12n
0007
(K12n
0007
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 11 5 6 12 7 9 10
Solving Sequence
5,7
8
4,11
6 9 12 3 2 1 10
c
7
c
4
c
6
c
8
c
11
c
3
c
2
c
1
c
10
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−9.22578 × 10
81
u
32
+ 2.64829 × 10
82
u
31
+ ··· + 1.74190 × 10
85
b + 7.65657 × 10
83
,
1.64277 × 10
83
u
32
− 4.01664 × 10
83
u
31
+ ··· + 6.96761 × 10
85
a − 2.63055 × 10
86
,
u
33
− 2u
32
+ ··· − 1024u
2
+ 1024i
I
u
2
= hb, −2u
3
− u
2
+ a − 5u − 1, u
4
+ u
3
+ 3u
2
+ 2u + 1i
I
v
1
= ha, −152v
9
+ 36v
8
− 216v
7
+ 881v
6
− 468v
5
+ 684v
4
− 1376v
3
+ 252v
2
+ 115b − 144v + 219,
v
10
− v
9
+ 2v
8
− 7v
7
+ 8v
6
− 9v
5
+ 14v
4
− 10v
3
+ 5v
2
− 3v + 1i
* 3 irreducible components of dim
C
= 0, with total 47 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−9.23 × 10
81
u
32
+ 2.65 × 10
82
u
31
+ · · · + 1.74 × 10
85
b + 7.66 ×
10
83
, 1.64 × 10
83
u
32
− 4.02 × 10
83
u
31
+ · · · + 6.97 × 10
85
a − 2.63 ×
10
86
, u
33
− 2u
32
+ · · · − 1024u
2
+ 1024i
(i) Arc colorings
a
5
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
4
=
u
u
3
+ u
a
11
=
−0.00235772u
32
+ 0.00576474u
31
+ ··· − 1.78671u + 3.77540
0.000529638u
32
− 0.00152034u
31
+ ··· + 1.75463u − 0.0439552
a
6
=
0.00161857u
32
− 0.00336873u
31
+ ··· − 4.32267u + 2.88182
−0.000754427u
32
+ 0.00172427u
31
+ ··· + 0.455492u − 1.26583
a
9
=
0.000947588u
32
− 0.00205444u
31
+ ··· − 0.758148u + 2.00893
0.000267518u
32
− 0.000630392u
31
+ ··· + 0.294022u + 0.443178
a
12
=
−0.00169931u
32
+ 0.00435099u
31
+ ··· − 3.09719u + 4.67643
0.000267518u
32
− 0.000630392u
31
+ ··· + 0.294022u + 0.443178
a
3
=
−0.00169705u
32
+ 0.00482151u
31
+ ··· + 0.535570u − 2.78065
−0.000159267u
32
+ 0.000118296u
31
+ ··· + 2.00893u − 0.970330
a
2
=
−0.00169705u
32
+ 0.00482151u
31
+ ··· + 0.535570u − 2.78065
0.0000201791u
32
− 0.000817233u
31
+ ··· + 3.74671u − 2.43201
a
1
=
−0.00167929u
32
+ 0.00313362u
31
+ ··· + 6.43558u − 4.28239
−0.0000607181u
32
− 0.000235106u
31
+ ··· + 2.11291u − 1.40057
a
10
=
−0.00182808u
32
+ 0.00424440u
31
+ ··· − 0.0320791u + 3.73144
0.000529638u
32
− 0.00152034u
31
+ ··· + 1.75463u − 0.0439552
(ii) Obstruction class = −1
(iii) Cusp Shapes = −0.00699839u
32
+ 0.0259705u
31
+ ··· − 24.8101u + 1.26303
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
33
+ 5u
32
+ ··· + 3u − 1
c
2
, c
5
u
33
+ 7u
32
+ ··· + 5u + 1
c
3
u
33
− 7u
32
+ ··· − 29960u + 14308
c
4
, c
7
u
33
− 2u
32
+ ··· − 1024u
2
+ 1024
c
6
, c
10
u
33
+ 3u
32
+ ··· + 56u − 16
c
8
u
33
+ 4u
32
+ ··· + 2u + 1
c
9
, c
11
, c
12
u
33
+ 7u
32
+ ··· + 4u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
33
+ 53y
32
+ ··· + 3y −1
c
2
, c
5
y
33
+ 5y
32
+ ··· + 3y −1
c
3
y
33
+ 101y
32
+ ··· + 162370712y −204718864
c
4
, c
7
y
33
+ 60y
32
+ ··· + 2097152y −1048576
c
6
, c
10
y
33
+ 33y
32
+ ··· − 2496y −256
c
8
y
33
− 62y
32
+ ··· − 26y −1
c
9
, c
11
, c
12
y
33
− 39y
32
+ ··· − 124y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.736102 + 0.542727I
a = −0.637804 − 0.145676I
b = −0.263754 + 1.251930I
4.58725 + 2.79647I 1.21032 − 3.36441I
u = −0.736102 − 0.542727I
a = −0.637804 + 0.145676I
b = −0.263754 − 1.251930I
4.58725 − 2.79647I 1.21032 + 3.36441I
u = 0.624216 + 0.614071I
a = −0.451815 − 0.151872I
b = −0.581770 + 1.231250I
6.44848 + 5.59696I 6.32857 − 1.76937I
u = 0.624216 − 0.614071I
a = −0.451815 + 0.151872I
b = −0.581770 − 1.231250I
6.44848 − 5.59696I 6.32857 + 1.76937I
u = 0.296890 + 0.749665I
a = −0.46200 − 1.54808I
b = −0.396918 + 0.657378I
2.07164 − 0.96578I 7.05787 − 0.04324I
u = 0.296890 − 0.749665I
a = −0.46200 + 1.54808I
b = −0.396918 − 0.657378I
2.07164 + 0.96578I 7.05787 + 0.04324I
u = −0.479019 + 0.528772I
a = 2.34330 + 1.85274I
b = 0.056157 − 0.749682I
1.16540 − 2.90676I 5.52935 + 0.45206I
u = −0.479019 − 0.528772I
a = 2.34330 − 1.85274I
b = 0.056157 + 0.749682I
1.16540 + 2.90676I 5.52935 − 0.45206I
u = 0.639723 + 0.300028I
a = 0.463444 + 0.633798I
b = 0.345995 − 0.989797I
0.62465 + 2.03384I 1.49443 − 2.77231I
u = 0.639723 − 0.300028I
a = 0.463444 − 0.633798I
b = 0.345995 + 0.989797I
0.62465 − 2.03384I 1.49443 + 2.77231I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.680743
a = −1.22755
b = −0.999659
2.80941 2.91480
u = −0.445593 + 0.510958I
a = 0.788593 + 0.405247I
b = 0.389650 − 0.507994I
−0.634147 + 1.259730I −4.30303 − 4.77679I
u = −0.445593 − 0.510958I
a = 0.788593 − 0.405247I
b = 0.389650 + 0.507994I
−0.634147 − 1.259730I −4.30303 + 4.77679I
u = −0.304293 + 0.498677I
a = 1.074760 − 0.330459I
b = −0.237073 − 0.267230I
−0.29538 + 1.55166I −2.33060 − 5.33546I
u = −0.304293 − 0.498677I
a = 1.074760 + 0.330459I
b = −0.237073 + 0.267230I
−0.29538 − 1.55166I −2.33060 + 5.33546I
u = 0.529499 + 0.187441I
a = 4.29355 − 1.67102I
b = 0.312701 + 0.419841I
1.84801 − 1.60722I 0.8190 + 15.0118I
u = 0.529499 − 0.187441I
a = 4.29355 + 1.67102I
b = 0.312701 − 0.419841I
1.84801 + 1.60722I 0.8190 − 15.0118I
u = −0.10031 + 1.67720I
a = −0.0503517 − 0.0297636I
b = −0.604916 + 0.020460I
7.91587 + 3.25842I 0
u = −0.10031 − 1.67720I
a = −0.0503517 + 0.0297636I
b = −0.604916 − 0.020460I
7.91587 − 3.25842I 0
u = −0.85403 + 1.58938I
a = −0.454673 − 0.911109I
b = 0.21829 + 1.69561I
9.42003 − 4.56175I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.85403 − 1.58938I
a = −0.454673 + 0.911109I
b = 0.21829 − 1.69561I
9.42003 + 4.56175I 0
u = 1.61914 + 1.36310I
a = −0.415764 + 0.566639I
b = −0.06793 − 1.82415I
11.88220 − 1.87850I 0
u = 1.61914 − 1.36310I
a = −0.415764 − 0.566639I
b = −0.06793 + 1.82415I
11.88220 + 1.87850I 0
u = 0.47200 + 2.29182I
a = −0.313254 + 1.103940I
b = −0.34561 − 1.60946I
13.8782 − 7.1833I 0
u = 0.47200 − 2.29182I
a = −0.313254 − 1.103940I
b = −0.34561 + 1.60946I
13.8782 + 7.1833I 0
u = 0.91656 + 2.15918I
a = 0.567229 − 0.874265I
b = 0.84459 + 1.62798I
−18.3854 − 12.6657I 0
u = 0.91656 − 2.15918I
a = 0.567229 + 0.874265I
b = 0.84459 − 1.62798I
−18.3854 + 12.6657I 0
u = 0.01301 + 2.40418I
a = −0.078585 − 1.110070I
b = −0.18185 + 1.65385I
14.1748 − 0.2842I 0
u = 0.01301 − 2.40418I
a = −0.078585 + 1.110070I
b = −0.18185 − 1.65385I
14.1748 + 0.2842I 0
u = −0.27091 + 2.43864I
a = 0.0842297 + 0.0547107I
b = 1.72368 + 0.09458I
16.4290 + 3.7881I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.27091 − 2.43864I
a = 0.0842297 − 0.0547107I
b = 1.72368 − 0.09458I
16.4290 − 3.7881I 0
u = −0.58040 + 2.51179I
a = 0.362910 + 0.854969I
b = 0.78859 − 1.76638I
−17.4300 + 5.1499I 0
u = −0.58040 − 2.51179I
a = 0.362910 − 0.854969I
b = 0.78859 + 1.76638I
−17.4300 − 5.1499I 0
8
II. I
u
2
= hb, −2u
3
− u
2
+ a − 5u − 1, u
4
+ u
3
+ 3u
2
+ 2u + 1i
(i) Arc colorings
a
5
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
4
=
u
u
3
+ u
a
11
=
2u
3
+ u
2
+ 5u + 1
0
a
6
=
1
0
a
9
=
u
2
+ 1
u
2
a
12
=
2u
3
+ 5u
−u
2
a
3
=
u
3
+ 2u
u
3
+ u
a
2
=
u
3
+ 2u
u
3
+ u
2
+ 2u + 1
a
1
=
−u
2
− 1
−u
2
a
10
=
2u
3
+ u
2
+ 5u + 1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u
3
− 11u
2
− 22u − 11
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
4
− u
3
+ 3u
2
− 2u + 1
c
2
u
4
− u
3
+ u
2
+ 1
c
3
u
4
+ u
3
+ 5u
2
− u + 2
c
5
u
4
+ u
3
+ u
2
+ 1
c
6
, c
10
u
4
c
7
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
8
u
4
− 5u
3
+ 7u
2
− 2u + 1
c
9
(u + 1)
4
c
11
, c
12
(u − 1)
4
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
7
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
2
, c
5
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
3
y
4
+ 9y
3
+ 31y
2
+ 19y + 4
c
6
, c
10
y
4
c
8
y
4
− 11y
3
+ 31y
2
+ 10y + 1
c
9
, c
11
, c
12
(y −1)
4
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.395123 + 0.506844I
a = −0.59074 + 2.34806I
b = 0
1.43393 + 1.41510I −3.14142 − 7.60220I
u = −0.395123 − 0.506844I
a = −0.59074 − 2.34806I
b = 0
1.43393 − 1.41510I −3.14142 + 7.60220I
u = −0.10488 + 1.55249I
a = −0.409261 + 0.055548I
b = 0
8.43568 + 3.16396I 11.64142 − 1.04769I
u = −0.10488 − 1.55249I
a = −0.409261 − 0.055548I
b = 0
8.43568 − 3.16396I 11.64142 + 1.04769I
12
III. I
v
1
= ha, −152v
9
+ 36v
8
+ · · · + 115b + 219, v
10
− v
9
+ · · · − 3v + 1i
(i) Arc colorings
a
5
=
v
0
a
7
=
1
0
a
8
=
1
0
a
4
=
v
0
a
11
=
0
1.32174v
9
− 0.313043v
8
+ ··· + 1.25217v − 1.90435
a
6
=
1
1.35652v
9
− 0.373913v
8
+ ··· + 1.49565v − 0.191304
a
9
=
−1.35652v
9
+ 0.373913v
8
+ ··· − 1.49565v + 1.19130
−3.19130v
9
+ 0.834783v
8
+ ··· − 3.33913v + 3.07826
a
12
=
−1.83478v
9
+ 0.460870v
8
+ ··· − 1.84348v + 1.88696
−3.19130v
9
+ 0.834783v
8
+ ··· − 3.33913v + 3.07826
a
3
=
−0.982609v
9
+ 0.469565v
8
+ ··· − 2.87826v + 1.35652
−2.35652v
9
+ 1.37391v
8
+ ··· − 6.49565v + 3.19130
a
2
=
−0.469565v
9
+ 0.321739v
8
+ ··· − 2.28696v + 0.373913
−2.35652v
9
+ 1.37391v
8
+ ··· − 6.49565v + 3.19130
a
1
=
−1
−1.35652v
9
+ 0.373913v
8
+ ··· − 1.49565v + 0.191304
a
10
=
1.32174v
9
− 0.313043v
8
+ ··· + 1.25217v − 1.90435
1.32174v
9
− 0.313043v
8
+ ··· + 1.25217v − 1.90435
(ii) Obstruction class = 1
(iii) Cusp Shapes
= −
281
115
v
9
+
118
115
v
8
−
363
115
v
7
+
1693
115
v
6
−
959
115
v
5
+
977
115
v
4
−
2683
115
v
3
+
251
115
v
2
+
793
115
v +
622
115
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
5
c
2
(u
2
+ u + 1)
5
c
4
, c
7
u
10
c
6
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
8
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
9
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
10
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
11
, c
12
(u
5
+ u
4
− 2u
3
− u
2
+ u − 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)
5
c
4
, c
7
y
10
c
6
, c
10
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y − 1)
2
c
8
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
2
c
9
, c
11
, c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.219640 + 0.330957I
a = 0
b = 0.339110 − 0.822375I
0.329100 − 0.499304I 2.43337 − 0.47576I
v = 1.219640 − 0.330957I
a = 0
b = 0.339110 + 0.822375I
0.329100 + 0.499304I 2.43337 + 0.47576I
v = −0.323203 + 1.221720I
a = 0
b = 0.339110 + 0.822375I
0.32910 − 3.56046I −1.41726 + 7.41465I
v = −0.323203 − 1.221720I
a = 0
b = 0.339110 − 0.822375I
0.32910 + 3.56046I −1.41726 − 7.41465I
v = 0.575710 + 0.191698I
a = 0
b = −0.455697 + 1.200150I
5.87256 + 2.37095I 7.21285 − 1.44195I
v = 0.575710 − 0.191698I
a = 0
b = −0.455697 − 1.200150I
5.87256 − 2.37095I 7.21285 + 1.44195I
v = −0.121840 + 0.594429I
a = 0
b = −0.455697 − 1.200150I
5.87256 − 6.43072I 1.90884 + 7.88634I
v = −0.121840 − 0.594429I
a = 0
b = −0.455697 + 1.200150I
5.87256 + 6.43072I 1.90884 − 7.88634I
v = −0.85031 + 1.47278I
a = 0
b = −0.766826
2.40108 + 2.02988I −0.13779 − 5.66929I
v = −0.85031 − 1.47278I
a = 0
b = −0.766826
2.40108 − 2.02988I −0.13779 + 5.66929I
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
4
− u
3
+ 3u
2
− 2u + 1)(u
33
+ 5u
32
+ ··· + 3u − 1)
c
2
((u
2
+ u + 1)
5
)(u
4
− u
3
+ u
2
+ 1)(u
33
+ 7u
32
+ ··· + 5u + 1)
c
3
(u
2
− u + 1)
5
(u
4
+ u
3
+ 5u
2
− u + 2)
· (u
33
− 7u
32
+ ··· − 29960u + 14308)
c
4
u
10
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
33
− 2u
32
+ ··· − 1024u
2
+ 1024)
c
5
((u
2
− u + 1)
5
)(u
4
+ u
3
+ u
2
+ 1)(u
33
+ 7u
32
+ ··· + 5u + 1)
c
6
u
4
(u
5
+ u
4
+ ··· + u + 1)
2
(u
33
+ 3u
32
+ ··· + 56u − 16)
c
7
u
10
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
33
− 2u
32
+ ··· − 1024u
2
+ 1024)
c
8
(u
4
− 5u
3
+ 7u
2
− 2u + 1)(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
· (u
33
+ 4u
32
+ ··· + 2u + 1)
c
9
((u + 1)
4
)(u
5
− u
4
+ ··· + u + 1)
2
(u
33
+ 7u
32
+ ··· + 4u − 1)
c
10
u
4
(u
5
− u
4
+ ··· + u − 1)
2
(u
33
+ 3u
32
+ ··· + 56u − 16)
c
11
, c
12
((u − 1)
4
)(u
5
+ u
4
+ ··· + u − 1)
2
(u
33
+ 7u
32
+ ··· + 4u − 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
4
+ 5y
3
+ ··· + 2y + 1)(y
33
+ 53y
32
+ ··· + 3y − 1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
33
+ 5y
32
+ ··· + 3y − 1)
c
3
(y
2
+ y + 1)
5
(y
4
+ 9y
3
+ 31y
2
+ 19y + 4)
· (y
33
+ 101y
32
+ ··· + 162370712y − 204718864)
c
4
, c
7
y
10
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
33
+ 60y
32
+ ··· + 2097152y − 1048576)
c
6
, c
10
y
4
(y
5
+ 3y
4
+ ··· − y − 1)
2
(y
33
+ 33y
32
+ ··· − 2496y − 256)
c
8
(y
4
− 11y
3
+ 31y
2
+ 10y + 1)(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y − 1)
2
· (y
33
− 62y
32
+ ··· − 26y − 1)
c
9
, c
11
, c
12
(y − 1)
4
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y − 1)
2
· (y
33
− 39y
32
+ ··· − 124y − 1)
18
