12n
0594
(K12n
0594
)
A knot diagram
1
Linearized knot diagam
3 7 11 8 10 2 5 12 3 5 9 8
Solving Sequence
8,12
9
1,5
4 7 11 3 2 6 10
c
8
c
12
c
4
c
7
c
11
c
3
c
2
c
6
c
10
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−4u
19
− 11u
18
+ ··· + b − 6, −2u
19
− 3u
18
+ ··· + a − 7, u
20
+ 3u
19
+ ··· + 3u + 1i
I
u
2
= h−2u
13
+ 8u
12
+ ··· + b + 3,
− u
13
+ 2u
12
− 6u
11
+ 7u
10
− 10u
9
+ 8u
8
− 4u
7
+ 4u
6
+ 3u
5
+ 2u
4
+ 2u
3
− 2u
2
+ a − 2,
u
14
− 4u
13
+ 13u
12
− 28u
11
+ 50u
10
− 72u
9
+ 86u
8
− 89u
7
+ 76u
6
− 59u
5
+ 39u
4
− 23u
3
+ 12u
2
− 4u + 1i
* 2 irreducible components of dim
C
= 0, with total 34 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−4u
19
−11u
18
+· · ·+b−6, −2u
19
−3u
18
+· · ·+a−7, u
20
+3u
19
+· · ·+3u+1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
u
2
a
1
=
−u
u
a
5
=
2u
19
+ 3u
18
+ ··· − u + 7
4u
19
+ 11u
18
+ ··· + 5u + 6
a
4
=
6u
19
+ 14u
18
+ ··· + 4u + 13
4u
19
+ 11u
18
+ ··· + 5u + 6
a
7
=
−u
19
− u
18
+ ··· − u + 1
−u
19
− 4u
18
+ ··· − 4u − 2
a
11
=
u
u
3
+ u
a
3
=
3u
19
+ 7u
18
+ ··· + 4u + 12
4u
19
+ 10u
18
+ ··· + 2u + 3
a
2
=
u
19
+ u
18
+ ··· + 2u + 1
u
19
+ 4u
18
+ ··· + 4u + 1
a
6
=
−2u
19
− 2u
18
+ ··· + 3u − 6
−4u
19
− 12u
18
+ ··· − 5u − 6
a
10
=
u
18
+ 2u
17
+ ··· + 2u + 1
−u
19
− 3u
18
+ ··· − 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 20u
19
+ 55u
18
+ 219u
17
+ 423u
16
+ 917u
15
+ 1354u
14
+ 1901u
13
+ 2075u
12
+ 1606u
11
+
752u
10
−1039u
9
−2337u
8
−3525u
7
−3656u
6
−2870u
5
−1973u
4
−773u
3
−277u
2
+20u+23
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 10u
19
+ ··· + 29696u + 4096
c
2
, c
6
u
20
− 16u
19
+ ··· + 224u − 64
c
3
u
20
+ 2u
19
+ ··· − 135u − 31
c
4
, c
7
u
20
− 4u
19
+ ··· + u − 1
c
5
, c
9
, c
10
u
20
− u
19
+ ··· − 3u − 1
c
8
, c
11
, c
12
u
20
− 3u
19
+ ··· − 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
+ 102y
19
+ ··· − 1112539136y + 16777216
c
2
, c
6
y
20
− 10y
19
+ ··· − 29696y + 4096
c
3
y
20
+ 44y
19
+ ··· − 12831y + 961
c
4
, c
7
y
20
+ 42y
19
+ ··· − 29y + 1
c
5
, c
9
, c
10
y
20
+ 49y
19
+ ··· + 9y + 1
c
8
, c
11
, c
12
y
20
+ 15y
19
+ ··· − 21y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.042660 + 0.036790I
a = −0.041706 − 0.146779I
b = −1.09913 + 2.60338I
15.4500 + 5.4797I −10.60978 − 2.00835I
u = −1.042660 − 0.036790I
a = −0.041706 + 0.146779I
b = −1.09913 − 2.60338I
15.4500 − 5.4797I −10.60978 + 2.00835I
u = 1.08731
a = 0.282292
b = −0.147050
−5.86045 −4.92500
u = −0.176041 + 0.850242I
a = −0.272973 + 1.045810I
b = 0.746840 − 0.265541I
−0.414648 + 1.016570I −12.17893 − 0.72142I
u = −0.176041 − 0.850242I
a = −0.272973 − 1.045810I
b = 0.746840 + 0.265541I
−0.414648 − 1.016570I −12.17893 + 0.72142I
u = 0.222003 + 1.123880I
a = −0.155980 − 0.668566I
b = −0.185769 + 0.435768I
2.27216 − 2.01728I −6.94050 + 3.50947I
u = 0.222003 − 1.123880I
a = −0.155980 + 0.668566I
b = −0.185769 − 0.435768I
2.27216 + 2.01728I −6.94050 − 3.50947I
u = −0.153005 + 1.219290I
a = 0.53928 + 1.84965I
b = 0.69118 − 1.43719I
5.80153 + 4.79888I −3.91055 − 4.91239I
u = −0.153005 − 1.219290I
a = 0.53928 − 1.84965I
b = 0.69118 + 1.43719I
5.80153 − 4.79888I −3.91055 + 4.91239I
u = −0.112420 + 1.234960I
a = −0.73519 − 1.51536I
b = −0.356045 + 1.326390I
6.14605 − 0.99742I −6.22294 + 0.30113I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.112420 − 1.234960I
a = −0.73519 + 1.51536I
b = −0.356045 − 1.326390I
6.14605 + 0.99742I −6.22294 − 0.30113I
u = 0.538721 + 1.305800I
a = 0.121911 + 0.351718I
b = 0.036915 − 0.257923I
−1.83437 − 5.75000I −9.64448 − 0.35618I
u = 0.538721 − 1.305800I
a = 0.121911 − 0.351718I
b = 0.036915 + 0.257923I
−1.83437 + 5.75000I −9.64448 + 0.35618I
u = −0.54565 + 1.32924I
a = 2.33430 + 1.17622I
b = −1.00453 − 2.47780I
19.4483 + 0.1620I −7.85194 − 0.70320I
u = −0.54565 − 1.32924I
a = 2.33430 − 1.17622I
b = −1.00453 + 2.47780I
19.4483 − 0.1620I −7.85194 + 0.70320I
u = −0.50245 + 1.36268I
a = −0.91739 − 2.80304I
b = −1.11454 + 2.73435I
−19.6493 + 10.9702I −7.71994 − 4.45500I
u = −0.50245 − 1.36268I
a = −0.91739 + 2.80304I
b = −1.11454 − 2.73435I
−19.6493 − 10.9702I −7.71994 + 4.45500I
u = −0.401727 + 0.043777I
a = 0.05735 + 1.66973I
b = 0.241130 + 1.099340I
2.31319 − 2.74078I −11.17617 + 6.82485I
u = −0.401727 − 0.043777I
a = 0.05735 − 1.66973I
b = 0.241130 − 1.099340I
2.31319 + 2.74078I −11.17617 − 6.82485I
u = 0.259155
a = −1.14150
b = 0.234939
−0.567617 −17.5650
6
II.
I
u
2
= h−2u
13
+8u
12
+· · ·+b+3, −u
13
+2u
12
+· · ·+a−2, u
14
−4u
13
+· · ·−4u+1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
u
2
a
1
=
−u
u
a
5
=
u
13
− 2u
12
+ ··· + 2u
2
+ 2
2u
13
− 8u
12
+ ··· + 11u − 3
a
4
=
3u
13
− 10u
12
+ ··· + 11u − 1
2u
13
− 8u
12
+ ··· + 11u − 3
a
7
=
u
13
− 2u
12
+ ··· − 13u + 6
u
13
− 5u
12
+ ··· + 11u − 2
a
11
=
u
u
3
+ u
a
3
=
2u
13
− 6u
12
+ ··· − 15u
2
+ 8u
2u
13
− 8u
12
+ ··· + 9u − 2
a
2
=
u
13
− 2u
12
+ ··· − 11u + 6
u
13
− 5u
12
+ ··· + 10u − 1
a
6
=
3u
13
− 9u
12
+ ··· + 8u − 1
2u
13
− 10u
12
+ ··· + 15u − 5
a
10
=
2u
13
− 9u
12
+ ··· + 19u − 4
−u
13
+ 4u
12
+ ··· + 5u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes
= u
13
−u
12
+ u
11
+ 5u
10
−14u
9
+ 23u
8
−33u
7
+ 27u
6
−30u
5
+ 16u
4
−18u
3
+ 11u
2
−4u −5
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
− 10u
13
+ ··· − 13u + 1
c
2
u
14
+ 2u
13
+ ··· − u + 1
c
3
u
14
− u
13
+ ··· − 2u − 1
c
4
u
14
− 3u
13
+ ··· + 2u
2
− 1
c
5
, c
9
u
14
+ 2u
12
+ 4u
11
− u
10
+ 4u
9
+ 5u
8
− 4u
7
+ 4u
6
+ 5u
5
− 3u
4
+ 3u
2
− 1
c
6
u
14
− 2u
13
+ ··· + u + 1
c
7
u
14
+ 3u
13
+ ··· + 2u
2
− 1
c
8
u
14
− 4u
13
+ ··· − 4u + 1
c
10
u
14
+ 2u
12
− 4u
11
− u
10
− 4u
9
+ 5u
8
+ 4u
7
+ 4u
6
− 5u
5
− 3u
4
+ 3u
2
− 1
c
11
, c
12
u
14
+ 4u
13
+ ··· + 4u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 2y
13
+ ··· − 13y + 1
c
2
, c
6
y
14
− 10y
13
+ ··· − 13y + 1
c
3
y
14
+ 3y
13
+ ··· − 2y + 1
c
4
, c
7
y
14
+ 5y
13
+ ··· − 4y + 1
c
5
, c
9
, c
10
y
14
+ 4y
13
+ ··· − 6y + 1
c
8
, c
11
, c
12
y
14
+ 10y
13
+ ··· + 8y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.257957 + 1.025710I
a = −1.21422 + 1.52525I
b = −0.32410 − 1.59615I
4.34730 − 4.58698I −8.81634 + 3.98143I
u = 0.257957 − 1.025710I
a = −1.21422 − 1.52525I
b = −0.32410 + 1.59615I
4.34730 + 4.58698I −8.81634 − 3.98143I
u = 1.16118
a = −0.0729981
b = 0.525903
−6.14347 −34.2700
u = 0.813113
a = −0.251527
b = −1.09435
−3.21147 −11.3390
u = −0.388843 + 0.655973I
a = 1.45724 − 0.21090I
b = −0.540730 + 0.491931I
2.53558 + 1.80767I −8.41467 − 3.02752I
u = −0.388843 − 0.655973I
a = 1.45724 + 0.21090I
b = −0.540730 − 0.491931I
2.53558 − 1.80767I −8.41467 + 3.02752I
u = −0.082992 + 1.265620I
a = 0.16854 − 1.56029I
b = 0.608944 + 1.028170I
7.63621 + 2.52748I −1.68376 − 3.30171I
u = −0.082992 − 1.265620I
a = 0.16854 + 1.56029I
b = 0.608944 − 1.028170I
7.63621 − 2.52748I −1.68376 + 3.30171I
u = 0.382406 + 1.302770I
a = 0.432449 + 1.237680I
b = −1.079670 − 0.598913I
0.89066 − 4.29944I −6.20629 + 3.86373I
u = 0.382406 − 1.302770I
a = 0.432449 − 1.237680I
b = −1.079670 + 0.598913I
0.89066 + 4.29944I −6.20629 − 3.86373I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.60307 + 1.35028I
a = −0.114629 − 0.604014I
b = 0.503402 + 0.404837I
−2.00359 − 6.20208I −16.6975 + 15.2700I
u = 0.60307 − 1.35028I
a = −0.114629 + 0.604014I
b = 0.503402 − 0.404837I
−2.00359 + 6.20208I −16.6975 − 15.2700I
u = 0.241260 + 0.439152I
a = 1.93287 + 0.62822I
b = −0.383623 + 1.096450I
2.78585 + 2.08540I −5.87726 + 0.45245I
u = 0.241260 − 0.439152I
a = 1.93287 − 0.62822I
b = −0.383623 − 1.096450I
2.78585 − 2.08540I −5.87726 − 0.45245I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
14
− 10u
13
+ ··· − 13u + 1)(u
20
+ 10u
19
+ ··· + 29696u + 4096)
c
2
(u
14
+ 2u
13
+ ··· − u + 1)(u
20
− 16u
19
+ ··· + 224u − 64)
c
3
(u
14
− u
13
+ ··· − 2u − 1)(u
20
+ 2u
19
+ ··· − 135u − 31)
c
4
(u
14
− 3u
13
+ ··· + 2u
2
− 1)(u
20
− 4u
19
+ ··· + u − 1)
c
5
, c
9
(u
14
+ 2u
12
+ 4u
11
− u
10
+ 4u
9
+ 5u
8
− 4u
7
+ 4u
6
+ 5u
5
− 3u
4
+ 3u
2
− 1)
· (u
20
− u
19
+ ··· − 3u − 1)
c
6
(u
14
− 2u
13
+ ··· + u + 1)(u
20
− 16u
19
+ ··· + 224u − 64)
c
7
(u
14
+ 3u
13
+ ··· + 2u
2
− 1)(u
20
− 4u
19
+ ··· + u − 1)
c
8
(u
14
− 4u
13
+ ··· − 4u + 1)(u
20
− 3u
19
+ ··· − 3u + 1)
c
10
(u
14
+ 2u
12
− 4u
11
− u
10
− 4u
9
+ 5u
8
+ 4u
7
+ 4u
6
− 5u
5
− 3u
4
+ 3u
2
− 1)
· (u
20
− u
19
+ ··· − 3u − 1)
c
11
, c
12
(u
14
+ 4u
13
+ ··· + 4u + 1)(u
20
− 3u
19
+ ··· − 3u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
14
− 2y
13
+ ··· − 13y + 1)
· (y
20
+ 102y
19
+ ··· − 1112539136y + 16777216)
c
2
, c
6
(y
14
− 10y
13
+ ··· − 13y + 1)(y
20
− 10y
19
+ ··· − 29696y + 4096)
c
3
(y
14
+ 3y
13
+ ··· − 2y + 1)(y
20
+ 44y
19
+ ··· − 12831y + 961)
c
4
, c
7
(y
14
+ 5y
13
+ ··· − 4y + 1)(y
20
+ 42y
19
+ ··· − 29y + 1)
c
5
, c
9
, c
10
(y
14
+ 4y
13
+ ··· − 6y + 1)(y
20
+ 49y
19
+ ··· + 9y + 1)
c
8
, c
11
, c
12
(y
14
+ 10y
13
+ ··· + 8y + 1)(y
20
+ 15y
19
+ ··· − 21y + 1)
13
