12n
0694
(K12n
0694
)
A knot diagram
1
Linearized knot diagam
4 5 9 2 12 10 11 3 7 5 7 10
Solving Sequence
1,4
2
5,10
12 6 7 9 3 8 11
c
1
c
4
c
12
c
5
c
6
c
9
c
3
c
8
c
11
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h5542348185u
14
+ 12483915678u
13
+ ··· + 193710435976b − 48755007008,
− 57969081u
14
− 531745184u
13
+ ··· + 22789463056a − 38728287076,
u
15
+ 3u
14
+ ··· + 60u − 16i
I
u
2
= hu
4
+ 2u
3
− u
2
+ b − 2u + 2, u
4
+ 3u
3
+ a − 6u − 4, u
5
+ 4u
4
+ 3u
3
− 4u
2
− 4u + 1i
I
u
3
= h13a
3
u
2
+ 2a
3
u + 3a
2
u
2
− 9a
3
− 3a
2
u + 24u
2
a + a
2
+ 11au + 14u
2
+ 5b − 22a − 4u + 3,
− 2a
3
u
2
+ a
4
− 2a
3
u + 2a
2
u
2
− a
3
+ 3a
2
u − 32u
2
a + 4a
2
− 53au + 17u
2
− 38a + 31u + 23, u
3
+ u
2
− 1i
I
u
4
= hb − u, 2u
2
+ a + u − 1, u
3
− u + 1i
I
u
5
= hb − 2a + 1, 4a
2
− 6a + 1, u − 1i
* 5 irreducible components of dim
C
= 0, with total 37 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
= h5.54 × 10
9
u
14
+ 1.25 × 10
10
u
13
+ · · · + 1.94× 10
11
b − 4.88 ×10
10
, −5.80 ×
10
7
u
14
−5.32×10
8
u
13
+· · ·+2.28×10
10
a−3.87×10
10
, u
15
+3u
14
+· · ·+60u−16i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
10
=
0.00254368u
14
+ 0.0233329u
13
+ ··· + 0.0452708u + 1.69939
−0.0286115u
14
− 0.0644463u
13
+ ··· − 1.99339u + 0.251690
a
12
=
0.0380093u
14
+ 0.0802367u
13
+ ··· + 2.56494u + 0.802099
0.0511056u
14
+ 0.0966535u
13
+ ··· + 3.44007u − 0.786612
a
6
=
−0.0455457u
14
− 0.0977825u
13
+ ··· − 5.28838u + 0.536706
−0.107164u
14
− 0.218718u
13
+ ··· − 7.08098u + 1.73158
a
7
=
−0.00280296u
14
− 0.0172477u
13
+ ··· − 1.86151u − 0.654871
−0.0399111u
14
− 0.0794444u
13
+ ··· − 2.27957u + 0.756192
a
9
=
0.00251495u
14
+ 0.0394902u
13
+ ··· + 0.759483u + 1.78965
0.120131u
14
+ 0.203886u
13
+ ··· + 7.47958u − 1.83534
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
8
=
−0.00985312u
14
− 0.00127324u
13
+ ··· + 1.07235u + 1.36528
0.102369u
14
+ 0.176801u
13
+ ··· + 5.54963u − 1.50976
a
11
=
0.0247410u
14
+ 0.0688107u
13
+ ··· + 1.90500u + 1.26192
−0.0282390u
14
− 0.0646220u
13
+ ··· − 2.23111u + 0.351338
(ii) Obstruction class = −1
(iii) Cusp Shapes =
22363926543
45578926112
u
14
+
10216429583
11394731528
u
13
+ ··· +
485831949075
11394731528
u −
73524844069
2848682882
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
15
− 3u
14
+ ··· + 60u + 16
c
3
, c
8
u
15
+ 8u
14
+ ··· − 112u − 64
c
5
, c
6
, c
9
u
15
+ 9u
12
+ ··· + 7u
2
− 1
c
7
, c
10
, c
11
u
15
− u
14
+ ··· + u + 1
c
12
u
15
− 14u
14
+ ··· + 68u − 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
15
− 15y
14
+ ··· + 5232y −256
c
3
, c
8
y
15
− 12y
14
+ ··· + 41728y −4096
c
5
, c
6
, c
9
y
15
− 4y
13
+ ··· + 14y −1
c
7
, c
10
, c
11
y
15
+ 23y
14
+ ··· − 23y −1
c
12
y
15
− 68y
14
+ ··· + 2576y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.07635
a = 1.19704
b = 1.58782
−10.6859 −41.6000
u = 1.178640 + 0.172680I
a = 0.627487 − 0.433643I
b = 0.282219 − 0.202348I
−1.55215 − 0.88269I −10.53205 + 2.04290I
u = 1.178640 − 0.172680I
a = 0.627487 + 0.433643I
b = 0.282219 + 0.202348I
−1.55215 + 0.88269I −10.53205 − 2.04290I
u = 0.160834 + 0.708843I
a = 0.014785 + 0.187812I
b = 0.520261 − 0.168254I
1.32905 − 2.33965I −9.51656 + 6.09486I
u = 0.160834 − 0.708843I
a = 0.014785 − 0.187812I
b = 0.520261 + 0.168254I
1.32905 + 2.33965I −9.51656 − 6.09486I
u = −1.360170 + 0.309898I
a = 0.509328 + 0.442124I
b = 0.714313 + 0.230460I
−3.48368 + 6.07143I −13.1701 − 10.7080I
u = −1.360170 − 0.309898I
a = 0.509328 − 0.442124I
b = 0.714313 − 0.230460I
−3.48368 − 6.07143I −13.1701 + 10.7080I
u = −0.12171 + 1.44460I
a = −0.280558 − 0.267780I
b = 2.37166 + 0.47398I
9.74508 − 5.81410I −9.65717 + 3.18656I
u = −0.12171 − 1.44460I
a = −0.280558 + 0.267780I
b = 2.37166 − 0.47398I
9.74508 + 5.81410I −9.65717 − 3.18656I
u = −1.42958 + 0.70949I
a = 1.01628 + 1.25193I
b = 2.03512 − 0.49412I
5.6640 + 13.2045I −12.63072 − 5.98469I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.42958 − 0.70949I
a = 1.01628 − 1.25193I
b = 2.03512 + 0.49412I
5.6640 − 13.2045I −12.63072 + 5.98469I
u = 0.219796
a = 1.78361
b = −0.198958
−0.592779 −16.9910
u = 1.68228 + 0.91262I
a = 0.793163 − 0.925065I
b = 3.00988 + 0.43262I
4.36534 − 2.50283I −8.50423 + 2.90053I
u = 1.68228 − 0.91262I
a = 0.793163 + 0.925065I
b = 3.00988 − 0.43262I
4.36534 + 2.50283I −8.50423 − 2.90053I
u = −2.36403
a = −1.34162
b = −5.25577
−19.2116 16.8620
6
II.
I
u
2
= hu
4
+2u
3
−u
2
+b−2u+2, u
4
+3u
3
+a−6u−4, u
5
+4u
4
+3u
3
−4u
2
−4u+1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
10
=
−u
4
− 3u
3
+ 6u + 4
−u
4
− 2u
3
+ u
2
+ 2u − 2
a
12
=
−2u
4
− 5u
3
+ u
2
+ 10u + 7
−4u
4
− 7u
3
+ 4u
2
+ 7u − 4
a
6
=
4u
4
+ 17u
3
+ 17u
2
− 7u − 12
−2u
4
− 6u
3
− 3u
2
+ 3u + 3
a
7
=
2u
4
+ 8u
3
+ 7u
2
− 4u − 5
−u
4
− 3u
3
− u
2
+ 2u + 1
a
9
=
u
3
+ 2u
2
− u − 2
−2u
4
− 5u
3
+ u
2
+ 6u − 1
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
8
=
3u
4
+ 7u
3
− u
2
− 8u
6u
4
+ 10u
3
− 5u
2
− 12u + 3
a
11
=
−u
4
− 2u
3
+ u
2
+ 5u + 4
−4u
4
− 7u
3
+ 4u
2
+ 7u − 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = −10u
4
− 33u
3
− 29u
2
+ 3u − 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
u
5
+ 4u
4
+ 3u
3
− 4u
2
− 4u + 1
c
3
u
5
− 6u
3
+ 11u
2
− 6u + 1
c
4
u
5
− 4u
4
+ 3u
3
+ 4u
2
− 4u − 1
c
5
, c
9
u
5
+ u
4
− u
3
− 2u
2
− u + 1
c
6
u
5
− u
4
− u
3
+ 2u
2
− u − 1
c
7
, c
10
u
5
+ u
4
− 2u
3
+ u
2
+ u − 1
c
8
u
5
− 6u
3
− 11u
2
− 6u − 1
c
11
u
5
− u
4
− 2u
3
− u
2
+ u + 1
c
12
u
5
+ 10u
4
+ 34u
3
+ 55u
2
+ 46u + 17
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
5
− 10y
4
+ 33y
3
− 48y
2
+ 24y −1
c
3
, c
8
y
5
− 12y
4
+ 24y
3
− 49y
2
+ 14y −1
c
5
, c
6
, c
9
y
5
− 3y
4
+ 3y
3
− 4y
2
+ 5y −1
c
7
, c
10
, c
11
y
5
− 5y
4
+ 4y
3
− 3y
2
+ 3y −1
c
12
y
5
− 32y
4
+ 148y
3
− 237y
2
+ 246y −289
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.935978
a = 6.38849
b = −1.65940
−5.47443 −63.3310
u = −1.41748 + 0.38647I
a = −0.124879 − 0.421155I
b = −0.807993 − 0.790836I
−3.56702 + 5.27138I −13.75047 − 1.28258I
u = −1.41748 − 0.38647I
a = −0.124879 + 0.421155I
b = −0.807993 + 0.790836I
−3.56702 − 5.27138I −13.75047 + 1.28258I
u = 0.213816
a = 5.25148
b = −1.54829
−4.24110 −7.02780
u = −2.31483
a = −1.39021
b = −5.17632
−19.3390 −46.1400
10
III.
I
u
3
= h13a
3
u
2
+3a
2
u
2
+· · ·−22a+3, −2a
3
u
2
+2a
2
u
2
+· · ·−38a+23, u
3
+u
2
−1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
u
2
+ u − 1
a
10
=
a
−2.60000a
3
u
2
− 0.600000a
2
u
2
+ ··· + 4.40000a − 0.600000
a
12
=
−
2
5
a
3
u
2
+
1
5
a
2
u
2
+ ··· −
2
5
a −
4
5
−2.60000a
3
u
2
− 0.600000a
2
u
2
+ ··· + 4.40000a − 1.60000
a
6
=
−a
3
u
2
−
3
5
a
2
u
2
+ ··· − a −
3
5
7
5
a
3
u
2
+
8
5
a
2
u
2
+ ··· −
18
5
a −
17
5
a
7
=
−
4
5
a
2
u
2
−
2
5
u
2
+ ··· +
2
5
a
2
+
6
5
9
5
a
3
u
2
+
2
5
a
2
u
2
+ ··· −
11
5
a +
2
5
a
9
=
−u
2u
2
+ u − 2
a
3
=
−u
2
+ 1
u
2
− u + 1
a
8
=
−2u + 1
5u
2
+ 2u − 4
a
11
=
2
5
a
3
u
2
+
7
5
a
2
u
2
+ ··· −
13
5
a −
18
5
−7.40000a
3
u
2
− 2.40000a
2
u
2
+ ··· + 15.6000a + 1.60000
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u − 14
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
(u
3
− u
2
+ 1)
4
c
3
, c
8
(u
3
− u
2
+ 2u − 1)
4
c
5
, c
6
, c
9
u
12
− 3u
11
+ ··· − 2u − 59
c
7
, c
10
, c
11
u
12
+ 3u
11
+ ··· − 314u − 121
c
12
(u
2
+ u − 1)
6
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y
3
− y
2
+ 2y −1)
4
c
3
, c
8
(y
3
+ 3y
2
+ 2y −1)
4
c
5
, c
6
, c
9
y
12
− y
11
+ ··· + 5424y + 3481
c
7
, c
10
, c
11
y
12
+ 11y
11
+ ··· − 50196y + 14641
c
12
(y
2
− 3y + 1)
6
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.037366 − 0.810507I
b = 0.618034
6.97197 + 2.82812I −10.49024 − 2.97945I
u = −0.877439 + 0.744862I
a = −0.127901 − 1.361650I
b = −1.61803
−0.92371 + 2.82812I −10.49024 − 2.97945I
u = −0.877439 + 0.744862I
a = −0.397503 − 0.457922I
b = −1.61803
−0.92371 + 2.82812I −10.49024 − 2.97945I
u = −0.877439 + 0.744862I
a = 0.23805 + 1.50552I
b = 0.618034
6.97197 + 2.82812I −10.49024 − 2.97945I
u = −0.877439 − 0.744862I
a = −0.037366 + 0.810507I
b = 0.618034
6.97197 − 2.82812I −10.49024 + 2.97945I
u = −0.877439 − 0.744862I
a = −0.127901 + 1.361650I
b = −1.61803
−0.92371 − 2.82812I −10.49024 + 2.97945I
u = −0.877439 − 0.744862I
a = −0.397503 + 0.457922I
b = −1.61803
−0.92371 − 2.82812I −10.49024 + 2.97945I
u = −0.877439 − 0.744862I
a = 0.23805 − 1.50552I
b = 0.618034
6.97197 − 2.82812I −10.49024 + 2.97945I
u = 0.754878
a = 0.603863
b = −1.61803
−5.06130 −17.0200
u = 0.754878
a = −1.12774 + 4.03110I
b = 0.618034
2.83439 −17.0200
14
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.754878
a = −1.12774 − 4.03110I
b = 0.618034
2.83439 −17.0200
u = 0.754878
a = 5.30105
b = −1.61803
−5.06130 −17.0200
15
IV. I
u
4
= hb − u, 2u
2
+ a + u − 1, u
3
− u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
1
a
10
=
−2u
2
− u + 1
u
a
12
=
u
2
+ u − 1
−u
2
a
6
=
−u + 1
u
2
− u + 1
a
7
=
−u
2
+ 2
u
2
a
9
=
−u
2
+ 1
u
2
a
3
=
−u
2
+ 1
u
2
+ u
a
8
=
0
−u
a
11
=
−u
2
− u
−u
2
+ 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
2
+ 5u − 14
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
u
3
− u + 1
c
3
(u + 1)
3
c
4
u
3
− u − 1
c
5
, c
9
u
3
+ 2u
2
+ u + 1
c
6
u
3
− 2u
2
+ u − 1
c
7
, c
10
u
3
− u
2
+ 2u − 1
c
8
(u − 1)
3
c
11
u
3
+ u
2
+ 2u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
12
y
3
− 2y
2
+ y −1
c
3
, c
8
(y −1)
3
c
5
, c
6
, c
9
y
3
− 2y
2
− 3y −1
c
7
, c
10
, c
11
y
3
+ 3y
2
+ 2y −1
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.662359 + 0.562280I
a = 0.09252 − 2.05200I
b = 0.662359 + 0.562280I
2.83014 − 0.94271I −10.44308 + 4.30112I
u = 0.662359 − 0.562280I
a = 0.09252 + 2.05200I
b = 0.662359 − 0.562280I
2.83014 + 0.94271I −10.44308 − 4.30112I
u = −1.32472
a = −1.18504
b = −1.32472
−8.95014 −17.1140
19
V. I
u
5
= hb − 2a + 1, 4a
2
− 6a + 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
10
=
a
2a − 1
a
12
=
−2a +
3
2
−2a
a
6
=
−3a
−6a + 1
a
7
=
−0.5
−2a
a
9
=
0
−4a
a
3
=
0
1
a
8
=
0
−4a
a
11
=
−a + 1
2a − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =
45
2
a − 20
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
8
u
2
c
4
(u + 1)
2
c
5
, c
6
u
2
− 3u + 1
c
7
u
2
+ u − 1
c
9
u
2
+ 3u + 1
c
10
, c
11
, c
12
u
2
− u − 1
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
2
c
3
, c
8
y
2
c
5
, c
6
, c
9
y
2
− 7y + 1
c
7
, c
10
, c
11
c
12
y
2
− 3y + 1
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.30902
b = 1.61803
−10.5276 9.45290
u = 1.00000
a = 0.190983
b = −0.618034
−2.63189 −15.7030
23
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
(u
3
− u + 1)(u
3
− u
2
+ 1)
4
(u
5
+ 4u
4
+ 3u
3
− 4u
2
− 4u + 1)
· (u
15
− 3u
14
+ ··· + 60u + 16)
c
3
u
2
(u + 1)
3
(u
3
− u
2
+ 2u − 1)
4
(u
5
− 6u
3
+ 11u
2
− 6u + 1)
· (u
15
+ 8u
14
+ ··· − 112u − 64)
c
4
(u + 1)
2
(u
3
− u − 1)(u
3
− u
2
+ 1)
4
(u
5
− 4u
4
+ 3u
3
+ 4u
2
− 4u − 1)
· (u
15
− 3u
14
+ ··· + 60u + 16)
c
5
(u
2
− 3u + 1)(u
3
+ 2u
2
+ u + 1)(u
5
+ u
4
− u
3
− 2u
2
− u + 1)
· (u
12
− 3u
11
+ ··· − 2u − 59)(u
15
+ 9u
12
+ ··· + 7u
2
− 1)
c
6
(u
2
− 3u + 1)(u
3
− 2u
2
+ u − 1)(u
5
− u
4
− u
3
+ 2u
2
− u − 1)
· (u
12
− 3u
11
+ ··· − 2u − 59)(u
15
+ 9u
12
+ ··· + 7u
2
− 1)
c
7
(u
2
+ u − 1)(u
3
− u
2
+ 2u − 1)(u
5
+ u
4
− 2u
3
+ u
2
+ u − 1)
· (u
12
+ 3u
11
+ ··· − 314u − 121)(u
15
− u
14
+ ··· + u + 1)
c
8
u
2
(u − 1)
3
(u
3
− u
2
+ 2u − 1)
4
(u
5
− 6u
3
− 11u
2
− 6u − 1)
· (u
15
+ 8u
14
+ ··· − 112u − 64)
c
9
(u
2
+ 3u + 1)(u
3
+ 2u
2
+ u + 1)(u
5
+ u
4
− u
3
− 2u
2
− u + 1)
· (u
12
− 3u
11
+ ··· − 2u − 59)(u
15
+ 9u
12
+ ··· + 7u
2
− 1)
c
10
(u
2
− u − 1)(u
3
− u
2
+ 2u − 1)(u
5
+ u
4
− 2u
3
+ u
2
+ u − 1)
· (u
12
+ 3u
11
+ ··· − 314u − 121)(u
15
− u
14
+ ··· + u + 1)
c
11
(u
2
− u − 1)(u
3
+ u
2
+ 2u + 1)(u
5
− u
4
− 2u
3
− u
2
+ u + 1)
· (u
12
+ 3u
11
+ ··· − 314u − 121)(u
15
− u
14
+ ··· + u + 1)
c
12
(u
2
− u − 1)(u
2
+ u − 1)
6
(u
3
− u + 1)
· (u
5
+ 10u
4
+ ··· + 46u + 17)(u
15
− 14u
14
+ ··· + 68u − 8)
24
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
2
(y
3
− 2y
2
+ y −1)(y
3
− y
2
+ 2y −1)
4
· (y
5
− 10y
4
+ ··· + 24y −1)(y
15
− 15y
14
+ ··· + 5232y −256)
c
3
, c
8
y
2
(y −1)
3
(y
3
+ 3y
2
+ 2y −1)
4
(y
5
− 12y
4
+ ··· + 14y −1)
· (y
15
− 12y
14
+ ··· + 41728y −4096)
c
5
, c
6
, c
9
(y
2
− 7y + 1)(y
3
− 2y
2
− 3y −1)(y
5
− 3y
4
+ 3y
3
− 4y
2
+ 5y −1)
· (y
12
− y
11
+ ··· + 5424y + 3481)(y
15
− 4y
13
+ ··· + 14y −1)
c
7
, c
10
, c
11
(y
2
− 3y + 1)(y
3
+ 3y
2
+ 2y −1)(y
5
− 5y
4
+ 4y
3
− 3y
2
+ 3y −1)
· (y
12
+ 11y
11
+ ··· − 50196y + 14641)(y
15
+ 23y
14
+ ··· − 23y −1)
c
12
(y
2
− 3y + 1)
7
(y
3
− 2y
2
+ y −1)
· (y
5
− 32y
4
+ 148y
3
− 237y
2
+ 246y −289)
· (y
15
− 68y
14
+ ··· + 2576y −64)
25
