12n
0054
(K12n
0054
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 11 10 5 12 6 7 9
Solving Sequence
6,10
11 7 8
2,12
5 3 1 4 9
c
10
c
6
c
7
c
11
c
5
c
2
c
1
c
4
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
23
+ 2u
22
+ ··· + 2b − 2, u
20
− u
19
+ ··· + 2a − 1, u
24
− 3u
23
+ ··· − u − 1i
I
u
2
= hu
4
a + au + b − a + u, u
4
a + u
3
a + u
4
− 2u
2
a + 2u
3
+ a
2
− au − u
2
+ a − 3u, u
5
+ u
4
− 2u
3
− u
2
+ u − 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−2u
23
+2u
22
+· · ·+2b−2, u
20
−u
19
+· · ·+2a−1, u
24
−3u
23
+· · ·−u−1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
8
=
−u
3
+ 2u
−u
3
+ u
a
2
=
−
1
2
u
20
+
1
2
u
19
+ ··· −
7
2
u +
1
2
u
23
− u
22
+ ··· +
1
2
u + 1
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
5
=
−u
22
+ 2u
21
+ ··· −
5
2
u −
3
2
−u
23
+
3
2
u
22
+ ··· −
3
2
u − 1
a
3
=
−u
23
+ 13u
21
+ ··· −
7
2
u −
5
2
−
1
2
u
20
+ 5u
18
+ ··· + 3u
2
−
1
2
u
a
1
=
u
10
− 5u
8
+ 8u
6
− 3u
4
− u
2
− 1
−u
12
+ 6u
10
− 12u
8
+ 8u
6
− u
4
+ 2u
2
a
4
=
−u
23
+ 13u
21
+ ··· −
7
2
u −
5
2
−4u
23
+
11
2
u
22
+ ··· −
9
2
u − 3
a
9
=
u
6
− 3u
4
+ 2u
2
+ 1
−u
8
+ 4u
6
− 4u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
5
2
u
23
+
7
2
u
22
+ 25u
21
−29u
20
−
215
2
u
19
+ 83u
18
+ 261u
17
−59u
16
−
775
2
u
15
−
305
2
u
14
+
325u
13
+
601
2
u
12
−66u
11
−
251
2
u
10
−
195
2
u
9
−33u
8
+2u
7
−2u
6
+27u
5
−8u
4
+50u
3
+7u
2
−2u+
5
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
24
+ 2u
23
+ ··· − 22u
2
+ 1
c
2
, c
5
u
24
+ 6u
23
+ ··· + 4u + 1
c
3
u
24
− 6u
23
+ ··· + 22568u + 2857
c
4
, c
8
u
24
− u
23
+ ··· + 2048u − 1024
c
6
, c
10
, c
11
u
24
− 3u
23
+ ··· − u − 1
c
7
u
24
+ 9u
23
+ ··· + 193u + 37
c
9
, c
12
u
24
− u
23
+ ··· + 3u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
24
+ 46y
23
+ ··· − 44y + 1
c
2
, c
5
y
24
+ 2y
23
+ ··· − 22y
2
+ 1
c
3
y
24
+ 90y
23
+ ··· + 73696224y + 8162449
c
4
, c
8
y
24
− 55y
23
+ ··· + 3145728y + 1048576
c
6
, c
10
, c
11
y
24
− 25y
23
+ ··· − 7y + 1
c
7
y
24
− 21y
23
+ ··· − 29479y + 1369
c
9
, c
12
y
24
+ 43y
23
+ ··· − 7y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.576252 + 0.762796I
a = 0.17468 − 1.56325I
b = 1.184550 − 0.408250I
15.5271 + 1.3395I 7.10412 + 0.50033I
u = −0.576252 − 0.762796I
a = 0.17468 + 1.56325I
b = 1.184550 + 0.408250I
15.5271 − 1.3395I 7.10412 − 0.50033I
u = −0.514997 + 0.789630I
a = −1.57008 − 0.30044I
b = −1.76900 − 0.98677I
15.3349 − 6.5027I 6.70839 + 4.44626I
u = −0.514997 − 0.789630I
a = −1.57008 + 0.30044I
b = −1.76900 + 0.98677I
15.3349 + 6.5027I 6.70839 − 4.44626I
u = 1.225150 + 0.076811I
a = 0.083295 + 0.422369I
b = 1.39234 − 0.93644I
2.02860 + 0.55793I 3.98398 + 0.47568I
u = 1.225150 − 0.076811I
a = 0.083295 − 0.422369I
b = 1.39234 + 0.93644I
2.02860 − 0.55793I 3.98398 − 0.47568I
u = 0.386232 + 0.611238I
a = 0.587872 − 0.504692I
b = 0.568781 − 0.120868I
0.95423 + 1.88035I 5.59254 − 3.14019I
u = 0.386232 − 0.611238I
a = 0.587872 + 0.504692I
b = 0.568781 + 0.120868I
0.95423 − 1.88035I 5.59254 + 3.14019I
u = −1.368080 + 0.114668I
a = 0.221757 − 0.722378I
b = 0.31647 + 2.25463I
3.28861 − 3.53789I 6.68532 + 4.97474I
u = −1.368080 − 0.114668I
a = 0.221757 + 0.722378I
b = 0.31647 − 2.25463I
3.28861 + 3.53789I 6.68532 − 4.97474I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.45120 + 0.24568I
a = −0.508403 + 0.116308I
b = −1.21041 − 0.93814I
6.85595 − 5.06667I 9.81977 + 2.58134I
u = −1.45120 − 0.24568I
a = −0.508403 − 0.116308I
b = −1.21041 + 0.93814I
6.85595 + 5.06667I 9.81977 − 2.58134I
u = 1.47737 + 0.05442I
a = −0.580000 − 0.650952I
b = −1.53636 + 0.50809I
6.55976 + 3.21841I 9.78805 − 2.36901I
u = 1.47737 − 0.05442I
a = −0.580000 + 0.650952I
b = −1.53636 − 0.50809I
6.55976 − 3.21841I 9.78805 + 2.36901I
u = 0.519837
a = −1.00613
b = 0.367248
1.05585 10.2690
u = −1.50378
a = 0.695944
b = 0.281103
7.71062 11.8740
u = 0.073930 + 0.488820I
a = −1.54481 + 0.21964I
b = −1.121570 + 0.795516I
−1.23888 + 1.54124I −1.90845 − 5.21623I
u = 0.073930 − 0.488820I
a = −1.54481 − 0.21964I
b = −1.121570 − 0.795516I
−1.23888 − 1.54124I −1.90845 + 5.21623I
u = 1.53161 + 0.28371I
a = 0.782365 + 0.603573I
b = 1.95290 − 1.78945I
−17.4846 + 10.4436I 9.58510 − 4.63962I
u = 1.53161 − 0.28371I
a = 0.782365 − 0.603573I
b = 1.95290 + 1.78945I
−17.4846 − 10.4436I 9.58510 + 4.63962I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.55820 + 0.25583I
a = 0.626457 − 0.749741I
b = −0.376301 − 0.227538I
−16.9383 + 2.4136I 10.12159 − 0.67000I
u = 1.55820 − 0.25583I
a = 0.626457 + 0.749741I
b = −0.376301 + 0.227538I
−16.9383 − 2.4136I 10.12159 + 0.67000I
u = −0.349993 + 0.170334I
a = 2.38196 − 1.38000I
b = 0.774418 + 0.002185I
0.46862 − 2.38365I 3.44790 + 2.07617I
u = −0.349993 − 0.170334I
a = 2.38196 + 1.38000I
b = 0.774418 − 0.002185I
0.46862 + 2.38365I 3.44790 − 2.07617I
7
II.
I
u
2
= hu
4
a + au + b − a + u, u
4
a + u
4
+ · · · + a
2
+ a, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
8
=
−u
3
+ 2u
−u
3
+ u
a
2
=
a
−u
4
a − au + a − u
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
5
=
u
4
+ u
3
− 2u
2
+ a − u + 1
−u
4
a + u
4
+ u
2
a − au − 2u
2
+ a
a
3
=
u
4
+ u
3
− 2u
2
+ a − u + 1
−u
4
a + u
4
− au − 2u
2
+ a − u
a
1
=
0
−u
a
4
=
u
4
+ u
3
− 2u
2
+ a − u + 1
−u
4
a + u
4
+ u
2
a − au − 2u
2
+ a
a
9
=
−u
3
+ 2u
−u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
4
a − u
3
a − 2u
2
a + 5u
3
+ 5au − 9u + 10
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
5
c
2
(u
2
+ u + 1)
5
c
4
, c
8
u
10
c
6
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
7
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
c
9
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
10
, c
11
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
c
12
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
5
c
4
, c
8
y
10
c
6
, c
10
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
7
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
c
9
, c
12
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.21774
a = −0.410598 + 0.711177I
b = −0.22546 − 1.71868I
2.40108 − 2.02988I 6.62546 + 2.50057I
u = 1.21774
a = −0.410598 − 0.711177I
b = −0.22546 + 1.71868I
2.40108 + 2.02988I 6.62546 − 2.50057I
u = 0.309916 + 0.549911I
a = −1.58413 + 0.01647I
b = −1.51295 + 0.11095I
0.329100 − 0.499304I 5.04069 − 0.50981I
u = 0.309916 + 0.549911I
a = 0.80632 + 1.36366I
b = 0.863922 + 0.161516I
0.32910 + 3.56046I 2.53179 − 8.01848I
u = 0.309916 − 0.549911I
a = −1.58413 − 0.01647I
b = −1.51295 − 0.11095I
0.329100 + 0.499304I 5.04069 + 0.50981I
u = 0.309916 − 0.549911I
a = 0.80632 − 1.36366I
b = 0.863922 − 0.161516I
0.32910 − 3.56046I 2.53179 + 8.01848I
u = −1.41878 + 0.21917I
a = 0.252108 + 0.649344I
b = −0.291925 − 0.343564I
5.87256 − 2.37095I 9.19707 + 1.05452I
u = −1.41878 + 0.21917I
a = 0.436295 − 0.543004I
b = 2.16641 + 1.32455I
5.87256 − 6.43072I 6.60498 + 6.63374I
u = −1.41878 − 0.21917I
a = 0.252108 − 0.649344I
b = −0.291925 + 0.343564I
5.87256 + 2.37095I 9.19707 − 1.05452I
u = −1.41878 − 0.21917I
a = 0.436295 + 0.543004I
b = 2.16641 − 1.32455I
5.87256 + 6.43072I 6.60498 − 6.63374I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
24
+ 2u
23
+ ··· − 22u
2
+ 1)
c
2
((u
2
+ u + 1)
5
)(u
24
+ 6u
23
+ ··· + 4u + 1)
c
3
((u
2
− u + 1)
5
)(u
24
− 6u
23
+ ··· + 22568u + 2857)
c
4
, c
8
u
10
(u
24
− u
23
+ ··· + 2048u − 1024)
c
5
((u
2
− u + 1)
5
)(u
24
+ 6u
23
+ ··· + 4u + 1)
c
6
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
24
− 3u
23
+ ··· − u − 1)
c
7
((u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
)(u
24
+ 9u
23
+ ··· + 193u + 37)
c
9
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
24
− u
23
+ ··· + 3u − 1)
c
10
, c
11
((u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
)(u
24
− 3u
23
+ ··· − u − 1)
c
12
((u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
)(u
24
− u
23
+ ··· + 3u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
24
+ 46y
23
+ ··· − 44y + 1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
24
+ 2y
23
+ ··· − 22y
2
+ 1)
c
3
((y
2
+ y + 1)
5
)(y
24
+ 90y
23
+ ··· + 7.36962 × 10
7
y + 8162449)
c
4
, c
8
y
10
(y
24
− 55y
23
+ ··· + 3145728y + 1048576)
c
6
, c
10
, c
11
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
)(y
24
− 25y
23
+ ··· − 7y + 1)
c
7
((y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
)(y
24
− 21y
23
+ ··· − 29479y + 1369)
c
9
, c
12
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
24
+ 43y
23
+ ··· − 7y + 1)
13
