12n
0258
(K12n
0258
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 1 10 12 4 7 1 5 7
Solving Sequence
3,5
2 1
6,10
7 11 4 9 12 8
c
2
c
1
c
5
c
6
c
10
c
4
c
9
c
12
c
7
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h102949u
15
+ 169425u
14
+ ··· + 344734b − 488230,
475025u
15
+ 757448u
14
+ ··· + 689468a − 2984519, u
16
+ 2u
15
+ ··· − 15u − 4i
I
u
2
= hu
5
a + 2u
4
a + u
5
+ 4u
4
− 2u
2
a + 3u
3
− 2au − 2u
2
+ 3b − 2a − 3u − 1, 4u
4
− 2u
2
a − u
3
+ a
2
− 7u
2
+ 2a + 5,
u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
I
u
3
= hau + b + u + 1, a
2
+ au + 3, u
2
+ u − 1i
I
u
4
= h2b + 1, a − 1, u − 1i
* 4 irreducible components of dim
C
= 0, with total 33 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
= h1.03 × 10
5
u
15
+ 1 .69 × 10
5
u
14
+ · · · + 3.45 × 10
5
b − 4.88 × 10
5
, 4.75 ×
10
5
u
15
+7.57×10
5
u
14
+· · ·+6.89×10
5
a−2.98×10
6
, u
16
+2u
15
+· · ·−15u−4i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
6
=
u
5
− 2u
3
+ u
u
5
− u
3
+ u
a
10
=
−0.688973u
15
− 1.09860u
14
+ ··· + 10.8060u + 4.32873
−0.298633u
15
− 0.491466u
14
+ ··· + 4.30076u + 1.41625
a
7
=
0.527611u
15
+ 0.764711u
14
+ ··· − 7.10340u − 2.57154
0.0757251u
15
+ 0.157434u
14
+ ··· − 0.695406u − 0.399317
a
11
=
−0.396324u
15
− 0.532866u
14
+ ··· + 5.76866u + 2.60958
−0.296495u
15
− 0.408933u
14
+ ··· + 4.60606u + 1.80754
a
4
=
u
−u
3
+ u
a
9
=
−1.08530u
15
− 1.63146u
14
+ ··· + 16.5746u + 5.93830
−0.447127u
15
− 0.705112u
14
+ ··· + 6.59538u + 2.18466
a
12
=
−0.396324u
15
− 0.532866u
14
+ ··· + 5.76866u + 2.60958
−0.148494u
15
− 0.213646u
14
+ ··· + 2.29462u + 0.768413
a
8
=
0.896155u
15
+ 1.26037u
14
+ ··· − 12.0524u − 4.14979
0.122193u
15
+ 0.231518u
14
+ ··· − 1.42447u − 0.424931
(ii) Obstruction class = −1
(iii) Cusp Shapes =
2007761
689468
u
15
+
2194435
689468
u
14
+ ··· −
28351265
689468
u −
1282717
172367
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
+ 10u
15
+ ··· + 65u + 16
c
2
, c
4
u
16
− 2u
15
+ ··· + 15u − 4
c
3
, c
8
u
16
− 3u
15
+ ··· + 6u + 8
c
5
u
16
− 3u
15
+ ··· + 52u + 64
c
6
, c
7
, c
9
c
11
, c
12
u
16
− u
15
+ ··· − u − 1
c
10
u
16
− 12u
15
+ ··· − 50u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 6y
15
+ ··· + 1343y + 256
c
2
, c
4
y
16
− 10y
15
+ ··· − 65y + 16
c
3
, c
8
y
16
+ 3y
15
+ ··· − 52y + 64
c
5
y
16
+ 67y
15
+ ··· − 140304y + 4096
c
6
, c
7
, c
9
c
11
, c
12
y
16
+ 31y
15
+ ··· − 9y + 1
c
10
y
16
− 36y
15
+ ··· − 2908y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.891782
a = 0.236674
b = −0.553091
−1.30085 −8.84520
u = 1.091810 + 0.330825I
a = −0.147665 + 0.051010I
b = 0.210193 − 0.600885I
−3.22144 − 1.13355I −4.59977 + 1.25337I
u = 1.091810 − 0.330825I
a = −0.147665 − 0.051010I
b = 0.210193 + 0.600885I
−3.22144 + 1.13355I −4.59977 − 1.25337I
u = −1.094020 + 0.526074I
a = −1.054950 − 0.663309I
b = 0.141861 − 0.569038I
−1.87639 + 6.13504I 0.06427 − 7.98576I
u = −1.094020 − 0.526074I
a = −1.054950 + 0.663309I
b = 0.141861 + 0.569038I
−1.87639 − 6.13504I 0.06427 + 7.98576I
u = −0.029850 + 1.272860I
a = 4.37033 − 0.53285I
b = 3.61404 − 0.37068I
19.7225 − 4.8173I −2.30205 + 1.82839I
u = −0.029850 − 1.272860I
a = 4.37033 + 0.53285I
b = 3.61404 + 0.37068I
19.7225 + 4.8173I −2.30205 − 1.82839I
u = −1.31580
a = 0.254567
b = −1.49207
−0.940295 −7.17020
u = −0.296508 + 0.600916I
a = 0.432297 − 1.059770I
b = −0.140186 − 0.553162I
0.34157 − 1.65330I 2.57923 + 4.74775I
u = −0.296508 − 0.600916I
a = 0.432297 + 1.059770I
b = −0.140186 + 0.553162I
0.34157 + 1.65330I 2.57923 − 4.74775I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.537130 + 0.286418I
a = 0.60885 + 1.51474I
b = −0.016541 + 0.547073I
0.96164 + 1.16578I 5.62628 − 5.26913I
u = −0.537130 − 0.286418I
a = 0.60885 − 1.51474I
b = −0.016541 − 0.547073I
0.96164 − 1.16578I 5.62628 + 5.26913I
u = −1.42350 + 0.64934I
a = −1.89897 + 2.53374I
b = −3.64218 + 0.84571I
15.4087 + 11.5569I −4.10768 − 4.68861I
u = −1.42350 − 0.64934I
a = −1.89897 − 2.53374I
b = −3.64218 − 0.84571I
15.4087 − 11.5569I −4.10768 + 4.68861I
u = 1.50121 + 0.66208I
a = −1.43051 − 3.06754I
b = −3.39461 − 1.98308I
15.0197 − 2.0565I −4.37753 + 0.79044I
u = 1.50121 − 0.66208I
a = −1.43051 + 3.06754I
b = −3.39461 + 1.98308I
15.0197 + 2.0565I −4.37753 − 0.79044I
6
II. I
u
2
= hu
5
a + u
5
+ · · · − 2a − 1, 4u
4
− 2u
2
a − u
3
+ a
2
− 7u
2
+ 2a + 5, u
6
+
u
5
− u
4
− 2u
3
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
6
=
u
5
− 2u
3
+ u
u
5
− u
3
+ u
a
10
=
a
−
1
3
u
5
a −
1
3
u
5
+ ··· +
2
3
a +
1
3
a
7
=
1
3
u
5
a +
4
3
u
5
+ ··· +
1
3
a +
5
3
−
1
3
u
5
a +
2
3
u
5
+ ··· +
2
3
a +
1
3
a
11
=
1
3
u
4
a +
1
3
u
5
+ ··· +
2
3
a + 1
−
2
3
u
5
a −
2
3
u
4
a + ··· +
2
3
a +
2
3
a
4
=
u
−u
3
+ u
a
9
=
−
1
3
u
4
a −
1
3
u
5
+ ··· +
1
3
a −
2
3
u
1
3
u
4
a −
2
3
u
5
+ ··· −
1
3
a −
1
3
u
a
12
=
1
3
u
4
a +
1
3
u
5
+ ··· +
2
3
a + 1
−
1
3
u
5
a +
1
3
u
5
+ ··· + a +
1
3
a
8
=
1
3
u
5
a +
1
3
u
5
+ ··· +
1
3
a +
2
3
2
3
u
5
a +
2
3
u
4
a + ··· +
1
3
a +
1
3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
4
− 4u
2
− 4u − 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
c
2
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
c
3
, c
8
u
12
+ 3u
10
+ 5u
8
+ 4u
6
+ 2u
4
+ u
2
+ 1
c
4
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
c
6
, c
7
, c
9
c
11
, c
12
(u
2
+ 1)
6
c
10
u
12
− 12u
11
+ ··· − 60u + 9
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
c
2
, c
4
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
c
3
, c
8
(y
6
+ 3y
5
+ 5y
4
+ 4y
3
+ 2y
2
+ y + 1)
2
c
6
, c
7
, c
9
c
11
, c
12
(y + 1)
12
c
10
y
12
− 14y
11
+ ··· − 108y + 81
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002190 + 0.295542I
a = 0.387926 + 1.194620I
b = 1.74846 − 0.37806I
−5.18047 − 0.92430I −9.71672 + 0.79423I
u = 1.002190 + 0.295542I
a = −0.553835 − 0.009862I
b = −0.37806 − 1.74846I
−5.18047 − 0.92430I −9.71672 + 0.79423I
u = 1.002190 − 0.295542I
a = 0.387926 − 1.194620I
b = 1.74846 + 0.37806I
−5.18047 + 0.92430I −9.71672 − 0.79423I
u = 1.002190 − 0.295542I
a = −0.553835 + 0.009862I
b = −0.37806 + 1.74846I
−5.18047 + 0.92430I −9.71672 − 0.79423I
u = −0.428243 + 0.664531I
a = −2.09808 + 1.60703I
b = −1.188690 + 0.647273I
−1.39926 − 0.92430I −2.28328 + 0.79423I
u = −0.428243 + 0.664531I
a = −0.41834 − 2.74535I
b = −0.647273 − 1.188690I
−1.39926 − 0.92430I −2.28328 + 0.79423I
u = −0.428243 − 0.664531I
a = −2.09808 − 1.60703I
b = −1.188690 − 0.647273I
−1.39926 + 0.92430I −2.28328 − 0.79423I
u = −0.428243 − 0.664531I
a = −0.41834 + 2.74535I
b = −0.647273 + 1.188690I
−1.39926 + 0.92430I −2.28328 − 0.79423I
u = −1.073950 + 0.558752I
a = 1.42256 − 0.62619I
b = 1.114040 + 0.351534I
−3.28987 + 5.69302I −6.00000 − 5.51057I
u = −1.073950 + 0.558752I
a = −1.74023 − 1.77409I
b = 0.351534 − 1.114040I
−3.28987 + 5.69302I −6.00000 − 5.51057I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.073950 − 0.558752I
a = 1.42256 + 0.62619I
b = 1.114040 − 0.351534I
−3.28987 − 5.69302I −6.00000 + 5.51057I
u = −1.073950 − 0.558752I
a = −1.74023 + 1.77409I
b = 0.351534 + 1.114040I
−3.28987 − 5.69302I −6.00000 + 5.51057I
11
III. I
u
3
= hau + b + u + 1, a
2
+ au + 3, u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u − 1
a
1
=
u
u − 1
a
6
=
2u − 1
4u − 2
a
10
=
a
−au − u − 1
a
7
=
−u − 1
−a + 1
a
11
=
a + u
−au − 2
a
4
=
u
−u + 1
a
9
=
−u
u
a
12
=
a + u
−a − 2u − 1
a
8
=
2u − 1
−3u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
2
+ 3u + 1)
2
c
2
, c
4
(u
2
− u − 1)
2
c
3
, c
8
(u
2
+ u − 1)
2
c
6
, c
7
, c
9
c
11
, c
12
u
4
+ 3u
3
+ 10u
2
+ 6u + 9
c
10
(u + 1)
4
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
2
− 7y + 1)
2
c
2
, c
3
, c
4
c
8
(y
2
− 3y + 1)
2
c
6
, c
7
, c
9
c
11
, c
12
y
4
+ 11y
3
+ 82y
2
+ 144y + 81
c
10
(y − 1)
4
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −0.30902 + 1.70426I
b = −1.42705 − 1.05329I
−4.27683 −6.00000
u = 0.618034
a = −0.30902 − 1.70426I
b = −1.42705 + 1.05329I
−4.27683 −6.00000
u = −1.61803
a = 0.80902 + 1.53150I
b = 1.92705 + 2.47802I
−12.1725 −6.00000
u = −1.61803
a = 0.80902 − 1.53150I
b = 1.92705 − 2.47802I
−12.1725 −6.00000
15
IV. I
u
4
= h2b + 1, a − 1, u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
1
a
2
=
1
−1
a
1
=
0
−1
a
6
=
0
1
a
10
=
1
−0.5
a
7
=
1
0.5
a
11
=
1
0.5
a
4
=
1
0
a
9
=
2
0
a
12
=
1
−0.5
a
8
=
2
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2.25
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
9
c
11
, c
12
u − 1
c
3
, c
5
, c
8
u
c
4
, c
6
, c
7
c
10
u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
7
, c
9
c
10
, c
11
, c
12
y − 1
c
3
, c
5
, c
8
y
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = −0.500000
0 2.25000
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
2
+ 3u + 1)
2
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
· (u
16
+ 10u
15
+ ··· + 65u + 16)
c
2
(u − 1)(u
2
− u − 1)
2
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
· (u
16
− 2u
15
+ ··· + 15u − 4)
c
3
, c
8
u(u
2
+ u − 1)
2
(u
12
+ 3u
10
+ 5u
8
+ 4u
6
+ 2u
4
+ u
2
+ 1)
· (u
16
− 3u
15
+ ··· + 6u + 8)
c
4
(u + 1)(u
2
− u − 1)
2
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
· (u
16
− 2u
15
+ ··· + 15u − 4)
c
5
u(u
2
+ 3u + 1)
2
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
· (u
16
− 3u
15
+ ··· + 52u + 64)
c
6
, c
7
(u + 1)(u
2
+ 1)
6
(u
4
+ 3u
3
+ ··· + 6u + 9)(u
16
− u
15
+ ··· − u − 1)
c
9
, c
11
, c
12
(u − 1)(u
2
+ 1)
6
(u
4
+ 3u
3
+ ··· + 6u + 9)(u
16
− u
15
+ ··· − u − 1)
c
10
((u + 1)
5
)(u
12
− 12u
11
+ ··· − 60u + 9)(u
16
− 12u
15
+ ··· − 50u + 4)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y − 1)(y
2
− 7y + 1)
2
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
· (y
16
− 6y
15
+ ··· + 1343y + 256)
c
2
, c
4
(y − 1)(y
2
− 3y + 1)
2
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
· (y
16
− 10y
15
+ ··· − 65y + 16)
c
3
, c
8
y(y
2
− 3y + 1)
2
(y
6
+ 3y
5
+ 5y
4
+ 4y
3
+ 2y
2
+ y + 1)
2
· (y
16
+ 3y
15
+ ··· − 52y + 64)
c
5
y(y
2
− 7y + 1)
2
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
· (y
16
+ 67y
15
+ ··· − 140304y + 4096)
c
6
, c
7
, c
9
c
11
, c
12
(y − 1)(y + 1)
12
(y
4
+ 11y
3
+ 82y
2
+ 144y + 81)
· (y
16
+ 31y
15
+ ··· − 9y + 1)
c
10
((y − 1)
5
)(y
12
− 14y
11
+ ··· − 108y + 81)
· (y
16
− 36y
15
+ ··· − 2908y + 16)
21
