12n
0156
(K12n
0156
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 10 12 3 5 6 9 7 11
Solving Sequence
5,10 3,6
2 1 4 9 11 8 7 12
c
5
c
2
c
1
c
4
c
9
c
10
c
8
c
7
c
12
c
3
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
9
+ u
8
+ 5u
7
+ 3u
6
+ 9u
5
+ 7u
4
+ 6u
3
+ 5u
2
+ 4b + 3,
− u
9
+ 3u
8
− 5u
7
+ 5u
6
− 5u
5
+ u
4
+ 2u
3
− 13u
2
+ 8a + 8u − 11,
u
10
+ 4u
8
+ 2u
7
+ 6u
6
+ 6u
5
+ 3u
4
+ 7u
3
− u
2
+ 3u + 1i
I
u
2
= hb + 1, −u
3
+ u
2
+ 2a + 1, u
4
+ u
2
+ u + 1i
I
u
3
= h−4u
15
+ 18u
14
+ ··· + 33b + 38, 29u
15
− 48u
14
+ ··· + 33a − 61, u
16
− 2u
15
+ ··· − 2u + 1i
I
u
4
= hb + 1, u
5
− u
4
+ u
3
− u
2
+ a + u, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
* 4 irreducible components of dim
C
= 0, with total 36 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
= hu
9
+u
8
+· · · +4b + 3, −u
9
+3u
8
+· · · +8a − 11, u
10
+4u
8
+· · · +3u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
1
8
u
9
−
3
8
u
8
+ ··· − u +
11
8
−
1
4
u
9
−
1
4
u
8
+ ··· −
5
4
u
2
−
3
4
a
6
=
1
−u
2
a
2
=
−
1
8
u
9
−
5
8
u
8
+ ··· − u +
5
8
−
1
4
u
9
−
1
4
u
8
+ ··· −
5
4
u
2
−
3
4
a
1
=
−u
9
− 3u
7
− 2u
6
− 4u
5
− 4u
4
− 3u
2
−u
9
− 2u
7
− 2u
6
− 2u
5
− 4u
4
+ 2u
3
− 3u
2
a
4
=
13
8
u
9
−
7
8
u
8
+ ··· − 2u +
7
8
3
4
u
9
−
1
4
u
8
+ ··· +
3
4
u
2
−
3
4
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
5
+ u
3
+ u
a
8
=
u
3
u
3
+ u
a
7
=
u
8
+ 3u
6
+ 2u
5
+ 3u
4
+ 4u
3
− u
2
+ 3u
u
8
+ 3u
6
+ 2u
5
+ 4u
4
+ 4u
3
+ 3u + 1
a
12
=
−u
9
− 3u
7
− 2u
6
− 3u
5
− 4u
4
− 3u
2
−u
9
− 3u
7
− 2u
6
− 3u
5
− 4u
4
+ u
3
− 3u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
33
16
u
9
−
53
16
u
8
−
141
16
u
7
−
227
16
u
6
−
325
16
u
5
−
367
16
u
4
−
159
8
u
3
−
237
16
u
2
− 6u −
147
16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
+ 17u
9
+ ··· + 353u + 16
c
2
, c
4
u
10
− 3u
9
− 4u
8
+ 13u
7
+ 22u
6
− 70u
5
+ 56u
4
− 7u
3
− 47u
2
+ 27u − 4
c
3
, c
7
u
10
− 3u
9
+ ··· − 48u − 64
c
5
, c
6
, c
9
c
11
u
10
+ 4u
8
+ 2u
7
+ 6u
6
+ 6u
5
+ 3u
4
+ 7u
3
− u
2
+ 3u + 1
c
8
u
10
+ 6u
9
+ 9u
8
− 6u
7
− 21u
6
− 14u
5
+ 21u
3
+ 13u
2
+ 16u + 4
c
10
, c
12
u
10
+ 8u
9
+ ··· − 11u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 13y
9
+ ··· − 56161y + 256
c
2
, c
4
y
10
− 17y
9
+ ··· − 353y + 16
c
3
, c
7
y
10
− 21y
9
+ ··· + 16128y + 4096
c
5
, c
6
, c
9
c
11
y
10
+ 8y
9
+ ··· − 11y + 1
c
8
y
10
− 18y
9
+ ··· − 152y + 16
c
10
, c
12
y
10
− 8y
9
+ ··· − 191y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.10322
a = 0.0897722
b = 2.01806
−13.1809 −5.01890
u = −0.434341 + 1.157890I
a = 0.168075 − 0.547483I
b = 0.424180 − 0.920028I
−3.22150 − 6.17796I −7.18871 + 5.57381I
u = −0.434341 − 1.157890I
a = 0.168075 + 0.547483I
b = 0.424180 + 0.920028I
−3.22150 + 6.17796I −7.18871 − 5.57381I
u = 0.453609 + 0.609493I
a = 0.755807 + 0.185749I
b = 0.306434 − 0.019942I
0.54459 + 1.46281I 2.01320 − 4.52195I
u = 0.453609 − 0.609493I
a = 0.755807 − 0.185749I
b = 0.306434 + 0.019942I
0.54459 − 1.46281I 2.01320 + 4.52195I
u = 0.126773 + 1.317690I
a = −1.87389 + 0.33187I
b = −1.95902 + 1.19918I
−9.42139 + 3.00890I −13.74651 − 2.98751I
u = 0.126773 − 1.317690I
a = −1.87389 − 0.33187I
b = −1.95902 − 1.19918I
−9.42139 − 3.00890I −13.74651 + 2.98751I
u = 0.53944 + 1.37745I
a = 1.77401 + 1.11600I
b = 2.12782 − 0.47355I
17.6402 + 11.6714I −10.01565 − 5.34252I
u = 0.53944 − 1.37745I
a = 1.77401 − 1.11600I
b = 2.12782 + 0.47355I
17.6402 − 11.6714I −10.01565 + 5.34252I
u = −0.267745
a = 1.76223
b = −0.816874
−1.19281 −8.35580
5
II. I
u
2
= hb + 1, −u
3
+ u
2
+ 2a + 1, u
4
+ u
2
+ u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
1
2
u
3
−
1
2
u
2
−
1
2
−1
a
6
=
1
−u
2
a
2
=
1
2
u
3
−
1
2
u
2
−
3
2
−1
a
1
=
−1
0
a
4
=
1
2
u
3
−
1
2
u
2
−
1
2
−1
a
9
=
−u
u
3
+ u
a
11
=
−u
3
−u
2
a
8
=
u
3
u
3
+ u
a
7
=
u
3
u
3
+ u
a
12
=
−u
3
− u
2
− u − 1
−u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =
11
4
u
3
−
21
4
u
2
−
1
2
u −
31
4
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
7
u
4
c
4
(u + 1)
4
c
5
, c
6
u
4
+ u
2
+ u + 1
c
8
u
4
− 3u
3
+ 4u
2
− 3u + 2
c
9
, c
11
u
4
+ u
2
− u + 1
c
10
, c
12
u
4
+ 2u
3
+ 3u
2
+ u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
7
y
4
c
5
, c
6
, c
9
c
11
y
4
+ 2y
3
+ 3y
2
+ y + 1
c
8
y
4
− y
3
+ 2y
2
+ 7y + 4
c
10
, c
12
y
4
+ 2y
3
+ 7y
2
+ 5y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.547424 + 0.585652I
a = −0.278726 + 0.483420I
b = −1.00000
−0.66484 − 1.39709I −6.15099 + 3.96898I
u = −0.547424 − 0.585652I
a = −0.278726 − 0.483420I
b = −1.00000
−0.66484 + 1.39709I −6.15099 − 3.96898I
u = 0.547424 + 1.120870I
a = −0.971274 − 0.813859I
b = −1.00000
−4.26996 + 7.64338I −8.22401 − 8.10462I
u = 0.547424 − 1.120870I
a = −0.971274 + 0.813859I
b = −1.00000
−4.26996 − 7.64338I −8.22401 + 8.10462I
9
III. I
u
3
= h−4u
15
+ 18u
14
+ · · · + 33b + 38, 29u
15
− 48u
14
+ · · · + 33a −
61, u
16
− 2u
15
+ · · · − 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
−0.878788u
15
+ 1.45455u
14
+ ··· − 0.636364u + 1.84848
0.121212u
15
− 0.545455u
14
+ ··· + 0.363636u − 1.15152
a
6
=
1
−u
2
a
2
=
−0.757576u
15
+ 0.909091u
14
+ ··· − 0.272727u + 0.696970
0.121212u
15
− 0.545455u
14
+ ··· + 0.363636u − 1.15152
a
1
=
1.27273u
15
− 2.72727u
14
+ ··· + 3.81818u − 2.09091
1.78788u
15
− 1.54545u
14
+ ··· + 0.363636u − 1.48485
a
4
=
−0.393939u
15
+ 1.27273u
14
+ ··· − 0.181818u + 1.24242
1.36364u
15
− 1.63636u
14
+ ··· + 2.09091u − 2.45455
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
5
+ u
3
+ u
a
8
=
u
3
u
3
+ u
a
7
=
−0.0909091u
15
− 0.0909091u
14
+ ··· + 2.72727u + 1.36364
1.21212u
15
− 1.45455u
14
+ ··· + 1.63636u − 2.51515
a
12
=
2.24242u
15
− 3.09091u
14
+ ··· + 4.72727u − 3.30303
−2.75758u
15
+ 2.90909u
14
+ ··· − 3.27273u + 2.69697
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
19
11
u
15
+
25
11
u
14
−
135
11
u
13
+
153
11
u
12
−
410
11
u
11
+
412
11
u
10
−
603
11
u
9
+
543
11
u
8
− 29u
7
+
303
11
u
6
+
127
11
u
5
+
60
11
u
4
+
67
11
u
3
+
56
11
u
2
−
90
11
u −
34
11
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
8
+ 16u
7
+ 98u
6
+ 283u
5
+ 381u
4
+ 191u
3
− 45u
2
+ 10u + 1)
2
c
2
, c
4
(u
8
− 4u
7
+ 13u
5
− 3u
4
− 15u
3
+ 3u
2
− 2u − 1)
2
c
3
, c
7
(u
8
+ u
7
− 10u
6
− 7u
5
+ 19u
4
− 23u
3
− 12u + 8)
2
c
5
, c
6
, c
9
c
11
u
16
− 2u
15
+ ··· − 2u + 1
c
8
(u
8
− 2u
7
− 7u
6
+ 12u
5
+ 7u
4
− 2u
3
− 2u
2
+ 3u − 1)
2
c
10
, c
12
u
16
+ 10u
15
+ ··· + 14u
2
+ 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
8
− 60y
7
+ ··· − 190y + 1)
2
c
2
, c
4
(y
8
− 16y
7
+ 98y
6
− 283y
5
+ 381y
4
− 191y
3
− 45y
2
− 10y + 1)
2
c
3
, c
7
(y
8
− 21y
7
+ 152y
6
− 383y
5
+ 79y
4
− 857y
3
− 248y
2
− 144y + 64)
2
c
5
, c
6
, c
9
c
11
y
16
+ 10y
15
+ ··· + 14y
2
+ 1
c
8
(y
8
− 18y
7
+ 111y
6
− 254y
5
+ 135y
4
− 90y
3
+ 2y
2
− 5y + 1)
2
c
10
, c
12
y
16
− 10y
15
+ ··· + 28y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.381176 + 0.988501I
a = 0.400335 + 0.134981I
b = 0.202560 + 0.429200I
−0.54882 + 2.12062I −1.41411 − 2.85603I
u = 0.381176 − 0.988501I
a = 0.400335 − 0.134981I
b = 0.202560 − 0.429200I
−0.54882 − 2.12062I −1.41411 + 2.85603I
u = −0.175038 + 1.044950I
a = 0.881139 + 0.709579I
b = −0.266855
−3.96569 −10.71257 + 0.I
u = −0.175038 − 1.044950I
a = 0.881139 − 0.709579I
b = −0.266855
−3.96569 −10.71257 + 0.I
u = 1.097050 + 0.006514I
a = 0.0528548 + 0.1140410I
b = 2.08865 − 0.23775I
−17.5075 + 5.8605I −7.51154 − 2.72065I
u = 1.097050 − 0.006514I
a = 0.0528548 − 0.1140410I
b = 2.08865 + 0.23775I
−17.5075 − 5.8605I −7.51154 + 2.72065I
u = −0.087856 + 1.180370I
a = −1.94399 + 0.56029I
b = −1.251300 − 0.394571I
−4.42998 − 1.32248I −7.15537 + 1.48485I
u = −0.087856 − 1.180370I
a = −1.94399 − 0.56029I
b = −1.251300 + 0.394571I
−4.42998 + 1.32248I −7.15537 − 1.48485I
u = −0.579676 + 0.232048I
a = 0.915955 − 0.352378I
b = 0.202560 + 0.429200I
−0.54882 + 2.12062I −1.41411 − 2.85603I
u = −0.579676 − 0.232048I
a = 0.915955 + 0.352378I
b = 0.202560 − 0.429200I
−0.54882 − 2.12062I −1.41411 + 2.85603I
13
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.54916 + 1.38012I
a = 1.49564 + 1.24562I
b = 2.18705
17.6846 −10.12539 + 0.I
u = 0.54916 − 1.38012I
a = 1.49564 − 1.24562I
b = 2.18705
17.6846 −10.12539 + 0.I
u = −0.54464 + 1.38261I
a = 1.62067 − 1.13338I
b = 2.08865 + 0.23775I
−17.5075 − 5.8605I −7.51154 + 2.72065I
u = −0.54464 − 1.38261I
a = 1.62067 + 1.13338I
b = 2.08865 − 0.23775I
−17.5075 + 5.8605I −7.51154 − 2.72065I
u = 0.359826 + 0.343977I
a = 2.07740 − 0.65398I
b = −1.251300 + 0.394571I
−4.42998 + 1.32248I −7.15537 − 1.48485I
u = 0.359826 − 0.343977I
a = 2.07740 + 0.65398I
b = −1.251300 − 0.394571I
−4.42998 − 1.32248I −7.15537 + 1.48485I
14
IV.
I
u
4
= hb + 1, u
5
− u
4
+ u
3
− u
2
+ a + u, u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
3
=
−u
5
+ u
4
− u
3
+ u
2
− u
−1
a
6
=
1
−u
2
a
2
=
−u
5
+ u
4
− u
3
+ u
2
− u − 1
−1
a
1
=
−1
0
a
4
=
−u
5
+ u
4
− u
3
+ u
2
− u
−1
a
9
=
−u
u
3
+ u
a
11
=
−u
3
u
5
+ u
3
+ u
a
8
=
u
3
u
3
+ u
a
7
=
u
3
u
3
+ u
a
12
=
−u
4
− u
2
+ u − 1
2u
5
− u
4
+ 3u
3
− 2u
2
+ 3u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
5
+ 5u
3
− u
2
+ 5u − 10
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
7
u
6
c
4
(u + 1)
6
c
5
, c
6
u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1
c
8
(u
3
+ u
2
− 1)
2
c
9
, c
11
u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1
c
10
, c
12
u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
6
c
3
, c
7
y
6
c
5
, c
6
, c
9
c
11
y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1
c
8
(y
3
− y
2
+ 2y − 1)
2
c
10
, c
12
y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.498832 + 1.001300I
a = −0.767394 + 0.943705I
b = −1.00000
−1.91067 − 2.82812I −6.15260 + 3.54173I
u = −0.498832 − 1.001300I
a = −0.767394 − 0.943705I
b = −1.00000
−1.91067 + 2.82812I −6.15260 − 3.54173I
u = 0.284920 + 1.115140I
a = −1.37744 − 1.47725I
b = −1.00000
−6.04826 −10.69479 + 0.I
u = 0.284920 − 1.115140I
a = −1.37744 + 1.47725I
b = −1.00000
−6.04826 −10.69479 + 0.I
u = 0.713912 + 0.305839I
a = −0.355167 − 0.198843I
b = −1.00000
−1.91067 − 2.82812I −6.15260 + 3.54173I
u = 0.713912 − 0.305839I
a = −0.355167 + 0.198843I
b = −1.00000
−1.91067 + 2.82812I −6.15260 − 3.54173I
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
10
· (u
8
+ 16u
7
+ 98u
6
+ 283u
5
+ 381u
4
+ 191u
3
− 45u
2
+ 10u + 1)
2
· (u
10
+ 17u
9
+ ··· + 353u + 16)
c
2
(u − 1)
10
(u
8
− 4u
7
+ 13u
5
− 3u
4
− 15u
3
+ 3u
2
− 2u − 1)
2
· (u
10
− 3u
9
− 4u
8
+ 13u
7
+ 22u
6
− 70u
5
+ 56u
4
− 7u
3
− 47u
2
+ 27u − 4)
c
3
, c
7
u
10
(u
8
+ u
7
− 10u
6
− 7u
5
+ 19u
4
− 23u
3
− 12u + 8)
2
· (u
10
− 3u
9
+ ··· − 48u − 64)
c
4
(u + 1)
10
(u
8
− 4u
7
+ 13u
5
− 3u
4
− 15u
3
+ 3u
2
− 2u − 1)
2
· (u
10
− 3u
9
− 4u
8
+ 13u
7
+ 22u
6
− 70u
5
+ 56u
4
− 7u
3
− 47u
2
+ 27u − 4)
c
5
, c
6
(u
4
+ u
2
+ u + 1)(u
6
− u
5
+ 2u
4
− 2u
3
+ 2u
2
− 2u + 1)
· (u
10
+ 4u
8
+ 2u
7
+ 6u
6
+ 6u
5
+ 3u
4
+ 7u
3
− u
2
+ 3u + 1)
· (u
16
− 2u
15
+ ··· − 2u + 1)
c
8
(u
3
+ u
2
− 1)
2
(u
4
− 3u
3
+ 4u
2
− 3u + 2)
· (u
8
− 2u
7
− 7u
6
+ 12u
5
+ 7u
4
− 2u
3
− 2u
2
+ 3u − 1)
2
· (u
10
+ 6u
9
+ 9u
8
− 6u
7
− 21u
6
− 14u
5
+ 21u
3
+ 13u
2
+ 16u + 4)
c
9
, c
11
(u
4
+ u
2
− u + 1)(u
6
+ u
5
+ 2u
4
+ 2u
3
+ 2u
2
+ 2u + 1)
· (u
10
+ 4u
8
+ 2u
7
+ 6u
6
+ 6u
5
+ 3u
4
+ 7u
3
− u
2
+ 3u + 1)
· (u
16
− 2u
15
+ ··· − 2u + 1)
c
10
, c
12
(u
4
+ 2u
3
+ 3u
2
+ u + 1)(u
6
+ 3u
5
+ 4u
4
+ 2u
3
+ 1)
· (u
10
+ 8u
9
+ ··· − 11u + 1)(u
16
+ 10u
15
+ ··· + 14u
2
+ 1)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
10
)(y
8
− 60y
7
+ ··· − 190y + 1)
2
· (y
10
− 13y
9
+ ··· − 56161y + 256)
c
2
, c
4
(y − 1)
10
· (y
8
− 16y
7
+ 98y
6
− 283y
5
+ 381y
4
− 191y
3
− 45y
2
− 10y + 1)
2
· (y
10
− 17y
9
+ ··· − 353y + 16)
c
3
, c
7
y
10
· (y
8
− 21y
7
+ 152y
6
− 383y
5
+ 79y
4
− 857y
3
− 248y
2
− 144y + 64)
2
· (y
10
− 21y
9
+ ··· + 16128y + 4096)
c
5
, c
6
, c
9
c
11
(y
4
+ 2y
3
+ 3y
2
+ y + 1)(y
6
+ 3y
5
+ 4y
4
+ 2y
3
+ 1)
· (y
10
+ 8y
9
+ ··· − 11y + 1)(y
16
+ 10y
15
+ ··· + 14y
2
+ 1)
c
8
(y
3
− y
2
+ 2y − 1)
2
(y
4
− y
3
+ 2y
2
+ 7y + 4)
· (y
8
− 18y
7
+ 111y
6
− 254y
5
+ 135y
4
− 90y
3
+ 2y
2
− 5y + 1)
2
· (y
10
− 18y
9
+ ··· − 152y + 16)
c
10
, c
12
(y
4
+ 2y
3
+ 7y
2
+ 5y + 1)(y
6
− y
5
+ 4y
4
− 2y
3
+ 8y
2
+ 1)
· (y
10
− 8y
9
+ ··· − 191y + 1)(y
16
− 10y
15
+ ··· + 28y + 1)
20
