12n
0856
(K12n
0856
)
A knot diagram
1
Linearized knot diagam
4 7 9 12 11 10 3 12 2 6 5 9
Solving Sequence
6,11
5 12
2,4
1 10 7 9 3 8
c
5
c
11
c
4
c
1
c
10
c
6
c
9
c
3
c
7
c
2
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
15
− 5u
14
+ ··· + 2b − 4, −u
15
− 4u
14
+ ··· + 2a + 1, u
16
+ 5u
15
+ ··· + 30u + 4i
I
u
2
= h−74u
4
a
3
+ 62u
4
a
2
+ ··· + 20a − 170, 2u
4
a
3
− 11u
4
a + ··· − 23a − 23, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
I
u
3
= h−u
5
− u
4
− 3u
3
− 3u
2
+ b − u − 1, u
8
+ 7u
6
− u
5
+ 15u
4
− 4u
3
+ 10u
2
+ a − 4u + 2,
u
9
+ 7u
7
+ 16u
5
+ 13u
3
+ 3u + 1i
* 3 irreducible components of dim
C
= 0, with total 45 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−u
15
−5u
14
+· · ·+2b−4, −u
15
−4u
14
+· · ·+2a+1, u
16
+5u
15
+· · ·+30u+4i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
5
=
1
−u
2
a
12
=
−u
u
3
+ u
a
2
=
1
2
u
15
+ 2u
14
+ ··· −
5
2
u −
1
2
1
2
u
15
+
5
2
u
14
+ ··· +
25
2
u + 2
a
4
=
u
2
+ 1
−u
4
− 2u
2
a
1
=
u
15
+
9
2
u
14
+ ··· + 32u +
11
2
1
2
u
15
+
5
2
u
14
+ ··· +
33
2
u + 2
a
10
=
u
u
a
7
=
u
2
+ 1
u
2
a
9
=
3
4
u
15
+
13
4
u
14
+ ··· +
89
4
u + 3
−
1
2
u
15
−
5
2
u
14
+ ··· −
41
2
u − 3
a
3
=
−u
15
−
9
2
u
14
+ ··· − 33u −
9
2
−
1
2
u
15
−
5
2
u
14
+ ··· −
31
2
u − 2
a
8
=
1
4
u
15
+
3
4
u
14
+ ··· −
33
4
u − 1
−
1
2
u
15
−
5
2
u
14
+ ··· −
39
2
u − 3
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
15
− 5u
14
− 23u
13
− 70u
12
− 184u
11
− 388u
10
− 709u
9
−
1086u
8
− 1443u
7
− 1613u
6
− 1539u
5
− 1210u
4
− 775u
3
− 384u
2
− 138u − 26
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
− 15u
15
+ ··· − 128u + 32
c
2
, c
7
, c
9
u
16
+ u
15
+ ··· − 3u
2
+ 1
c
3
, c
8
, c
12
u
16
+ 10u
14
+ ··· − u + 1
c
4
, c
5
, c
6
c
10
, c
11
u
16
+ 5u
15
+ ··· + 30u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 7y
15
+ ··· + 13824y + 1024
c
2
, c
7
, c
9
y
16
− 13y
15
+ ··· − 6y + 1
c
3
, c
8
, c
12
y
16
+ 20y
15
+ ··· + 25y + 1
c
4
, c
5
, c
6
c
10
, c
11
y
16
+ 21y
15
+ ··· + 116y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.623980 + 0.651550I
a = −0.423845 − 0.017311I
b = 0.571868 + 0.505425I
−1.44391 − 0.70743I 3.72290 + 0.24608I
u = −0.623980 − 0.651550I
a = −0.423845 + 0.017311I
b = 0.571868 − 0.505425I
−1.44391 + 0.70743I 3.72290 − 0.24608I
u = −0.463938 + 1.039450I
a = 0.077415 + 0.436847I
b = 0.01364 + 1.74946I
1.07382 + 9.25014I 4.41993 − 6.63902I
u = −0.463938 − 1.039450I
a = 0.077415 − 0.436847I
b = 0.01364 − 1.74946I
1.07382 − 9.25014I 4.41993 + 6.63902I
u = −0.740661 + 0.206858I
a = 1.043460 + 0.692415I
b = −0.181803 − 0.371994I
−2.76230 + 5.19350I 1.00902 − 5.12835I
u = −0.740661 − 0.206858I
a = 1.043460 − 0.692415I
b = −0.181803 + 0.371994I
−2.76230 − 5.19350I 1.00902 + 5.12835I
u = −0.128783 + 1.242080I
a = −0.456102 − 0.659720I
b = −0.214071 − 1.326220I
5.44979 + 1.79581I 4.75482 − 3.53630I
u = −0.128783 − 1.242080I
a = −0.456102 + 0.659720I
b = −0.214071 + 1.326220I
5.44979 − 1.79581I 4.75482 + 3.53630I
u = −0.16118 + 1.52906I
a = −0.619375 − 0.523872I
b = −0.733331 − 0.959617I
5.66087 + 2.21773I 8.32065 − 1.76776I
u = −0.16118 − 1.52906I
a = −0.619375 + 0.523872I
b = −0.733331 + 0.959617I
5.66087 − 2.21773I 8.32065 + 1.76776I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.210462 + 0.369049I
a = 0.457557 − 0.529485I
b = 0.070506 + 0.348052I
0.056580 + 0.803599I 1.57400 − 8.67162I
u = −0.210462 − 0.369049I
a = 0.457557 + 0.529485I
b = 0.070506 − 0.348052I
0.056580 − 0.803599I 1.57400 + 8.67162I
u = −0.13072 + 1.73050I
a = −0.34920 − 2.38942I
b = 0.03210 − 3.29161I
10.8004 + 11.7066I 5.77296 − 5.44740I
u = −0.13072 − 1.73050I
a = −0.34920 + 2.38942I
b = 0.03210 + 3.29161I
10.8004 − 11.7066I 5.77296 + 5.44740I
u = −0.04027 + 1.78867I
a = 0.27009 + 1.91061I
b = −0.05891 + 2.52256I
16.5308 + 2.6330I 2.42571 − 2.64676I
u = −0.04027 − 1.78867I
a = 0.27009 − 1.91061I
b = −0.05891 − 2.52256I
16.5308 − 2.6330I 2.42571 + 2.64676I
6
II. I
u
2
= h−74u
4
a
3
+ 62u
4
a
2
+ · · · + 20a − 170, 2u
4
a
3
− 11u
4
a + · · · − 23a −
23, u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
5
=
1
−u
2
a
12
=
−u
u
3
+ u
a
2
=
a
1.17460a
3
u
4
− 0.984127a
2
u
4
+ ··· − 0.317460a + 2.69841
a
4
=
u
2
+ 1
−u
4
− 2u
2
a
1
=
−0.206349a
3
u
4
+ 0.253968a
2
u
4
+ ··· + 1.92063a + 2.17460
1.65079a
3
u
4
− 1.03175a
2
u
4
+ ··· − 2.36508a + 2.60317
a
10
=
u
u
a
7
=
u
2
+ 1
u
2
a
9
=
0.0634921a
3
u
4
+ 0.0793651a
2
u
4
+ ··· − 1.20635a − 0.507937
0.253968a
3
u
4
− 0.634921a
2
u
4
+ ··· + 0.174603a + 4.06349
a
3
=
−0.206349a
3
u
4
+ 0.253968a
2
u
4
+ ··· + 0.920635a + 2.17460
2
3
u
4
a
3
−
2
3
u
4
a
2
+ ··· −
2
3
a +
8
3
a
8
=
−0.0634921a
3
u
4
+ 0.682540a
2
u
4
+ ··· + 1.20635a + 0.0317460
−0.253968a
3
u
4
+ 0.682540a
2
u
4
+ ··· − 0.174603a − 3.96825
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
4
+ 4u
3
− 16u
2
+ 12u − 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
+ u − 1)
10
c
2
, c
7
, c
9
u
20
+ u
19
+ ··· − 62u − 89
c
3
, c
8
, c
12
u
20
− u
19
+ ··· − 152u − 29
c
4
, c
5
, c
6
c
10
, c
11
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
2
− 3y + 1)
10
c
2
, c
7
, c
9
y
20
− 9y
19
+ ··· − 42648y + 7921
c
3
, c
8
, c
12
y
20
+ 11y
19
+ ··· − 24612y + 841
c
4
, c
5
, c
6
c
10
, c
11
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.233677 + 0.885557I
a = −1.040300 − 0.283528I
b = −0.783355 + 0.908585I
5.76765 − 2.21397I 4.88568 + 4.22289I
u = 0.233677 + 0.885557I
a = 1.240690 + 0.223658I
b = −0.281646 + 0.312488I
−2.12804 − 2.21397I 4.88568 + 4.22289I
u = 0.233677 + 0.885557I
a = 0.775190 + 0.996664I
b = 0.95815 + 1.93963I
−2.12804 − 2.21397I 4.88568 + 4.22289I
u = 0.233677 + 0.885557I
a = 0.270298 − 0.182593I
b = 0.52495 − 1.76882I
5.76765 − 2.21397I 4.88568 + 4.22289I
u = 0.233677 − 0.885557I
a = −1.040300 + 0.283528I
b = −0.783355 − 0.908585I
5.76765 + 2.21397I 4.88568 − 4.22289I
u = 0.233677 − 0.885557I
a = 1.240690 − 0.223658I
b = −0.281646 − 0.312488I
−2.12804 + 2.21397I 4.88568 − 4.22289I
u = 0.233677 − 0.885557I
a = 0.775190 − 0.996664I
b = 0.95815 − 1.93963I
−2.12804 + 2.21397I 4.88568 − 4.22289I
u = 0.233677 − 0.885557I
a = 0.270298 + 0.182593I
b = 0.52495 + 1.76882I
5.76765 + 2.21397I 4.88568 − 4.22289I
u = 0.416284
a = −1.26489
b = 0.932768
3.06566 −3.60880
u = 0.416284
a = −1.99317 + 1.58726I
b = −0.739269 − 0.509493I
−4.83002 −3.60880
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.416284
a = −1.99317 − 1.58726I
b = −0.739269 + 0.509493I
−4.83002 −3.60880
u = 0.416284
a = 2.78754
b = −0.368017
3.06566 −3.60880
u = 0.05818 + 1.69128I
a = 0.632434 − 0.451910I
b = 1.58486 − 0.66741I
7.01045 − 3.33174I 5.91874 + 2.36228I
u = 0.05818 + 1.69128I
a = 0.65235 − 1.40305I
b = 0.17965 − 2.05845I
14.9061 − 3.3317I 5.91874 + 2.36228I
u = 0.05818 + 1.69128I
a = −0.64368 + 2.64197I
b = −0.51264 + 3.62749I
14.9061 − 3.3317I 5.91874 + 2.36228I
u = 0.05818 + 1.69128I
a = −0.65514 − 2.79163I
b = −0.71308 − 3.44038I
7.01045 − 3.33174I 5.91874 + 2.36228I
u = 0.05818 − 1.69128I
a = 0.632434 + 0.451910I
b = 1.58486 + 0.66741I
7.01045 + 3.33174I 5.91874 − 2.36228I
u = 0.05818 − 1.69128I
a = 0.65235 + 1.40305I
b = 0.17965 + 2.05845I
14.9061 + 3.3317I 5.91874 − 2.36228I
u = 0.05818 − 1.69128I
a = −0.64368 − 2.64197I
b = −0.51264 − 3.62749I
14.9061 + 3.3317I 5.91874 − 2.36228I
u = 0.05818 − 1.69128I
a = −0.65514 + 2.79163I
b = −0.71308 + 3.44038I
7.01045 + 3.33174I 5.91874 − 2.36228I
11
III. I
u
3
= h−u
5
− u
4
− 3u
3
− 3u
2
+ b − u − 1, u
8
+ 7u
6
+ · · · + a + 2, u
9
+
7u
7
+ 16u
5
+ 13u
3
+ 3u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
5
=
1
−u
2
a
12
=
−u
u
3
+ u
a
2
=
−u
8
− 7u
6
+ u
5
− 15u
4
+ 4u
3
− 10u
2
+ 4u − 2
u
5
+ u
4
+ 3u
3
+ 3u
2
+ u + 1
a
4
=
u
2
+ 1
−u
4
− 2u
2
a
1
=
−u
8
− 6u
6
+ u
5
− 11u
4
+ 4u
3
− 7u
2
+ 3u − 2
u
6
+ u
5
+ 5u
4
+ 4u
3
+ 6u
2
+ 2u + 1
a
10
=
u
u
a
7
=
u
2
+ 1
u
2
a
9
=
u
6
− u
5
+ 5u
4
− 4u
3
+ 7u
2
− 3u + 3
−u
8
+ u
7
− 6u
6
+ 5u
5
− 11u
4
+ 7u
3
− 6u
2
+ 3u
a
3
=
−u
8
− 6u
6
+ u
5
− 11u
4
+ 4u
3
− 6u
2
+ 4u − 1
u
6
+ u
5
+ 4u
4
+ 3u
3
+ 4u
2
+ u + 1
a
8
=
−u
5
+ u
4
− 4u
3
+ 4u
2
− 3u + 3
u
7
− u
6
+ 5u
5
− 4u
4
+ 7u
3
− 3u
2
+ 3u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
8
+ 2u
7
+ 24u
6
+ 10u
5
+ 44u
4
+ 14u
3
+ 25u
2
+ 3u + 10
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
− 4u
8
+ 6u
7
− 8u
6
+ 13u
5
− 9u
4
+ u
3
− 5u
2
+ 3u + 3
c
2
, c
9
u
9
+ u
8
− 2u
7
− 2u
6
+ u
5
+ u
3
+ 2u
2
+ 1
c
3
, c
8
u
9
+ 2u
7
+ u
6
+ u
4
− 2u
3
− 2u
2
+ u + 1
c
4
, c
5
, c
6
u
9
+ 7u
7
+ 16u
5
+ 13u
3
+ 3u + 1
c
7
u
9
− u
8
− 2u
7
+ 2u
6
+ u
5
+ u
3
− 2u
2
− 1
c
10
, c
11
u
9
+ 7u
7
+ 16u
5
+ 13u
3
+ 3u − 1
c
12
u
9
+ 2u
7
− u
6
− u
4
− 2u
3
+ 2u
2
+ u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
− 4y
8
− 2y
7
+ 22y
6
+ 3y
5
− 75y
4
+ 37y
3
+ 35y
2
+ 39y −9
c
2
, c
7
, c
9
y
9
− 5y
8
+ 10y
7
− 6y
6
− 7y
5
+ 8y
4
+ 5y
3
− 4y
2
− 4y −1
c
3
, c
8
, c
12
y
9
+ 4y
8
+ 4y
7
− 5y
6
− 8y
5
+ 7y
4
+ 6y
3
− 10y
2
+ 5y −1
c
4
, c
5
, c
6
c
10
, c
11
y
9
+ 14y
8
+ 81y
7
+ 250y
6
+ 444y
5
+ 458y
4
+ 265y
3
+ 78y
2
+ 9y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.176178 + 1.056080I
a = −0.772923 − 0.371228I
b = −0.67471 − 1.53869I
7.25976 + 1.49693I 10.69582 − 1.07320I
u = −0.176178 − 1.056080I
a = −0.772923 + 0.371228I
b = −0.67471 + 1.53869I
7.25976 − 1.49693I 10.69582 + 1.07320I
u = 0.252500 + 0.604050I
a = 0.76253 + 1.27109I
b = −0.323758 + 0.973050I
−3.19201 − 0.85520I 1.77424 + 0.81850I
u = 0.252500 − 0.604050I
a = 0.76253 − 1.27109I
b = −0.323758 − 0.973050I
−3.19201 + 0.85520I 1.77424 − 0.81850I
u = 0.09972 + 1.60032I
a = 0.374760 − 1.005510I
b = 0.801978 − 1.133290I
4.49433 − 2.25221I 0.93167 + 1.22444I
u = 0.09972 − 1.60032I
a = 0.374760 + 1.005510I
b = 0.801978 + 1.133290I
4.49433 + 2.25221I 0.93167 − 1.22444I
u = −0.255288
a = −3.80617
b = 0.893478
3.76466 10.8130
u = −0.04840 + 1.76025I
a = 0.53872 + 2.02200I
b = 0.24975 + 2.75063I
17.5195 + 2.4733I 11.19170 − 0.90094I
u = −0.04840 − 1.76025I
a = 0.53872 − 2.02200I
b = 0.24975 − 2.75063I
17.5195 − 2.4733I 11.19170 + 0.90094I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u − 1)
10
)(u
9
− 4u
8
+ ··· + 3u + 3)
· (u
16
− 15u
15
+ ··· − 128u + 32)
c
2
, c
9
(u
9
+ u
8
+ ··· + 2u
2
+ 1)(u
16
+ u
15
+ ··· − 3u
2
+ 1)
· (u
20
+ u
19
+ ··· − 62u − 89)
c
3
, c
8
(u
9
+ 2u
7
+ ··· + u + 1)(u
16
+ 10u
14
+ ··· − u + 1)
· (u
20
− u
19
+ ··· − 152u − 29)
c
4
, c
5
, c
6
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
4
(u
9
+ 7u
7
+ 16u
5
+ 13u
3
+ 3u + 1)
· (u
16
+ 5u
15
+ ··· + 30u + 4)
c
7
(u
9
− u
8
+ ··· − 2u
2
− 1)(u
16
+ u
15
+ ··· − 3u
2
+ 1)
· (u
20
+ u
19
+ ··· − 62u − 89)
c
10
, c
11
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)
4
(u
9
+ 7u
7
+ 16u
5
+ 13u
3
+ 3u − 1)
· (u
16
+ 5u
15
+ ··· + 30u + 4)
c
12
(u
9
+ 2u
7
+ ··· + u − 1)(u
16
+ 10u
14
+ ··· − u + 1)
· (u
20
− u
19
+ ··· − 152u − 29)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
− 3y + 1)
10
· (y
9
− 4y
8
− 2y
7
+ 22y
6
+ 3y
5
− 75y
4
+ 37y
3
+ 35y
2
+ 39y −9)
· (y
16
− 7y
15
+ ··· + 13824y + 1024)
c
2
, c
7
, c
9
(y
9
− 5y
8
+ 10y
7
− 6y
6
− 7y
5
+ 8y
4
+ 5y
3
− 4y
2
− 4y −1)
· (y
16
− 13y
15
+ ··· − 6y + 1)(y
20
− 9y
19
+ ··· − 42648y + 7921)
c
3
, c
8
, c
12
(y
9
+ 4y
8
+ 4y
7
− 5y
6
− 8y
5
+ 7y
4
+ 6y
3
− 10y
2
+ 5y −1)
· (y
16
+ 20y
15
+ ··· + 25y + 1)(y
20
+ 11y
19
+ ··· − 24612y + 841)
c
4
, c
5
, c
6
c
10
, c
11
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)
4
· (y
9
+ 14y
8
+ 81y
7
+ 250y
6
+ 444y
5
+ 458y
4
+ 265y
3
+ 78y
2
+ 9y −1)
· (y
16
+ 21y
15
+ ··· + 116y + 16)
17
