − 5.2268I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.257860 + 0.329615I
a = 0.896116 − 0.718720I
b = −0.35639 − 1.95698I
−1.93916 − 4.07565I 0
u = 1.257860 − 0.329615I
a = 0.896116 + 0.718720I
b = −0.35639 + 1.95698I
−1.93916 + 4.07565I 0
u = −1.255060 + 0.377694I
a = 1.75729 + 0.90621I
b = −0.56501 + 1.75769I
5.01586 + 5.40765I 0
u = −1.255060 − 0.377694I
a = 1.75729 − 0.90621I
b = −0.56501 − 1.75769I
5.01586 − 5.40765I 0
u = 0.322725 + 0.607838I
a = −0.233760 − 0.068743I
b = −0.438637 − 0.545582I
1.32290 − 1.49289I −6.61591 + 4.08272I
u = 0.322725 − 0.607838I
a = −0.233760 + 0.068743I
b = −0.438637 + 0.545582I
1.32290 + 1.49289I −6.61591 − 4.08272I
u = −1.329520 + 0.090325I
a = −0.290156 + 0.822947I
b = 0.390385 − 0.086461I
−5.31466 + 2.25967I 0
u = −1.329520 − 0.090325I
a = −0.290156 − 0.822947I
b = 0.390385 + 0.086461I
−5.31466 − 2.25967I 0
u = 1.282820 + 0.377841I
a = −1.19468 + 0.88899I
b = 0.09061 + 2.31978I
4.80427 − 3.30363I 0
u = 1.282820 − 0.377841I
a = −1.19468 − 0.88899I
b = 0.09061 − 2.31978I
4.80427 + 3.30363I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.297780 + 0.334051I
a = −1.219930 − 0.695265I
b = 0.89038 − 1.42707I
−2.27651 + 3.87333I 0
u = −1.297780 − 0.334051I
a = −1.219930 + 0.695265I
b = 0.89038 + 1.42707I
−2.27651 − 3.87333I 0
u = 1.293460 + 0.370170I
a = 1.13984 − 0.96168I
b = −0.19171 − 2.37881I
4.00605 − 9.19471I 0
u = 1.293460 − 0.370170I
a = 1.13984 + 0.96168I
b = −0.19171 + 2.37881I
4.00605 + 9.19471I 0
u = −1.309210 + 0.369347I
a = 1.24458 + 1.05046I
b = −1.00145 + 1.70429I
0.60671 + 7.25062I 0
u = −1.309210 − 0.369347I
a = 1.24458 − 1.05046I
b = −1.00145 − 1.70429I
0.60671 − 7.25062I 0
u = −1.357870 + 0.182843I
a = −0.369645 + 0.306246I
b = 0.887893 − 0.304238I
−3.98197 + 4.18596I 0
u = −1.357870 − 0.182843I
a = −0.369645 − 0.306246I
b = 0.887893 + 0.304238I
−3.98197 − 4.18596I 0
u = −1.381880 + 0.048424I
a = 0.062267 − 0.819525I
b = −0.281898 − 0.253910I
−6.78410 − 1.97007I 0
u = −1.381880 − 0.048424I
a = 0.062267 + 0.819525I
b = −0.281898 + 0.253910I
−6.78410 + 1.97007I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.337640 + 0.368476I
a = −0.97571 − 1.14358I
b = 1.24988 − 1.67951I
−3.03784 + 10.12450I 0
u = −1.337640 − 0.368476I
a = −0.97571 + 1.14358I
b = 1.24988 + 1.67951I
−3.03784 − 10.12450I 0
u = −1.334550 + 0.398433I
a = 1.11424 + 1.42325I
b = −1.23756 + 1.96013I
3.96873 + 9.71449I 0
u = −1.334550 − 0.398433I
a = 1.11424 − 1.42325I
b = −1.23756 − 1.96013I
3.96873 − 9.71449I 0
u = −1.387660 + 0.124640I
a = 0.129187 − 0.573699I
b = −0.730978 − 0.094246I
−9.63919 + 3.85228I 0
u = −1.387660 − 0.124640I
a = 0.129187 + 0.573699I
b = −0.730978 + 0.094246I
−9.63919 − 3.85228I 0
u = −1.386810 + 0.184853I
a = 0.193588 − 0.252518I
b = −1.046420 + 0.180944I
−5.02380 + 9.58444I 0
u = −1.386810 − 0.184853I
a = 0.193588 + 0.252518I
b = −1.046420 − 0.180944I
−5.02380 − 9.58444I 0
u = −1.343060 + 0.398637I
a = −1.03037 − 1.45790I
b = 1.31999 − 1.95979I
3.0695 + 15.6642I 0
u = −1.343060 − 0.398637I
a = −1.03037 + 1.45790I
b = 1.31999 + 1.95979I
3.0695 − 15.6642I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.283410 + 0.303037I
a = 0.523170 + 0.541674I
b = −0.021509 − 0.462591I
−0.336764 − 0.936515I −6.18984 + 7.26446I
u = 0.283410 − 0.303037I
a = 0.523170 − 0.541674I
b = −0.021509 + 0.462591I
−0.336764 + 0.936515I −6.18984 − 7.26446I
u = −0.163536 + 0.352944I
a = 2.36554 − 0.45097I
b = 0.576303 − 0.604664I
3.10141 − 1.94636I −0.51042 + 2.66215I
u = −0.163536 − 0.352944I
a = 2.36554 + 0.45097I
b = 0.576303 + 0.604664I
3.10141 + 1.94636I −0.51042 − 2.66215I
u = −0.220160 + 0.295017I
a = −2.70473 + 0.35585I
b = −0.655975 + 0.466796I
2.84761 + 3.57248I −0.90847 − 3.64285I
u = −0.220160 − 0.295017I
a = −2.70473 − 0.35585I
b = −0.655975 − 0.466796I
2.84761 − 3.57248I −0.90847 + 3.64285I
u = −0.165659
a = −3.43662
b = −0.466057
−1.33272 −6.61800
12
II. I
u
2
= hb, a
3
+ a
2
+ 2a + 1, u − 1i
(i) Arc colorings
a
1
=
0
1
a
9
=
1
0
a
8
=
1
−1
a
12
=
1
0
a
4
=
a
0
a
7
=
0
−1
a
3
=
a
−a
a
2
=
−a
2
a
2
+ 1
a
11
=
1
0
a
10
=
1
0
a
5
=
a
0
a
6
=
−a
2
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −7a
2
− 5a − 17
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
4
, c
9
, c
10
c
11
u
3
c
5
u
3
− u
2
+ 1
c
6
u
3
+ u
2
+ 2u + 1
c
7
, c
8
(u − 1)
3
c
12
(u + 1)
3
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
y
3
+ 3y
2
+ 2y −1
c
2
, c
5
y
3
− y
2
+ 2y −1
c
4
, c
9
, c
10
c
11
y
3
c
7
, c
8
, c
12
(y −1)
3
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.215080 + 1.307140I
b = 0
1.37919 + 2.82812I −4.28809 − 2.59975I
u = 1.00000
a = −0.215080 − 1.307140I
b = 0
1.37919 − 2.82812I −4.28809 + 2.59975I
u = 1.00000
a = −0.569840
b = 0
−2.75839 −16.4240
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
− u
2
+ 2u − 1)(u
77
+ 24u
76
+ ··· + 13u + 1)
c
2
(u
3
+ u
2
− 1)(u
77
+ 2u
76
+ ··· − 3u − 1)
c
3
(u
3
− u
2
+ 2u − 1)(u
77
− 2u
76
+ ··· + 15757u − 4753)
c
4
, c
10
u
3
(u
77
+ u
76
+ ··· − 36u − 8)
c
5
(u
3
− u
2
+ 1)(u
77
+ 2u
76
+ ··· − 3u − 1)
c
6
(u
3
+ u
2
+ 2u + 1)(u
77
+ 24u
76
+ ··· + 13u + 1)
c
7
, c
8
((u − 1)
3
)(u
77
− 4u
76
+ ··· + 2u − 1)
c
9
, c
11
u
3
(u
77
+ 21u
76
+ ··· − 432u − 64)
c
12
((u + 1)
3
)(u
77
− 4u
76
+ ··· + 2u − 1)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
3
+ 3y
2
+ 2y −1)(y
77
+ 60y
76
+ ··· + 61y −1)
c
2
, c
5
(y
3
− y
2
+ 2y −1)(y
77
− 24y
76
+ ··· + 13y −1)
c
3
(y
3
+ 3y
2
+ 2y −1)(y
77
+ 24y
76
+ ··· + 3.06821 × 10
8
y −2.25910 × 10
7
)
c
4
, c
10
y
3
(y
77
+ 21y
76
+ ··· − 432y −64)
c
7
, c
8
, c
12
((y −1)
3
)(y
77
− 62y
76
+ ··· + 26y −1)
c
9
, c
11
y
3
(y
77
+ 65y
76
+ ··· + 232704y −4096)
18
