12n
0278
(K12n
0278
)
A knot diagram
1
Linearized knot diagam
3 4 9 2 10 4 12 11 6 3 8 7
Solving Sequence
3,9 4,6
7 10 11 2 1 5 8 12
c
3
c
6
c
9
c
10
c
2
c
1
c
5
c
8
c
12
c
4
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−635380494822630u
39
− 689616041260383u
38
+ ··· + 1433600017911488b + 1677666964673313,
2246684406994641u
39
+ 1078394006282716u
38
+ ··· + 7168000089557440a + 9725037800310384,
u
40
+ u
39
+ ··· + 4u + 5i
I
u
2
= hu
3
b + 4u
2
b − u
3
+ b
2
− bu + u
2
− 2b − u − 4, −u
2
+ a, u
4
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 48 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−6.35×10
14
u
39
−6.90×10
14
u
38
+· · ·+1.43×10
15
b+1.68×10
15
, 2.25×
10
15
u
39
+1.08×10
15
u
38
+· · ·+7.17×10
15
a+9.73×10
15
, u
40
+u
39
+· · ·+4u+5i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
−0.313433u
39
− 0.150446u
38
+ ··· + 1.82893u − 1.35673
0.443206u
39
+ 0.481038u
38
+ ··· − 5.21478u − 1.17025
a
7
=
−0.109699u
39
+ 0.157341u
38
+ ··· − 2.47064u − 3.34191
0.331526u
39
+ 0.441625u
38
+ ··· − 3.77990u − 0.649983
a
10
=
−0.286938u
39
− 0.370199u
38
+ ··· + 5.00815u + 1.05379
0.190141u
39
− 0.0380655u
38
+ ··· − 2.69477u + 1.78981
a
11
=
−0.0967963u
39
− 0.408265u
38
+ ··· + 2.31338u + 2.84360
0.190141u
39
− 0.0380655u
38
+ ··· − 2.69477u + 1.78981
a
2
=
−u
2
+ 1
u
4
a
1
=
u
4
− u
2
+ 1
u
4
a
5
=
u
4
− u
2
+ 1
−u
6
− u
2
a
8
=
0.332958u
39
+ 0.469712u
38
+ ··· − 6.44378u − 1.37541
0.255919u
39
+ 0.714505u
38
+ ··· − 6.65923u − 4.34499
a
12
=
0.453947u
39
+ 0.662238u
38
+ ··· − 5.70833u − 2.30072
−0.0932185u
39
− 0.0614314u
38
+ ··· + 0.780423u + 1.26772
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
10623651294159
102400001279392
u
39
−
45699572803011
102400001279392
u
38
+ ··· +
228874377951323
25600000319848
u +
930610235166537
102400001279392
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
40
+ 41u
39
+ ··· − 4756u + 625
c
2
, c
4
u
40
− 11u
39
+ ··· − 216u + 25
c
3
u
40
+ u
39
+ ··· + 4u + 5
c
5
, c
9
u
40
+ u
39
+ ··· + 13u
2
+ 4
c
6
u
40
+ 5u
39
+ ··· − 11820u + 18731
c
7
, c
8
, c
11
c
12
u
40
− u
39
+ ··· − 8u + 1
c
10
u
40
− 3u
39
+ ··· + 1600u + 1984
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
40
− 79y
39
+ ··· + 16952964y + 390625
c
2
, c
4
y
40
+ 41y
39
+ ··· − 4756y + 625
c
3
y
40
− 11y
39
+ ··· − 216y + 25
c
5
, c
9
y
40
+ 13y
39
+ ··· + 104y + 16
c
6
y
40
+ 31y
39
+ ··· + 6061334998y + 350850361
c
7
, c
8
, c
11
c
12
y
40
+ 45y
39
+ ··· − 4y + 1
c
10
y
40
+ 15y
39
+ ··· + 108480512y + 3936256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.816138 + 0.601846I
a = −0.100989 + 0.484046I
b = 1.272200 + 0.371404I
9.03582 − 2.34742I −1.16092 + 4.21084I
u = −0.816138 − 0.601846I
a = −0.100989 − 0.484046I
b = 1.272200 − 0.371404I
9.03582 + 2.34742I −1.16092 − 4.21084I
u = 0.942774 + 0.425003I
a = 0.249731 + 0.561680I
b = 0.05704 − 1.50077I
1.43856 + 1.71130I −1.298482 + 0.550005I
u = 0.942774 − 0.425003I
a = 0.249731 − 0.561680I
b = 0.05704 + 1.50077I
1.43856 − 1.71130I −1.298482 − 0.550005I
u = 0.778485 + 0.532883I
a = 0.574413 + 1.007710I
b = 0.08684 − 1.66241I
1.51229 + 2.11244I −4.00016 − 4.74712I
u = 0.778485 − 0.532883I
a = 0.574413 − 1.007710I
b = 0.08684 + 1.66241I
1.51229 − 2.11244I −4.00016 + 4.74712I
u = −1.044590 + 0.339980I
a = −0.206550 − 0.804001I
b = 0.54339 + 2.06103I
1.66475 − 4.28233I 1.04865 + 9.16728I
u = −1.044590 − 0.339980I
a = −0.206550 + 0.804001I
b = 0.54339 − 2.06103I
1.66475 + 4.28233I 1.04865 − 9.16728I
u = −0.870167 + 0.752444I
a = 0.817881 − 0.905901I
b = 0.43266 + 2.06327I
6.83536 − 2.84310I 2.40205 + 2.94368I
u = −0.870167 − 0.752444I
a = 0.817881 + 0.905901I
b = 0.43266 − 2.06327I
6.83536 + 2.84310I 2.40205 − 2.94368I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.642627 + 0.538012I
a = 0.365174 − 0.413262I
b = 0.325214 − 0.141593I
0.58360 + 1.86035I −1.76602 − 5.14793I
u = 0.642627 − 0.538012I
a = 0.365174 + 0.413262I
b = 0.325214 + 0.141593I
0.58360 − 1.86035I −1.76602 + 5.14793I
u = −1.100630 + 0.389040I
a = 0.706000 − 0.374232I
b = −0.84428 + 1.49004I
7.52435 − 1.33305I 3.41871 + 0.66465I
u = −1.100630 − 0.389040I
a = 0.706000 + 0.374232I
b = −0.84428 − 1.49004I
7.52435 + 1.33305I 3.41871 − 0.66465I
u = 1.153370 + 0.265663I
a = −0.361327 + 0.909523I
b = 0.80159 − 2.60812I
8.28312 + 6.34531I 4.99183 − 5.94173I
u = 1.153370 − 0.265663I
a = −0.361327 − 0.909523I
b = 0.80159 + 2.60812I
8.28312 − 6.34531I 4.99183 + 5.94173I
u = −0.740318 + 0.931805I
a = −1.181910 + 0.030020I
b = −0.522469 − 0.429658I
0.02606 + 6.35035I 0.13702 − 2.71532I
u = −0.740318 − 0.931805I
a = −1.181910 − 0.030020I
b = −0.522469 + 0.429658I
0.02606 − 6.35035I 0.13702 + 2.71532I
u = −0.057006 + 0.799683I
a = 0.932252 − 0.995287I
b = 0.476313 + 0.200122I
4.23419 − 2.71559I −0.50842 + 3.06868I
u = −0.057006 − 0.799683I
a = 0.932252 + 0.995287I
b = 0.476313 − 0.200122I
4.23419 + 2.71559I −0.50842 − 3.06868I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.846654 + 0.893541I
a = 0.153750 − 0.996865I
b = −0.370640 + 1.101610I
−1.261380 + 0.472950I −0.86316 − 1.46415I
u = 0.846654 − 0.893541I
a = 0.153750 + 0.996865I
b = −0.370640 − 1.101610I
−1.261380 − 0.472950I −0.86316 + 1.46415I
u = 0.824369 + 0.917399I
a = −1.102340 + 0.038722I
b = −0.112606 + 0.389706I
−6.94288 − 2.71197I −3.03827 + 2.61066I
u = 0.824369 − 0.917399I
a = −1.102340 − 0.038722I
b = −0.112606 − 0.389706I
−6.94288 + 2.71197I −3.03827 − 2.61066I
u = −0.899972 + 0.889292I
a = −1.022400 − 0.107733I
b = 0.290479 − 0.280456I
−7.22089 − 2.24192I −3.59196 + 2.70686I
u = −0.899972 − 0.889292I
a = −1.022400 + 0.107733I
b = 0.290479 + 0.280456I
−7.22089 + 2.24192I −3.59196 − 2.70686I
u = 0.710509 + 0.109408I
a = −0.51036 − 1.45979I
b = −1.72253 + 1.95217I
11.47330 + 0.46021I 8.35615 + 1.19991I
u = 0.710509 − 0.109408I
a = −0.51036 + 1.45979I
b = −1.72253 − 1.95217I
11.47330 − 0.46021I 8.35615 − 1.19991I
u = −0.944065 + 0.869245I
a = 0.057186 + 1.042530I
b = −0.44747 − 1.75317I
−7.07974 − 4.25891I −3.44151 + 2.27415I
u = −0.944065 − 0.869245I
a = 0.057186 − 1.042530I
b = −0.44747 + 1.75317I
−7.07974 + 4.25891I −3.44151 − 2.27415I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.978094 + 0.843510I
a = −0.928803 + 0.191863I
b = 0.723604 + 0.076328I
−0.85183 + 5.95407I −0.43798 − 3.29902I
u = 0.978094 − 0.843510I
a = −0.928803 − 0.191863I
b = 0.723604 − 0.076328I
−0.85183 − 5.95407I −0.43798 + 3.29902I
u = 1.004740 + 0.837681I
a = −0.008007 − 1.067440I
b = −0.44665 + 2.27489I
−6.36974 + 9.19261I −1.78825 − 7.32691I
u = 1.004740 − 0.837681I
a = −0.008007 + 1.067440I
b = −0.44665 − 2.27489I
−6.36974 − 9.19261I −1.78825 + 7.32691I
u = −1.049340 + 0.798693I
a = −0.064393 + 1.081620I
b = −0.38142 − 2.75805I
1.00002 − 12.73120I 1.58755 + 7.13056I
u = −1.049340 − 0.798693I
a = −0.064393 − 1.081620I
b = −0.38142 + 2.75805I
1.00002 + 12.73120I 1.58755 − 7.13056I
u = −0.638290 + 0.118778I
a = −0.03687 − 1.63298I
b = −0.82110 + 1.58089I
3.42951 − 0.40840I 6.88413 − 1.45858I
u = −0.638290 − 0.118778I
a = −0.03687 + 1.63298I
b = −0.82110 − 1.58089I
3.42951 + 0.40840I 6.88413 + 1.45858I
u = −0.221109 + 0.591527I
a = 0.867559 + 0.657116I
b = 0.159826 − 0.034687I
−0.995483 + 0.739308I −6.93097 − 3.32361I
u = −0.221109 − 0.591527I
a = 0.867559 − 0.657116I
b = 0.159826 + 0.034687I
−0.995483 − 0.739308I −6.93097 + 3.32361I
8
II. I
u
2
= hu
3
b + 4u
2
b − u
3
+ b
2
− bu + u
2
− 2b − u − 4, −u
2
+ a, u
4
− u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
u
2
b
a
7
=
2u
2
+ b − 1
−u
2
b + b + 1
a
10
=
−u
3
+ u
−u
3
b + u
a
11
=
−u
3
b − u
3
+ 2u
−u
3
b + u
a
2
=
−u
2
+ 1
u
2
− 1
a
1
=
0
u
2
− 1
a
5
=
0
−u
2
+ 1
a
8
=
u
3
− 2u
2
− b − u + 1
u
3
b + 2u
3
− 2u
2
− b − 2u + 1
a
12
=
u
3
b + u
3
+ u
2
− 2u
2u
3
+ bu + 3u
2
+ b − u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
+ 8
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
3
(u
4
− u
2
+ 1)
2
c
5
, c
9
(u
2
+ 1)
4
c
6
u
8
− 4u
7
+ 7u
6
− 16u
5
+ 36u
4
− 50u
3
+ 55u
2
− 50u + 25
c
7
, c
8
, c
11
c
12
(u
4
+ 3u
2
+ 1)
2
c
10
u
8
− 2u
7
+ 3u
6
− 2u
5
− 4u
4
+ 20u
3
− 5u
2
+ 25
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y
2
+ y + 1)
4
c
3
(y
2
− y + 1)
4
c
5
, c
9
(y + 1)
8
c
6
y
8
− 2y
7
− 7y
6
− 42y
5
+ 116y
4
+ 210y
3
− 175y
2
+ 250y + 625
c
7
, c
8
, c
11
c
12
(y
2
+ 3y + 1)
4
c
10
y
8
+ 2y
7
− 7y
6
+ 42y
5
+ 116y
4
− 210y
3
− 175y
2
− 250y + 625
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.866025 + 0.500000I
a = 0.500000 + 0.866025I
b = −0.53523 − 1.42303I
2.63189 + 2.02988I 6.00000 − 3.46410I
u = 0.866025 + 0.500000I
a = 0.500000 + 0.866025I
b = 1.40126 − 2.54107I
10.52760 + 2.02988I 6.00000 − 3.46410I
u = 0.866025 − 0.500000I
a = 0.500000 − 0.866025I
b = −0.53523 + 1.42303I
2.63189 − 2.02988I 6.00000 + 3.46410I
u = 0.866025 − 0.500000I
a = 0.500000 − 0.866025I
b = 1.40126 + 2.54107I
10.52760 − 2.02988I 6.00000 + 3.46410I
u = −0.866025 + 0.500000I
a = 0.500000 − 0.866025I
b = −1.40126 + 0.92303I
10.52760 − 2.02988I 6.00000 + 3.46410I
u = −0.866025 + 0.500000I
a = 0.500000 − 0.866025I
b = 0.53523 + 2.04107I
2.63189 − 2.02988I 6.00000 + 3.46410I
u = −0.866025 − 0.500000I
a = 0.500000 + 0.866025I
b = −1.40126 − 0.92303I
10.52760 + 2.02988I 6.00000 − 3.46410I
u = −0.866025 − 0.500000I
a = 0.500000 + 0.866025I
b = 0.53523 − 2.04107I
2.63189 + 2.02988I 6.00000 − 3.46410I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
40
+ 41u
39
+ ··· − 4756u + 625)
c
2
((u
2
+ u + 1)
4
)(u
40
− 11u
39
+ ··· − 216u + 25)
c
3
((u
4
− u
2
+ 1)
2
)(u
40
+ u
39
+ ··· + 4u + 5)
c
4
((u
2
− u + 1)
4
)(u
40
− 11u
39
+ ··· − 216u + 25)
c
5
, c
9
((u
2
+ 1)
4
)(u
40
+ u
39
+ ··· + 13u
2
+ 4)
c
6
(u
8
− 4u
7
+ 7u
6
− 16u
5
+ 36u
4
− 50u
3
+ 55u
2
− 50u + 25)
· (u
40
+ 5u
39
+ ··· − 11820u + 18731)
c
7
, c
8
, c
11
c
12
((u
4
+ 3u
2
+ 1)
2
)(u
40
− u
39
+ ··· − 8u + 1)
c
10
(u
8
− 2u
7
+ 3u
6
− 2u
5
− 4u
4
+ 20u
3
− 5u
2
+ 25)
· (u
40
− 3u
39
+ ··· + 1600u + 1984)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
40
− 79y
39
+ ··· + 1.69530 × 10
7
y + 390625)
c
2
, c
4
((y
2
+ y + 1)
4
)(y
40
+ 41y
39
+ ··· − 4756y + 625)
c
3
((y
2
− y + 1)
4
)(y
40
− 11y
39
+ ··· − 216y + 25)
c
5
, c
9
((y + 1)
8
)(y
40
+ 13y
39
+ ··· + 104y + 16)
c
6
(y
8
− 2y
7
− 7y
6
− 42y
5
+ 116y
4
+ 210y
3
− 175y
2
+ 250y + 625)
· (y
40
+ 31y
39
+ ··· + 6061334998y + 350850361)
c
7
, c
8
, c
11
c
12
((y
2
+ 3y + 1)
4
)(y
40
+ 45y
39
+ ··· − 4y + 1)
c
10
(y
8
+ 2y
7
− 7y
6
+ 42y
5
+ 116y
4
− 210y
3
− 175y
2
− 250y + 625)
· (y
40
+ 15y
39
+ ··· + 108480512y + 3936256)
14
