12n
0338
(K12n
0338
)
A knot diagram
1
Linearized knot diagam
3 6 8 9 2 11 3 4 6 12 7 10
Solving Sequence
3,7
8 4 9
5,11
12 6 2 1 10
c
7
c
3
c
8
c
4
c
11
c
6
c
2
c
1
c
10
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−3004101246716u
20
− 373089817451u
19
+ ··· + 21082954445324b + 26346240192572,
1831272589986u
20
+ 602119374746u
19
+ ··· + 21082954445324a − 24452295430928,
u
21
+ u
20
+ ··· + 8u − 8i
I
u
2
= h4a
2
u + 6a
2
+ b − 1, 4a
3
+ 4au − 6a − 7u + 10, u
2
− 2i
I
v
1
= ha, b + v + 1, v
3
+ 2v
2
+ v + 1i
* 3 irreducible components of dim
C
= 0, with total 30 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−3.00×10
12
u
20
−3.73×10
11
u
19
+· · ·+2.11×10
13
b+2.63×10
13
, 1.83×
10
12
u
20
+6.02×10
11
u
19
+· · ·+2.11×10
13
a−2.45×10
13
, u
21
+u
20
+· · ·+8u−8i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
4
=
−u
−u
3
+ u
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
a
5
=
u
3
− 2u
u
5
− 3u
3
+ u
a
11
=
−0.0868603u
20
− 0.0285595u
19
+ ··· − 0.459641u + 1.15981
0.142490u
20
+ 0.0176963u
19
+ ··· + 2.40462u − 1.24965
a
12
=
−0.229350u
20
− 0.0462558u
19
+ ··· − 2.86426u + 2.40946
0.142490u
20
+ 0.0176963u
19
+ ··· + 2.40462u − 1.24965
a
6
=
−0.138686u
20
+ 0.00718333u
19
+ ··· − 0.505659u + 1.36119
−0.242033u
20
+ 0.00703747u
19
+ ··· − 5.49406u + 2.13835
a
2
=
−0.128794u
20
+ 0.0122843u
19
+ ··· − 3.05410u + 1.34108
−0.237410u
20
− 0.0250130u
19
+ ··· − 2.82817u + 2.12013
a
1
=
−0.128794u
20
+ 0.0122843u
19
+ ··· − 3.05410u + 1.34108
−0.111694u
20
− 0.00741087u
19
+ ··· − 0.669180u + 0.991499
a
10
=
−0.121424u
20
− 0.0196218u
19
+ ··· − 2.92331u + 2.24622
−0.185607u
20
− 0.0285721u
19
+ ··· − 2.65413u + 1.26626
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
29083336030791
21082954445324
u
20
−
503291539209
21082954445324
u
19
+ ··· +
136967234579540
5270738611331
u −
144077080803222
5270738611331
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
21
+ 36u
20
+ ··· + 11991u + 529
c
2
, c
5
u
21
+ 4u
20
+ ··· − 59u − 23
c
3
, c
4
, c
7
c
8
u
21
− u
20
+ ··· + 8u + 8
c
6
, c
11
u
21
− 2u
20
+ ··· + 8u
2
− 1
c
9
u
21
+ 2u
20
+ ··· − 144u − 52
c
10
, c
12
u
21
+ 10u
20
+ ··· + 16u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
21
− 92y
20
+ ··· + 39622923y −279841
c
2
, c
5
y
21
− 36y
20
+ ··· + 11991y −529
c
3
, c
4
, c
7
c
8
y
21
− 35y
20
+ ··· + 320y −64
c
6
, c
11
y
21
− 10y
20
+ ··· + 16y −1
c
9
y
21
− 66y
20
+ ··· + 76376y −2704
c
10
, c
12
y
21
+ 6y
20
+ ··· + 96y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.901926 + 0.051407I
a = 1.58428 + 0.44230I
b = 0.929270 − 0.621178I
−0.70179 + 4.44296I −15.1059 − 6.6514I
u = −0.901926 − 0.051407I
a = 1.58428 − 0.44230I
b = 0.929270 + 0.621178I
−0.70179 − 4.44296I −15.1059 + 6.6514I
u = −0.487980 + 0.535998I
a = −1.29130 − 1.86784I
b = −0.972602 + 0.217979I
−3.41098 + 0.72478I −18.1754 − 4.1507I
u = −0.487980 − 0.535998I
a = −1.29130 + 1.86784I
b = −0.972602 − 0.217979I
−3.41098 − 0.72478I −18.1754 + 4.1507I
u = 0.667009 + 0.250056I
a = 0.473812 − 0.352817I
b = 0.746002 − 0.517025I
−0.131566 + 0.215455I −13.55030 + 1.35945I
u = 0.667009 − 0.250056I
a = 0.473812 + 0.352817I
b = 0.746002 + 0.517025I
−0.131566 − 0.215455I −13.55030 − 1.35945I
u = −1.336150 + 0.286473I
a = −0.480443 − 0.210578I
b = 0.219656 + 0.684964I
−6.45258 + 0.58096I −15.0688 − 0.0562I
u = −1.336150 − 0.286473I
a = −0.480443 + 0.210578I
b = 0.219656 − 0.684964I
−6.45258 − 0.58096I −15.0688 + 0.0562I
u = −1.46037
a = −1.04497
b = −0.429188
−6.73694 −12.3530
u = 1.37773 + 0.63713I
a = 1.24070 − 1.07467I
b = 1.177670 + 0.522691I
−9.24475 − 5.31786I −17.8058 + 4.0813I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.37773 − 0.63713I
a = 1.24070 + 1.07467I
b = 1.177670 − 0.522691I
−9.24475 + 5.31786I −17.8058 − 4.0813I
u = 0.007014 + 0.428132I
a = 0.157773 + 1.318050I
b = −0.876187 − 0.694943I
2.12758 − 2.67655I −5.20055 + 2.49560I
u = 0.007014 − 0.428132I
a = 0.157773 − 1.318050I
b = −0.876187 + 0.694943I
2.12758 + 2.67655I −5.20055 − 2.49560I
u = 0.380480
a = 0.651164
b = 0.349133
−0.576083 −17.0820
u = 1.73579 + 0.22895I
a = −1.43373 + 0.00630I
b = −1.176150 + 0.419094I
−10.01720 + 3.13008I −18.2978 − 3.1992I
u = 1.73579 − 0.22895I
a = −1.43373 − 0.00630I
b = −1.176150 − 0.419094I
−10.01720 − 3.13008I −18.2978 + 3.1992I
u = 1.90238 + 0.14731I
a = −0.0278466 + 0.0602262I
b = −0.516255 + 1.062480I
−18.6129 − 3.3460I −15.4445 + 0.4593I
u = 1.90238 − 0.14731I
a = −0.0278466 − 0.0602262I
b = −0.516255 − 1.062480I
−18.6129 + 3.3460I −15.4445 − 0.4593I
u = −1.89336 + 0.24852I
a = −1.27473 − 0.71223I
b = −1.194710 + 0.746952I
18.7399 + 9.8858I −17.0280 − 4.3210I
u = −1.89336 − 0.24852I
a = −1.27473 + 0.71223I
b = −1.194710 − 0.746952I
18.7399 − 9.8858I −17.0280 + 4.3210I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −2.06112
a = 1.49678
b = 1.40668
13.3737 −19.2100
7
II. I
u
2
= h4a
2
u + 6a
2
+ b − 1, 4a
3
+ 4au − 6a − 7u + 10, u
2
− 2i
(i) Arc colorings
a
3
=
0
u
a
7
=
1
0
a
8
=
1
2
a
4
=
−u
−u
a
9
=
−1
0
a
5
=
0
−u
a
11
=
a
−4a
2
u − 6a
2
+ 1
a
12
=
4a
2
u + 6a
2
+ a − 1
−4a
2
u − 6a
2
+ 1
a
6
=
1
2
u
4a
2
u + 6a
2
+ au + 2a − 1
a
2
=
1
2
u
4a
2
u + 6a
2
+ au + 2a + u − 1
a
1
=
1
2
u
4a
2
u + 6a
2
+ au + 2a − 1
a
10
=
−3a
2
u − 4a
2
− au − a +
1
2
u − 1
au + 2a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −16a
2
u − 24a
2
− 16
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
6
c
2
(u + 1)
6
c
3
, c
4
, c
7
c
8
(u
2
− 2)
3
c
6
(u
3
− u
2
+ 1)
2
c
9
, c
10
(u
3
− u
2
+ 2u − 1)
2
c
11
(u
3
+ u
2
− 1)
2
c
12
(u
3
+ u
2
+ 2u + 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
7
c
8
(y −2)
6
c
6
, c
11
(y
3
− y
2
+ 2y −1)
2
c
9
, c
10
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = −0.388001
b = −0.754878
−7.69319 −23.0200
u = 1.41421
a = 0.194000 + 0.164688I
b = 0.877439 − 0.744862I
−3.55561 + 2.82812I −16.4902 − 2.9794I
u = 1.41421
a = 0.194000 − 0.164688I
b = 0.877439 + 0.744862I
−3.55561 − 2.82812I −16.4902 + 2.9794I
u = −1.41421
a = 1.13072 + 0.95987I
b = 0.877439 − 0.744862I
−3.55561 + 2.82812I −16.4902 − 2.9794I
u = −1.41421
a = 1.13072 − 0.95987I
b = 0.877439 + 0.744862I
−3.55561 − 2.82812I −16.4902 + 2.9794I
u = −1.41421
a = −2.26144
b = −0.754878
−7.69319 −23.0200
11
III. I
v
1
= ha, b + v + 1, v
3
+ 2v
2
+ v + 1i
(i) Arc colorings
a
3
=
v
0
a
7
=
1
0
a
8
=
1
0
a
4
=
v
0
a
9
=
1
0
a
5
=
v
0
a
11
=
0
−v − 1
a
12
=
v + 1
−v − 1
a
6
=
1
−v
2
− 2v − 1
a
2
=
v − 1
v
2
+ 2v + 1
a
1
=
−1
v
2
+ 2v + 1
a
10
=
−v
2
− 2v
v
2
+ v − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4v
2
+ 6v − 14
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
7
c
8
u
3
c
5
(u + 1)
3
c
6
u
3
+ u
2
− 1
c
9
, c
12
u
3
+ u
2
+ 2u + 1
c
10
u
3
− u
2
+ 2u − 1
c
11
u
3
− u
2
+ 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y − 1)
3
c
3
, c
4
, c
7
c
8
y
3
c
6
, c
11
y
3
− y
2
+ 2y − 1
c
9
, c
10
, c
12
y
3
+ 3y
2
+ 2y − 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.122561 + 0.744862I
a = 0
b = −0.877439 − 0.744862I
1.37919 − 2.82812I −16.8946 + 3.7388I
v = −0.122561 − 0.744862I
a = 0
b = −0.877439 + 0.744862I
1.37919 + 2.82812I −16.8946 − 3.7388I
v = −1.75488
a = 0
b = 0.754878
−2.75839 −12.2110
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
21
+ 36u
20
+ ··· + 11991u + 529)
c
2
((u − 1)
3
)(u + 1)
6
(u
21
+ 4u
20
+ ··· − 59u − 23)
c
3
, c
4
, c
7
c
8
u
3
(u
2
− 2)
3
(u
21
− u
20
+ ··· + 8u + 8)
c
5
((u − 1)
6
)(u + 1)
3
(u
21
+ 4u
20
+ ··· − 59u − 23)
c
6
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
21
− 2u
20
+ ··· + 8u
2
− 1)
c
9
((u
3
− u
2
+ 2u − 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
21
+ 2u
20
+ ··· − 144u − 52)
c
10
((u
3
− u
2
+ 2u − 1)
3
)(u
21
+ 10u
20
+ ··· + 16u + 1)
c
11
(u
3
− u
2
+ 1)(u
3
+ u
2
− 1)
2
(u
21
− 2u
20
+ ··· + 8u
2
− 1)
c
12
((u
3
+ u
2
+ 2u + 1)
3
)(u
21
+ 10u
20
+ ··· + 16u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
9
)(y
21
− 92y
20
+ ··· + 3.96229 × 10
7
y − 279841)
c
2
, c
5
((y − 1)
9
)(y
21
− 36y
20
+ ··· + 11991y − 529)
c
3
, c
4
, c
7
c
8
y
3
(y − 2)
6
(y
21
− 35y
20
+ ··· + 320y − 64)
c
6
, c
11
((y
3
− y
2
+ 2y − 1)
3
)(y
21
− 10y
20
+ ··· + 16y − 1)
c
9
((y
3
+ 3y
2
+ 2y − 1)
3
)(y
21
− 66y
20
+ ··· + 76376y − 2704)
c
10
, c
12
((y
3
+ 3y
2
+ 2y − 1)
3
)(y
21
+ 6y
20
+ ··· + 96y − 1)
17
