12a
0421
(K12a
0421
)
A knot diagram
1
Linearized knot diagam
3 6 10 7 2 5 11 12 4 1 9 8
Solving Sequence
2,6 3,10
4 1 11 5 7 8 9 12
c
2
c
3
c
1
c
10
c
5
c
6
c
7
c
9
c
12
c
4
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−39u
74
+ 70u
73
+ ··· + 4b + 8, 25u
74
− 114u
73
+ ··· + 4a + 33, u
75
− 4u
74
+ ··· + 5u − 1i
I
u
2
= hb + u, u
2
+ a + u, u
3
+ u
2
− 1i
I
u
3
= h−u
2
a + b, −u
2
a + a
2
− 2au + u
2
− a + 2u + 2, u
3
+ u
2
− 1i
* 3 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
= h−39u
74
+ 70u
73
+ · · · + 4b + 8, 25u
74
− 114u
73
+ · · · + 4a +
33, u
75
− 4u
74
+ · · · + 5u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
10
=
−6.25000u
74
+ 28.5000u
73
+ ··· + 42.5000u − 8.25000
39
4
u
74
−
35
2
u
73
+ ··· +
1
4
u − 2
a
4
=
u
5
+ u
u
5
− u
3
+ u
a
1
=
−u
2
+ 1
−u
4
a
11
=
−2u
74
+
27
4
u
73
+ ··· +
17
2
u +
1
4
7
2
u
74
−
35
4
u
73
+ ··· − 10u +
7
4
a
5
=
u
u
a
7
=
−u
3
−u
3
+ u
a
8
=
1
4
u
73
−
3
4
u
72
+ ··· −
1
4
u − 1
1
4
u
73
−
3
4
u
72
+ ··· +
1
4
u
2
+
7
4
u
a
9
=
−
7
4
u
74
+
7
2
u
73
+ ··· +
5
2
u +
5
4
5
4
u
74
−
11
2
u
73
+ ··· −
41
4
u + 2
a
12
=
−
5
4
u
73
+
7
4
u
72
+ ··· −
29
4
u + 3
−
1
4
u
74
−
1
2
u
73
+ ··· − 8u +
7
4
(ii) Obstruction class = −1
(iii) Cusp Shapes =
19
4
u
74
+
3
2
u
73
+ ··· +
97
4
u −
45
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
u
75
+ 18u
74
+ ··· + 21u + 1
c
2
, c
5
u
75
+ 4u
74
+ ··· + 5u + 1
c
3
, c
9
u
75
+ u
74
+ ··· + 2048u + 512
c
7
u
75
+ 4u
74
+ ··· − 4485u + 1153
c
8
, c
11
, c
12
u
75
− 4u
74
+ ··· − 3u + 1
c
10
u
75
− 14u
74
+ ··· − 2323u + 1251
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
y
75
+ 82y
74
+ ··· + 197y −1
c
2
, c
5
y
75
− 18y
74
+ ··· + 21y −1
c
3
, c
9
y
75
+ 49y
74
+ ··· − 4325376y −262144
c
7
y
75
+ 14y
74
+ ··· + 3138453y −1329409
c
8
, c
11
, c
12
y
75
+ 70y
74
+ ··· + 29y −1
c
10
y
75
+ 42y
74
+ ··· − 3150503y −1565001
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.981997 + 0.062495I
a = −0.158117 + 0.959495I
b = −0.0572768 − 0.0273762I
−1.33660 + 1.51490I −12.00000 + 0.I
u = 0.981997 − 0.062495I
a = −0.158117 − 0.959495I
b = −0.0572768 + 0.0273762I
−1.33660 − 1.51490I −12.00000 + 0.I
u = −0.925744 + 0.463913I
a = 1.45832 + 0.58863I
b = 1.102820 + 0.176212I
1.74127 + 3.47396I 0
u = −0.925744 − 0.463913I
a = 1.45832 − 0.58863I
b = 1.102820 − 0.176212I
1.74127 − 3.47396I 0
u = 1.039500 + 0.060819I
a = 0.146948 − 1.100270I
b = 0.0726854 − 0.0349259I
4.21274 + 4.43090I 0
u = 1.039500 − 0.060819I
a = 0.146948 + 1.100270I
b = 0.0726854 + 0.0349259I
4.21274 − 4.43090I 0
u = 0.906111 + 0.214366I
a = 0.546312 − 0.823518I
b = 0.124447 + 0.208700I
0.534887 − 0.279588I −13.49764 + 1.12643I
u = 0.906111 − 0.214366I
a = 0.546312 + 0.823518I
b = 0.124447 − 0.208700I
0.534887 + 0.279588I −13.49764 − 1.12643I
u = −0.986819 + 0.430211I
a = −1.67189 − 0.45818I
b = −1.237830 − 0.093040I
0.79249 + 7.26020I 0
u = −0.986819 − 0.430211I
a = −1.67189 + 0.45818I
b = −1.237830 + 0.093040I
0.79249 − 7.26020I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.857424 + 0.324976I
a = 1.81994 + 1.10430I
b = 1.309870 + 0.479319I
1.07508 + 4.38196I −12.0000 − 8.6783I
u = −0.857424 − 0.324976I
a = 1.81994 − 1.10430I
b = 1.309870 − 0.479319I
1.07508 − 4.38196I −12.0000 + 8.6783I
u = 0.813449 + 0.419843I
a = 0.978987 − 0.633068I
b = 0.035809 + 0.678441I
3.42977 − 5.57287I −9.66221 + 6.64681I
u = 0.813449 − 0.419843I
a = 0.978987 + 0.633068I
b = 0.035809 − 0.678441I
3.42977 + 5.57287I −9.66221 − 6.64681I
u = −0.784284 + 0.772305I
a = −0.154248 − 0.741875I
b = −0.320352 − 0.513152I
6.62851 + 1.39151I 0
u = −0.784284 − 0.772305I
a = −0.154248 + 0.741875I
b = −0.320352 + 0.513152I
6.62851 − 1.39151I 0
u = −1.020000 + 0.436284I
a = 1.70688 + 0.35049I
b = 1.263320 + 0.023389I
6.45080 + 10.69250I 0
u = −1.020000 − 0.436284I
a = 1.70688 − 0.35049I
b = 1.263320 − 0.023389I
6.45080 − 10.69250I 0
u = −0.975758 + 0.543303I
a = −1.317190 − 0.277849I
b = −0.999148 + 0.021410I
7.84793 + 1.37305I 0
u = −0.975758 − 0.543303I
a = −1.317190 + 0.277849I
b = −0.999148 − 0.021410I
7.84793 − 1.37305I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.804622 + 0.341592I
a = −0.825449 + 0.643517I
b = −0.009664 − 0.476551I
−1.71326 − 2.52197I −15.2219 + 7.1642I
u = 0.804622 − 0.341592I
a = −0.825449 − 0.643517I
b = −0.009664 + 0.476551I
−1.71326 + 2.52197I −15.2219 − 7.1642I
u = −0.437073 + 0.751160I
a = −0.242441 − 1.250960I
b = −0.725204 − 0.713923I
9.56548 + 3.33472I −2.00493 − 2.98614I
u = −0.437073 − 0.751160I
a = −0.242441 + 1.250960I
b = −0.725204 + 0.713923I
9.56548 − 3.33472I −2.00493 + 2.98614I
u = −0.877037 + 0.715044I
a = 0.404333 + 0.188560I
b = 0.346024 + 0.046454I
2.43349 + 2.73740I 0
u = −0.877037 − 0.715044I
a = 0.404333 − 0.188560I
b = 0.346024 − 0.046454I
2.43349 − 2.73740I 0
u = −0.756510 + 0.278228I
a = −1.71639 − 1.68146I
b = −1.23262 − 0.78437I
−2.12507 + 1.06185I −14.2779 − 5.9921I
u = −0.756510 − 0.278228I
a = −1.71639 + 1.68146I
b = −1.23262 + 0.78437I
−2.12507 − 1.06185I −14.2779 + 5.9921I
u = −0.286571 + 0.749374I
a = 0.161353 + 1.250890I
b = 0.819985 + 0.663073I
8.84242 − 6.41894I −2.81064 + 3.75509I
u = −0.286571 − 0.749374I
a = 0.161353 − 1.250890I
b = 0.819985 − 0.663073I
8.84242 + 6.41894I −2.81064 − 3.75509I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.942042 + 0.743552I
a = −0.619040 + 0.245049I
b = −0.421240 + 0.323729I
6.16024 + 4.33025I 0
u = −0.942042 − 0.743552I
a = −0.619040 − 0.245049I
b = −0.421240 − 0.323729I
6.16024 − 4.33025I 0
u = −0.401183 + 0.671168I
a = 0.241879 + 1.316680I
b = 0.732361 + 0.665991I
3.42074 + 0.68845I −5.14590 − 3.23104I
u = −0.401183 − 0.671168I
a = 0.241879 − 1.316680I
b = 0.732361 − 0.665991I
3.42074 − 0.68845I −5.14590 + 3.23104I
u = 0.885217 + 0.845939I
a = 0.53098 − 2.05656I
b = 3.24456 − 0.48084I
8.13392 + 0.93562I 0
u = 0.885217 − 0.845939I
a = 0.53098 + 2.05656I
b = 3.24456 + 0.48084I
8.13392 − 0.93562I 0
u = −0.891011 + 0.845695I
a = −0.339704 + 0.721513I
b = −0.088337 + 0.655456I
5.33943 + 1.26962I 0
u = −0.891011 − 0.845695I
a = −0.339704 − 0.721513I
b = −0.088337 − 0.655456I
5.33943 − 1.26962I 0
u = 0.906133 + 0.837441I
a = −0.74848 + 2.16155I
b = −3.32632 + 0.25159I
4.35553 − 3.11893I 0
u = 0.906133 − 0.837441I
a = −0.74848 − 2.16155I
b = −3.32632 − 0.25159I
4.35553 + 3.11893I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.842072 + 0.904011I
a = −0.45004 + 1.46873I
b = −2.75133 + 0.55585I
9.54844 + 4.90423I 0
u = 0.842072 − 0.904011I
a = −0.45004 − 1.46873I
b = −2.75133 − 0.55585I
9.54844 − 4.90423I 0
u = 0.834552 + 0.915946I
a = 0.46805 − 1.38257I
b = 2.69223 − 0.53550I
15.4454 + 8.6299I 0
u = 0.834552 − 0.915946I
a = 0.46805 + 1.38257I
b = 2.69223 + 0.53550I
15.4454 − 8.6299I 0
u = −0.304636 + 0.693805I
a = −0.157840 − 1.295690I
b = −0.782092 − 0.642140I
2.97970 − 3.17000I −6.43688 + 4.00657I
u = −0.304636 − 0.693805I
a = −0.157840 + 1.295690I
b = −0.782092 + 0.642140I
2.97970 + 3.17000I −6.43688 − 4.00657I
u = 0.860351 + 0.897542I
a = 0.51786 − 1.57748I
b = 2.83408 − 0.51095I
10.46520 + 0.47593I 0
u = 0.860351 − 0.897542I
a = 0.51786 + 1.57748I
b = 2.83408 + 0.51095I
10.46520 − 0.47593I 0
u = −0.922117 + 0.835418I
a = 0.454904 − 0.666000I
b = 0.201003 − 0.635777I
5.24277 + 4.98689I 0
u = −0.922117 − 0.835418I
a = 0.454904 + 0.666000I
b = 0.201003 + 0.635777I
5.24277 − 4.98689I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.889456 + 0.870616I
a = 0.352412 − 0.809727I
b = 0.076324 − 0.735989I
11.29310 − 1.63609I 0
u = −0.889456 − 0.870616I
a = 0.352412 + 0.809727I
b = 0.076324 + 0.735989I
11.29310 + 1.63609I 0
u = 0.927268 + 0.831795I
a = 0.96690 − 2.16128I
b = 3.28157 − 0.01724I
8.00279 − 7.18225I 0
u = 0.927268 − 0.831795I
a = 0.96690 + 2.16128I
b = 3.28157 + 0.01724I
8.00279 + 7.18225I 0
u = −0.937813 + 0.850575I
a = −0.516072 + 0.722653I
b = −0.241335 + 0.698457I
11.13980 + 8.01758I 0
u = −0.937813 − 0.850575I
a = −0.516072 − 0.722653I
b = −0.241335 − 0.698457I
11.13980 − 8.01758I 0
u = 0.878651 + 0.913361I
a = −0.66681 + 1.55279I
b = −2.81360 + 0.40519I
17.4252 − 1.9074I 0
u = 0.878651 − 0.913361I
a = −0.66681 − 1.55279I
b = −2.81360 − 0.40519I
17.4252 + 1.9074I 0
u = −0.682598 + 0.202478I
a = 1.58421 + 2.38354I
b = 1.11380 + 1.14569I
2.17296 − 2.32447I −5.79052 − 5.26965I
u = −0.682598 − 0.202478I
a = 1.58421 − 2.38354I
b = 1.11380 − 1.14569I
2.17296 + 2.32447I −5.79052 + 5.26965I
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.543130 + 0.459714I
a = −0.770471 + 0.389630I
b = 0.594716 − 0.431365I
4.27941 + 2.11715I −6.41999 − 0.25459I
u = 0.543130 − 0.459714I
a = −0.770471 − 0.389630I
b = 0.594716 + 0.431365I
4.27941 − 2.11715I −6.41999 + 0.25459I
u = 0.970822 + 0.847794I
a = 1.16946 − 1.91708I
b = 2.93004 + 0.10560I
10.11250 − 6.92850I 0
u = 0.970822 − 0.847794I
a = 1.16946 + 1.91708I
b = 2.93004 − 0.10560I
10.11250 + 6.92850I 0
u = 0.984627 + 0.840414I
a = −1.24021 + 1.90311I
b = −2.86576 − 0.17531I
9.0945 − 11.3508I 0
u = 0.984627 − 0.840414I
a = −1.24021 − 1.90311I
b = −2.86576 + 0.17531I
9.0945 + 11.3508I 0
u = 0.970887 + 0.869077I
a = −1.12355 + 1.83005I
b = −2.87405 − 0.00559I
17.1288 − 4.6645I 0
u = 0.970887 − 0.869077I
a = −1.12355 − 1.83005I
b = −2.87405 + 0.00559I
17.1288 + 4.6645I 0
u = 0.995376 + 0.841787I
a = 1.26684 − 1.86881I
b = 2.80356 + 0.17862I
14.9316 − 15.1163I 0
u = 0.995376 − 0.841787I
a = 1.26684 + 1.86881I
b = 2.80356 − 0.17862I
14.9316 + 15.1163I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.569082 + 0.163141I
a = 0.783151 − 0.295781I
b = −0.278782 + 0.143296I
−0.755875 − 0.043813I −11.52697 − 0.84981I
u = 0.569082 − 0.163141I
a = 0.783151 + 0.295781I
b = −0.278782 − 0.143296I
−0.755875 + 0.043813I −11.52697 + 0.84981I
u = 0.020011 + 0.415797I
a = −0.562604 + 1.289900I
b = 0.631869 + 0.282943I
3.06974 − 1.95978I −5.26928 + 3.75701I
u = 0.020011 − 0.415797I
a = −0.562604 − 1.289900I
b = 0.631869 − 0.282943I
3.06974 + 1.95978I −5.26928 − 3.75701I
u = 0.288433
a = 1.44160
b = −0.372279
−0.730103 −13.0940
12
II. I
u
2
= hb + u, u
2
+ a + u, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
10
=
−u
2
− u
−u
a
4
=
1
u
2
a
1
=
−u
2
+ 1
−u
2
− u + 1
a
11
=
−1
−1
a
5
=
u
u
a
7
=
u
2
− 1
u
2
+ u − 1
a
8
=
u
2
− u − 1
u
2
− 1
a
9
=
−u
2
− u
−u
a
12
=
u − 1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
− 7u − 13
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
8
u
3
− u
2
+ 2u − 1
c
2
, c
7
, c
10
u
3
+ u
2
− 1
c
3
, c
9
u
3
c
5
u
3
− u
2
+ 1
c
6
, c
11
, c
12
u
3
+ u
2
+ 2u + 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
c
2
, c
5
, c
7
c
10
y
3
− y
2
+ 2y −1
c
3
, c
9
y
3
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.662359 + 0.562280I
b = 0.877439 − 0.744862I
6.04826 + 5.65624I −6.64285 − 6.52117I
u = −0.877439 − 0.744862I
a = 0.662359 − 0.562280I
b = 0.877439 + 0.744862I
6.04826 − 5.65624I −6.64285 + 6.52117I
u = 0.754878
a = −1.32472
b = −0.754878
−2.22691 −17.7140
16
III. I
u
3
= h−u
2
a + b, −u
2
a + a
2
− 2au + u
2
− a + 2u + 2, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
10
=
a
u
2
a
a
4
=
1
u
2
a
1
=
−u
2
+ 1
−u
2
− u + 1
a
11
=
au
au
a
5
=
u
u
a
7
=
u
2
− 1
u
2
+ u − 1
a
8
=
−u
2
a − au + u
2
+ u
−u
2
a − au + u
2
+ 2u
a
9
=
a
u
2
a
a
12
=
u
2
− a + 2u + 2
au + u
2
− a + u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
2
a + au + a − 5u − 17
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
8
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
7
, c
10
(u
3
+ u
2
− 1)
2
c
3
, c
9
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
, c
11
, c
12
(u
3
+ u
2
+ 2u + 1)
2
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
, c
7
c
10
(y
3
− y
2
+ 2y −1)
2
c
3
, c
9
y
6
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.447279 + 0.744862I
b = 0.877439 + 0.744862I
6.04826 −7.95781 + 0.50299I
u = −0.877439 + 0.744862I
a = −0.092519 − 0.562280I
b = −0.754878
1.91067 + 2.82812I −16.7346 − 3.8621I
u = −0.877439 − 0.744862I
a = −0.447279 − 0.744862I
b = 0.877439 − 0.744862I
6.04826 −7.95781 − 0.50299I
u = −0.877439 − 0.744862I
a = −0.092519 + 0.562280I
b = −0.754878
1.91067 − 2.82812I −16.7346 + 3.8621I
u = 0.754878
a = 1.53980 + 1.30714I
b = 0.877439 + 0.744862I
1.91067 − 2.82812I −12.8076 + 6.7630I
u = 0.754878
a = 1.53980 − 1.30714I
b = 0.877439 − 0.744862I
1.91067 + 2.82812I −12.8076 − 6.7630I
20
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
((u
3
− u
2
+ 2u − 1)
3
)(u
75
+ 18u
74
+ ··· + 21u + 1)
c
2
((u
3
+ u
2
− 1)
3
)(u
75
+ 4u
74
+ ··· + 5u + 1)
c
3
, c
9
u
9
(u
75
+ u
74
+ ··· + 2048u + 512)
c
5
((u
3
− u
2
+ 1)
3
)(u
75
+ 4u
74
+ ··· + 5u + 1)
c
6
((u
3
+ u
2
+ 2u + 1)
3
)(u
75
+ 18u
74
+ ··· + 21u + 1)
c
7
((u
3
+ u
2
− 1)
3
)(u
75
+ 4u
74
+ ··· − 4485u + 1153)
c
8
((u
3
− u
2
+ 2u − 1)
3
)(u
75
− 4u
74
+ ··· − 3u + 1)
c
10
((u
3
+ u
2
− 1)
3
)(u
75
− 14u
74
+ ··· − 2323u + 1251)
c
11
, c
12
((u
3
+ u
2
+ 2u + 1)
3
)(u
75
− 4u
74
+ ··· − 3u + 1)
21
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
((y
3
+ 3y
2
+ 2y −1)
3
)(y
75
+ 82y
74
+ ··· + 197y −1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
3
)(y
75
− 18y
74
+ ··· + 21y −1)
c
3
, c
9
y
9
(y
75
+ 49y
74
+ ··· − 4325376y −262144)
c
7
((y
3
− y
2
+ 2y −1)
3
)(y
75
+ 14y
74
+ ··· + 3138453y −1329409)
c
8
, c
11
, c
12
((y
3
+ 3y
2
+ 2y −1)
3
)(y
75
+ 70y
74
+ ··· + 29y −1)
c
10
((y
3
− y
2
+ 2y −1)
3
)(y
75
+ 42y
74
+ ··· − 3150503y −1565001)
22
