12n
0451
(K12n
0451
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 2 12 10 3 6 3 7 9
Solving Sequence
3,6
2 1
5,10
4 9 8 7 12 11
c
2
c
1
c
5
c
4
c
9
c
8
c
7
c
12
c
11
c
3
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h13u
14
− 82u
13
+ ··· + 2b − 28, −u
14
+ 3u
13
+ ··· + 2a + 7, u
15
− 8u
14
+ ··· + 18u + 4i
I
u
2
= h−u
7
− u
6
+ 4u
5
+ 3u
4
− 5u
3
− 3u
2
+ b + 2u + 1,
2u
9
+ 6u
8
− u
7
− 18u
6
− 10u
5
+ 18u
4
+ 15u
3
− 4u
2
+ 3a − 4u + 1,
u
10
+ 3u
9
− 2u
8
− 12u
7
− 2u
6
+ 18u
5
+ 9u
4
− 11u
3
− 8u
2
+ 2u + 3i
I
u
3
= ha
3
u
2
+ 8a
3
u + 6a
2
u
2
+ 2a
3
+ 9a
2
u − a
2
− 3u
2
+ 13b − 11u − 6,
− 2a
3
u
2
+ a
4
− a
3
u − 3a
2
u
2
+ 4a
3
− 3a
2
u − 6u
2
a + 7a
2
− 3au − 16u
2
+ 13a − 9u + 37, u
3
+ u
2
− 2u − 1i
* 3 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
=
h13u
14
−82u
13
+· · ·+2b−28, −u
14
+3u
13
+· · ·+2a+7, u
15
−8u
14
+· · ·+18u+4i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
5
=
u
−u
3
+ u
a
10
=
1
2
u
14
−
3
2
u
13
+ ··· −
3
2
u −
7
2
−
13
2
u
14
+ 41u
13
+ ··· +
145
2
u + 14
a
4
=
19
4
u
14
−
61
2
u
13
+ ··· −
239
4
u − 11
1
2
u
14
− 3u
13
+ ··· −
9
2
u − 1
a
9
=
1
2
u
14
−
3
2
u
13
+ ··· −
3
2
u −
7
2
−
1
2
u
14
+ 6u
13
+ ··· +
51
2
u + 4
a
8
=
u
14
−
15
2
u
13
+ ··· − 27u −
15
2
−
1
2
u
14
+ 6u
13
+ ··· +
51
2
u + 4
a
7
=
−
1
2
u
14
+
7
2
u
13
+ ··· −
3
2
u −
3
2
−
13
2
u
14
+ 41u
13
+ ··· +
145
2
u + 14
a
12
=
−
11
4
u
14
+
35
2
u
13
+ ··· +
143
4
u + 9
−
15
2
u
14
+ 48u
13
+ ··· +
193
2
u + 19
a
11
=
6u
14
−
79
2
u
13
+ ··· − 71u −
21
2
13
2
u
14
− 41u
13
+ ··· −
145
2
u − 14
(ii) Obstruction class = −1
(iii) Cusp Shapes = −14u
14
+ 88u
13
− 169u
12
+ 49u
11
+ u
10
+ 512u
9
− 618u
8
−
173u
7
− 82u
6
+ 878u
5
− 86u
4
− 524u
3
− 44u
2
+ 156u + 46
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
+ 18u
14
+ ··· + 428u + 16
c
2
, c
5
u
15
+ 8u
14
+ ··· + 18u − 4
c
3
, c
10
u
15
− u
14
+ ··· − u − 1
c
4
, c
8
u
15
+ 15u
13
+ ··· − 5u
2
− 1
c
6
, c
11
u
15
− 8u
14
+ ··· + 44u − 8
c
7
u
15
+ 11u
14
+ ··· − 30u − 4
c
9
, c
12
u
15
+ u
14
+ ··· − 10u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
− 38y
14
+ ··· + 98672y −256
c
2
, c
5
y
15
− 18y
14
+ ··· + 428y −16
c
3
, c
10
y
15
− 19y
14
+ ··· + 23y −1
c
4
, c
8
y
15
+ 30y
14
+ ··· − 10y −1
c
6
, c
11
y
15
+ 8y
14
+ ··· + 144y −64
c
7
y
15
+ 5y
14
+ ··· + 108y −16
c
9
, c
12
y
15
+ 19y
14
+ ··· + 54y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.959509 + 0.354430I
a = −0.161134 + 0.086188I
b = 0.316905 + 0.548611I
−1.65521 − 1.25769I 0.52360 + 4.10357I
u = 0.959509 − 0.354430I
a = −0.161134 − 0.086188I
b = 0.316905 − 0.548611I
−1.65521 + 1.25769I 0.52360 − 4.10357I
u = −0.550498 + 0.994483I
a = −0.718395 + 0.573199I
b = −1.342500 + 0.128424I
2.97662 − 0.13213I 5.02719 − 0.29116I
u = −0.550498 − 0.994483I
a = −0.718395 − 0.573199I
b = −1.342500 − 0.128424I
2.97662 + 0.13213I 5.02719 + 0.29116I
u = −0.946416 + 0.902266I
a = 0.592748 − 0.766485I
b = 1.45247 + 0.51633I
1.73111 + 6.58453I 3.78417 − 5.80099I
u = −0.946416 − 0.902266I
a = 0.592748 + 0.766485I
b = 1.45247 − 0.51633I
1.73111 − 6.58453I 3.78417 + 5.80099I
u = −0.484300 + 0.231140I
a = 1.92354 − 0.88468I
b = 0.417351 + 0.091087I
−3.75387 − 2.45110I 11.17942 + 2.61787I
u = −0.484300 − 0.231140I
a = 1.92354 + 0.88468I
b = 0.417351 − 0.091087I
−3.75387 + 2.45110I 11.17942 − 2.61787I
u = 1.61285 + 0.22347I
a = −0.112650 − 0.865027I
b = −1.065320 + 0.236511I
−10.89750 + 0.20978I 2.15059 − 0.10319I
u = 1.61285 − 0.22347I
a = −0.112650 + 0.865027I
b = −1.065320 − 0.236511I
−10.89750 − 0.20978I 2.15059 + 0.10319I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.72721 + 0.30285I
a = −0.232010 + 0.919021I
b = 1.57691 − 0.30542I
−4.83321 − 4.76833I 4.97369 + 2.14836I
u = 1.72721 − 0.30285I
a = −0.232010 − 0.919021I
b = 1.57691 + 0.30542I
−4.83321 + 4.76833I 4.97369 − 2.14836I
u = −0.224907
a = −2.25152
b = −0.299515
0.730621 13.9140
u = 1.79410 + 0.24246I
a = 0.333662 − 1.121120I
b = −1.70606 + 0.72255I
−7.78478 − 11.23000I 2.90412 + 5.28147I
u = 1.79410 − 0.24246I
a = 0.333662 + 1.121120I
b = −1.70606 − 0.72255I
−7.78478 + 11.23000I 2.90412 − 5.28147I
6
II.
I
u
2
= h−u
7
− u
6
+ · · · +b + 1, 2u
9
+ 6u
8
+ · · · +3a + 1, u
10
+ 3u
9
+ · · · +2u + 3i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
5
=
u
−u
3
+ u
a
10
=
−
2
3
u
9
− 2u
8
+ ··· +
4
3
u −
1
3
u
7
+ u
6
− 4u
5
− 3u
4
+ 5u
3
+ 3u
2
− 2u − 1
a
4
=
1
3
u
9
−
8
3
u
7
+ ··· +
7
3
u +
8
3
−u
3
+ 2u − 1
a
9
=
−
2
3
u
9
− 2u
8
+ ··· +
4
3
u −
1
3
−u
9
− 2u
8
+ 3u
7
+ 7u
6
− 3u
5
− 9u
4
+ u
3
+ 4u
2
− 1
a
8
=
1
3
u
9
−
8
3
u
7
+ ··· +
4
3
u +
2
3
−u
9
− 2u
8
+ 3u
7
+ 7u
6
− 3u
5
− 9u
4
+ u
3
+ 4u
2
− 1
a
7
=
2
3
u
9
+ 2u
8
+ ··· −
10
3
u −
2
3
−u
7
− u
6
+ 4u
5
+ 3u
4
− 5u
3
− 3u
2
+ 2u + 1
a
12
=
2
3
u
9
+ 2u
8
+ ··· −
10
3
u −
2
3
u
9
+ 2u
8
− 3u
7
− 7u
6
+ 2u
5
+ 9u
4
+ 2u
3
− 5u
2
− 2u + 1
a
11
=
−
2
3
u
9
− 2u
8
+ ··· −
2
3
u −
4
3
u
7
+ u
6
− 4u
5
− 3u
4
+ 5u
3
+ 3u
2
− 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
9
− 8u
8
+ 5u
7
+ 23u
6
+ u
5
− 21u
4
− 5u
3
+ 3u
2
− 4u + 3
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
− 13u
9
+ ··· − 52u + 9
c
2
u
10
+ 3u
9
− 2u
8
− 12u
7
− 2u
6
+ 18u
5
+ 9u
4
− 11u
3
− 8u
2
+ 2u + 3
c
3
, c
10
u
10
− u
9
− 3u
8
+ 3u
7
− u
6
+ 2u
5
+ 5u
4
− 9u
3
+ 7u
2
− 2u + 1
c
4
, c
8
u
10
+ 3u
8
− 4u
7
− 11u
6
+ 11u
5
+ 13u
4
− 6u
3
− 2u
2
+ 3u + 1
c
5
u
10
− 3u
9
− 2u
8
+ 12u
7
− 2u
6
− 18u
5
+ 9u
4
+ 11u
3
− 8u
2
− 2u + 3
c
6
u
10
− u
9
+ 4u
8
− 3u
7
+ 7u
6
− 3u
5
+ 5u
4
− u
3
+ 2u
2
+ u + 1
c
7
u
10
+ 6u
9
+ 13u
8
+ 12u
7
+ 5u
6
+ 2u
5
+ 4u
4
+ 22u
3
+ 41u
2
+ 28u + 7
c
9
, c
12
u
10
− u
9
+ 4u
8
− 6u
7
+ 4u
6
− 6u
5
+ 10u
4
− 9u
3
+ 6u
2
− 3u + 1
c
11
u
10
+ u
9
+ 4u
8
+ 3u
7
+ 7u
6
+ 3u
5
+ 5u
4
+ u
3
+ 2u
2
− u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 25y
9
+ ··· + 212y + 81
c
2
, c
5
y
10
− 13y
9
+ ··· − 52y + 9
c
3
, c
10
y
10
− 7y
9
+ 13y
8
+ 11y
7
− 45y
6
− 4y
5
+ 53y
4
− 5y
3
+ 23y
2
+ 10y + 1
c
4
, c
8
y
10
+ 6y
9
+ ··· − 13y + 1
c
6
, c
11
y
10
+ 7y
9
+ ··· + 3y + 1
c
7
y
10
− 10y
9
+ ··· − 210y + 49
c
9
, c
12
y
10
+ 7y
9
+ 12y
8
+ 4y
7
+ 18y
6
− 20y
5
+ 12y
4
+ 11y
3
+ 2y
2
+ 3y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.794660 + 0.197895I
a = −1.21999 − 0.81881I
b = −0.033564 + 0.426607I
−4.30076 + 2.46712I −4.14664 − 2.94515I
u = 0.794660 − 0.197895I
a = −1.21999 + 0.81881I
b = −0.033564 − 0.426607I
−4.30076 − 2.46712I −4.14664 + 2.94515I
u = −1.154300 + 0.430931I
a = −0.174417 + 0.481835I
b = −1.57328 − 0.24809I
3.14814 + 4.53334I 4.18402 − 3.64382I
u = −1.154300 − 0.430931I
a = −0.174417 − 0.481835I
b = −1.57328 + 0.24809I
3.14814 − 4.53334I 4.18402 + 3.64382I
u = −0.563250 + 0.505340I
a = −0.146187 − 0.911256I
b = 1.55694 − 0.21156I
5.07321 − 0.90406I 8.90897 − 1.12395I
u = −0.563250 − 0.505340I
a = −0.146187 + 0.911256I
b = 1.55694 + 0.21156I
5.07321 + 0.90406I 8.90897 + 1.12395I
u = 1.228530 + 0.260062I
a = 0.760530 + 0.065247I
b = −0.634254 − 0.504382I
−1.184580 − 0.336842I 3.77741 − 2.24920I
u = 1.228530 − 0.260062I
a = 0.760530 − 0.065247I
b = −0.634254 + 0.504382I
−1.184580 + 0.336842I 3.77741 + 2.24920I
u = −1.80564 + 0.05386I
a = 0.113401 − 1.082570I
b = 0.184156 + 1.137500I
−14.2505 − 0.9863I −1.223765 + 0.251668I
u = −1.80564 − 0.05386I
a = 0.113401 + 1.082570I
b = 0.184156 − 1.137500I
−14.2505 + 0.9863I −1.223765 − 0.251668I
10
III. I
u
3
=
ha
3
u
2
+6a
2
u
2
+· · · − a
2
−6, −2a
3
u
2
−3a
2
u
2
+· · · + 13a + 37, u
3
+u
2
−2u −1i
(i) Arc colorings
a
3
=
1
0
a
6
=
0
u
a
2
=
1
−u
2
a
1
=
−u
2
+ 1
−u
2
a
5
=
u
u
2
− u − 1
a
10
=
a
−0.0769231a
3
u
2
− 0.461538a
2
u
2
+ ··· + 0.0769231a
2
+ 0.461538
a
4
=
−0.615385a
3
u
2
− 0.692308a
2
u
2
+ ··· + a + 1.69231
8
13
a
3
u
2
+
9
13
a
2
u
2
+ ··· − a −
48
13
a
9
=
a
−0.0769231a
3
u
2
− 0.461538a
2
u
2
+ ··· + 0.0769231a
2
+ 0.461538
a
8
=
1
13
a
3
u
2
+
6
13
a
2
u
2
+ ··· + a −
6
13
−0.0769231a
3
u
2
− 0.461538a
2
u
2
+ ··· + 0.0769231a
2
+ 0.461538
a
7
=
1
13
a
3
u
2
+
6
13
a
2
u
2
+ ··· − a −
6
13
0.307692a
3
u
2
− 0.153846a
2
u
2
+ ··· − 0.307692a
2
+ 0.153846
a
12
=
1
13
a
3
u
2
+
6
13
a
2
u
2
+ ··· − a −
32
13
−0.538462a
3
u
2
+ 0.769231a
2
u
2
+ ··· − 0.461538a
2
− 0.769231
a
11
=
1
13
a
3
u
2
+
6
13
a
2
u
2
+ ··· − a −
6
13
0.0769231a
3
u
2
+ 0.461538a
2
u
2
+ ··· − 0.0769231a
2
− 0.461538
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
28
13
a
3
u
2
−
16
13
a
3
u −
12
13
a
2
u
2
−
4
13
a
3
−
44
13
a
2
u −
24
13
a
2
+ 4au +
32
13
u
2
+
48
13
u +
38
13
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
3
+ 5u
2
+ 6u + 1)
4
c
2
, c
5
, c
7
(u
3
− u
2
− 2u + 1)
4
c
3
, c
10
u
12
− u
11
+ ··· − 34u + 13
c
4
, c
8
u
12
+ u
11
+ ··· + 208u + 139
c
6
, c
11
(u
2
+ u + 1)
6
c
9
, c
12
u
12
− u
11
+ ··· − 16u + 43
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
3
− 13y
2
+ 26y −1)
4
c
2
, c
5
, c
7
(y
3
− 5y
2
+ 6y −1)
4
c
3
, c
10
y
12
− 5y
11
+ ··· − 740y + 169
c
4
, c
8
y
12
+ 19y
11
+ ··· + 24568y + 19321
c
6
, c
11
(y
2
+ y + 1)
6
c
9
, c
12
y
12
+ 11y
11
+ ··· + 948y + 1849
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.24698
a = 1.002800 + 0.135184I
b = −0.601826 − 0.829682I
−1.40994 + 2.02988I 4.00000 − 3.46410I
u = 1.24698
a = 1.002800 − 0.135184I
b = −0.601826 + 0.829682I
−1.40994 − 2.02988I 4.00000 + 3.46410I
u = 1.24698
a = −0.824347 + 0.444265I
b = 1.225320 + 0.250234I
−1.40994 − 2.02988I 4.00000 + 3.46410I
u = 1.24698
a = −0.824347 − 0.444265I
b = 1.225320 − 0.250234I
−1.40994 + 2.02988I 4.00000 − 3.46410I
u = −0.445042
a = 0.52051 + 2.02175I
b = −1.64400 − 0.07581I
4.22983 + 2.02988I 4.00000 − 3.46410I
u = −0.445042
a = 0.52051 − 2.02175I
b = −1.64400 + 0.07581I
4.22983 − 2.02988I 4.00000 + 3.46410I
u = −0.445042
a = −2.54497 + 1.48471I
b = 1.42148 + 0.46123I
4.22983 + 2.02988I 4.00000 − 3.46410I
u = −0.445042
a = −2.54497 − 1.48471I
b = 1.42148 − 0.46123I
4.22983 − 2.02988I 4.00000 + 3.46410I
u = −1.80194
a = 0.248738 + 0.776723I
b = −0.526217 − 0.296115I
−12.68950 + 2.02988I 4.00000 − 3.46410I
u = −1.80194
a = 0.248738 − 0.776723I
b = −0.526217 + 0.296115I
−12.68950 − 2.02988I 4.00000 + 3.46410I
14
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.80194
a = 0.097273 + 1.376030I
b = −0.37475 − 1.85664I
−12.68950 − 2.02988I 4.00000 + 3.46410I
u = −1.80194
a = 0.097273 − 1.376030I
b = −0.37475 + 1.85664I
−12.68950 + 2.02988I 4.00000 − 3.46410I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
+ 5u
2
+ 6u + 1)
4
)(u
10
− 13u
9
+ ··· − 52u + 9)
· (u
15
+ 18u
14
+ ··· + 428u + 16)
c
2
(u
3
− u
2
− 2u + 1)
4
· (u
10
+ 3u
9
− 2u
8
− 12u
7
− 2u
6
+ 18u
5
+ 9u
4
− 11u
3
− 8u
2
+ 2u + 3)
· (u
15
+ 8u
14
+ ··· + 18u − 4)
c
3
, c
10
(u
10
− u
9
− 3u
8
+ 3u
7
− u
6
+ 2u
5
+ 5u
4
− 9u
3
+ 7u
2
− 2u + 1)
· (u
12
− u
11
+ ··· − 34u + 13)(u
15
− u
14
+ ··· − u − 1)
c
4
, c
8
(u
10
+ 3u
8
− 4u
7
− 11u
6
+ 11u
5
+ 13u
4
− 6u
3
− 2u
2
+ 3u + 1)
· (u
12
+ u
11
+ ··· + 208u + 139)(u
15
+ 15u
13
+ ··· − 5u
2
− 1)
c
5
(u
3
− u
2
− 2u + 1)
4
· (u
10
− 3u
9
− 2u
8
+ 12u
7
− 2u
6
− 18u
5
+ 9u
4
+ 11u
3
− 8u
2
− 2u + 3)
· (u
15
+ 8u
14
+ ··· + 18u − 4)
c
6
((u
2
+ u + 1)
6
)(u
10
− u
9
+ ··· + u + 1)
· (u
15
− 8u
14
+ ··· + 44u − 8)
c
7
(u
3
− u
2
− 2u + 1)
4
· (u
10
+ 6u
9
+ 13u
8
+ 12u
7
+ 5u
6
+ 2u
5
+ 4u
4
+ 22u
3
+ 41u
2
+ 28u + 7)
· (u
15
+ 11u
14
+ ··· − 30u − 4)
c
9
, c
12
(u
10
− u
9
+ 4u
8
− 6u
7
+ 4u
6
− 6u
5
+ 10u
4
− 9u
3
+ 6u
2
− 3u + 1)
· (u
12
− u
11
+ ··· − 16u + 43)(u
15
+ u
14
+ ··· − 10u − 1)
c
11
((u
2
+ u + 1)
6
)(u
10
+ u
9
+ ··· − u + 1)
· (u
15
− 8u
14
+ ··· + 44u − 8)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
− 13y
2
+ 26y −1)
4
)(y
10
− 25y
9
+ ··· + 212y + 81)
· (y
15
− 38y
14
+ ··· + 98672y −256)
c
2
, c
5
((y
3
− 5y
2
+ 6y −1)
4
)(y
10
− 13y
9
+ ··· − 52y + 9)
· (y
15
− 18y
14
+ ··· + 428y −16)
c
3
, c
10
(y
10
− 7y
9
+ 13y
8
+ 11y
7
− 45y
6
− 4y
5
+ 53y
4
− 5y
3
+ 23y
2
+ 10y + 1)
· (y
12
− 5y
11
+ ··· − 740y + 169)(y
15
− 19y
14
+ ··· + 23y −1)
c
4
, c
8
(y
10
+ 6y
9
+ ··· − 13y + 1)(y
12
+ 19y
11
+ ··· + 24568y + 19321)
· (y
15
+ 30y
14
+ ··· − 10y −1)
c
6
, c
11
((y
2
+ y + 1)
6
)(y
10
+ 7y
9
+ ··· + 3y + 1)(y
15
+ 8y
14
+ ··· + 144y −64)
c
7
((y
3
− 5y
2
+ 6y −1)
4
)(y
10
− 10y
9
+ ··· − 210y + 49)
· (y
15
+ 5y
14
+ ··· + 108y −16)
c
9
, c
12
(y
10
+ 7y
9
+ 12y
8
+ 4y
7
+ 18y
6
− 20y
5
+ 12y
4
+ 11y
3
+ 2y
2
+ 3y + 1)
· (y
12
+ 11y
11
+ ··· + 948y + 1849)(y
15
+ 19y
14
+ ··· + 54y −1)
17
