12n
0246
(K12n
0246
)
A knot diagram
1
Linearized knot diagam
3 5 10 2 10 11 1 12 3 7 6 8
Solving Sequence
7,10
11 6 12
3,5
2 1 4 9 8
c
10
c
6
c
11
c
5
c
2
c
1
c
4
c
9
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
14
+ u
13
+ 6u
12
+ 6u
11
+ 15u
10
+ 14u
9
+ 19u
8
+ 13u
7
+ 9u
6
+ 3u
5
− 5u
4
− 2u
3
− 5u
2
+ 2b + 1,
u
14
− 3u
13
+ ··· + 8a + 19,
u
15
+ 7u
13
+ 2u
12
+ 19u
11
+ 13u
10
+ 23u
9
+ 30u
8
+ 10u
7
+ 26u
6
− u
4
+ 3u
3
− 9u
2
+ 3u + 1i
I
u
2
= h−716717u
25
+ 780792u
24
+ ··· + 963947b + 791849,
1775750u
25
− 1897723u
24
+ ··· + 963947a − 1845988, u
26
− 2u
25
+ ··· − 2u + 1i
I
u
3
= hb, −u
2
+ 2a − u − 3, u
3
+ 2u − 1i
I
u
4
= hb, u
3
+ a + u + 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
* 4 irreducible components of dim
C
= 0, with total 48 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
14
+u
13
+· · ·+2b+1, u
14
−3u
13
+· · ·+8a+19, u
15
+7u
13
+· · ·+3u+1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
−u
u
3
+ u
a
12
=
u
2
+ 1
−u
4
− 2u
2
a
3
=
−
1
8
u
14
+
3
8
u
13
+ ··· + 5u −
19
8
−
1
2
u
14
−
1
2
u
13
+ ··· +
5
2
u
2
−
1
2
a
5
=
−u
3
− 2u
u
3
+ u
a
2
=
−
3
8
u
14
+
1
8
u
13
+ ··· + 4u −
25
8
u
3
+ u
a
1
=
−1
u
13
+ 6u
11
+ 2u
10
+ 13u
9
+ 11u
8
+ 10u
7
+ 19u
6
+ 7u
4
− 7u
2
+ 3u + 1
a
4
=
3
8
u
14
+
7
8
u
13
+ ··· + 5u −
15
8
−
1
2
u
14
−
1
2
u
13
+ ··· +
5
2
u
2
−
1
2
a
9
=
u
3
+ 2u
−u
14
− 6u
12
− 2u
11
− 13u
10
− 11u
9
− 10u
8
− 19u
7
− 8u
5
+ 5u
3
− 3u
2
a
8
=
u
−u
14
− 6u
12
− 2u
11
− 13u
10
− 11u
9
− 10u
8
− 19u
7
− 7u
5
+ 7u
3
− 3u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
49
16
u
14
−
45
16
u
13
− 22u
12
−
177
8
u
11
−
1069
16
u
10
−
579
8
u
9
−
1701
16
u
8
−
1839
16
u
7
−
1341
16
u
6
−
1299
16
u
5
−
199
16
u
4
−
29
8
u
3
+
267
16
u
2
+
37
2
u −
123
16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
+ 4u
14
+ ··· − 127u + 16
c
2
, c
4
u
15
− 2u
14
+ ··· − 11u + 4
c
3
, c
9
u
15
− 3u
14
+ ··· + 8u + 32
c
5
u
15
+ 6u
14
+ ··· + 16u + 4
c
6
, c
7
, c
8
c
10
, c
11
, c
12
u
15
+ 7u
13
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
+ 16y
14
+ ··· + 15841y −256
c
2
, c
4
y
15
− 4y
14
+ ··· − 127y −16
c
3
, c
9
y
15
+ 15y
14
+ ··· + 320y −1024
c
5
y
15
− 16y
14
+ ··· + 408y −16
c
6
, c
7
, c
8
c
10
, c
11
, c
12
y
15
+ 14y
14
+ ··· + 27y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.939067 + 0.076154I
a = 0.11949 − 2.06723I
b = 0.27318 + 1.76916I
9.05174 − 3.68246I 5.44943 + 2.70726I
u = −0.939067 − 0.076154I
a = 0.11949 + 2.06723I
b = 0.27318 − 1.76916I
9.05174 + 3.68246I 5.44943 − 2.70726I
u = −0.231015 + 1.209380I
a = 0.202894 − 0.516586I
b = 1.71816 + 0.21702I
−5.39974 − 5.30636I −4.44673 + 7.07969I
u = −0.231015 − 1.209380I
a = 0.202894 + 0.516586I
b = 1.71816 − 0.21702I
−5.39974 + 5.30636I −4.44673 − 7.07969I
u = 0.072090 + 1.233060I
a = −0.067666 + 0.756607I
b = −0.93944 − 1.48122I
−8.55605 + 1.99221I −8.96301 − 2.93013I
u = 0.072090 − 1.233060I
a = −0.067666 − 0.756607I
b = −0.93944 + 1.48122I
−8.55605 − 1.99221I −8.96301 + 2.93013I
u = 0.446281 + 1.234210I
a = −0.890631 − 0.905985I
b = −0.39141 + 1.88678I
1.89371 + 6.18917I −0.33163 − 4.59933I
u = 0.446281 − 1.234210I
a = −0.890631 + 0.905985I
b = −0.39141 − 1.88678I
1.89371 − 6.18917I −0.33163 + 4.59933I
u = 0.45365 + 1.36129I
a = 1.09428 + 1.03938I
b = 0.73653 − 1.65036I
0.04752 + 13.68620I −2.00699 − 7.52630I
u = 0.45365 − 1.36129I
a = 1.09428 − 1.03938I
b = 0.73653 + 1.65036I
0.04752 − 13.68620I −2.00699 + 7.52630I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.511100 + 0.177219I
a = −0.649618 − 0.329498I
b = 0.434596 + 0.530398I
1.055330 + 0.384583I 8.52259 − 2.33147I
u = 0.511100 − 0.177219I
a = −0.649618 + 0.329498I
b = 0.434596 − 0.530398I
1.055330 − 0.384583I 8.52259 + 2.33147I
u = −0.21189 + 1.50842I
a = −0.299323 − 0.077409I
b = −0.130865 − 0.674935I
−10.59470 − 5.64919I 0.00874 + 7.25798I
u = −0.21189 − 1.50842I
a = −0.299323 + 0.077409I
b = −0.130865 + 0.674935I
−10.59470 + 5.64919I 0.00874 − 7.25798I
u = −0.202297
a = −3.51885
b = −0.401516
−1.31450 −10.7150
6
II.
I
u
2
= h−7.17 × 10
5
u
25
+ 7.81 × 10
5
u
24
+ · · · + 9.64 × 10
5
b + 7.92 × 10
5
, 1.78 ×
10
6
u
25
−1.90×10
6
u
24
+· · · +9.64×10
5
a−1.85×10
6
, u
26
−2u
25
+· · · − 2u +1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
−u
u
3
+ u
a
12
=
u
2
+ 1
−u
4
− 2u
2
a
3
=
−1.84217u
25
+ 1.96870u
24
+ ··· − 7.48746u + 1.91503
0.743523u
25
− 0.809995u
24
+ ··· + 3.02101u − 0.821465
a
5
=
−u
3
− 2u
u
3
+ u
a
2
=
−0.800732u
25
+ 0.943263u
24
+ ··· − 3.90828u + 0.652479
u
3
+ u
a
1
=
−0.679349u
25
+ 0.982915u
24
+ ··· + 5.45700u + 1.72423
0.456267u
25
− 0.655675u
24
+ ··· + 0.0722156u − 1.37578
a
4
=
−2.58569u
25
+ 2.77870u
24
+ ··· − 10.5085u + 2.73650
0.743523u
25
− 0.809995u
24
+ ··· + 3.02101u − 0.821465
a
9
=
1.25686u
25
− 1.88879u
24
+ ··· + 1.53675u − 2.45627
−0.476873u
25
+ 0.609237u
24
+ ··· + 0.104718u + 0.401848
a
8
=
0.624218u
25
− 1.70470u
24
+ ··· + 1.36553u − 1.32065
0.632641u
25
− 0.184085u
24
+ ··· + 2.17122u − 1.13562
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1918707
963947
u
25
−
2455152
963947
u
24
+ ··· −
5103125
963947
u +
1698759
963947
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
13
+ 3u
12
+ ··· + 8u + 1)
2
c
2
, c
4
(u
13
− 3u
12
+ ··· − 2u + 1)
2
c
3
, c
9
(u
13
+ u
12
+ ··· + 4u − 4)
2
c
5
(u
13
− 2u
12
+ ··· + 3u − 1)
2
c
6
, c
7
, c
8
c
10
, c
11
, c
12
u
26
− 2u
25
+ ··· − 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
13
+ 17y
12
+ ··· + 8y −1)
2
c
2
, c
4
(y
13
− 3y
12
+ ··· + 8y −1)
2
c
3
, c
9
(y
13
+ 15y
12
+ ··· − 56y −16)
2
c
5
(y
13
− 16y
12
+ ··· + 5y −1)
2
c
6
, c
7
, c
8
c
10
, c
11
, c
12
y
26
+ 18y
25
+ ··· + 30y
2
+ 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.971054 + 0.087562I
a = 0.20122 − 1.96513I
b = −0.50699 + 1.66583I
4.58598 + 8.60203I 1.58542 − 5.32797I
u = 0.971054 − 0.087562I
a = 0.20122 + 1.96513I
b = −0.50699 − 1.66583I
4.58598 − 8.60203I 1.58542 + 5.32797I
u = −0.166889 + 1.040940I
a = 1.41794 − 1.71859I
b = 0.612460
−4.29290 −6.11820 + 0.I
u = −0.166889 − 1.040940I
a = 1.41794 + 1.71859I
b = 0.612460
−4.29290 −6.11820 + 0.I
u = 0.898765 + 0.068276I
a = −0.51103 − 2.01532I
b = 0.02169 + 1.76519I
5.49041 − 1.38297I 2.93425 + 0.71622I
u = 0.898765 − 0.068276I
a = −0.51103 + 2.01532I
b = 0.02169 − 1.76519I
5.49041 + 1.38297I 2.93425 − 0.71622I
u = −0.705153 + 0.526357I
a = 0.314624 − 0.599897I
b = −0.032142 + 0.650070I
−3.89003 − 2.36301I 2.56487 + 4.19898I
u = −0.705153 − 0.526357I
a = 0.314624 + 0.599897I
b = −0.032142 − 0.650070I
−3.89003 + 2.36301I 2.56487 − 4.19898I
u = −0.063428 + 1.135530I
a = 0.99806 + 1.21295I
b = 0.452299 − 0.637242I
−4.25522 − 0.99909I 0.456384 − 0.581912I
u = −0.063428 − 1.135530I
a = 0.99806 − 1.21295I
b = 0.452299 + 0.637242I
−4.25522 + 0.99909I 0.456384 + 0.581912I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.239526 + 1.122350I
a = −0.582484 − 0.652382I
b = −0.997974 + 0.288600I
−1.68175 + 2.52293I 2.35428 − 4.38707I
u = 0.239526 − 1.122350I
a = −0.582484 + 0.652382I
b = −0.997974 − 0.288600I
−1.68175 − 2.52293I 2.35428 + 4.38707I
u = −0.485725 + 1.232220I
a = 0.968192 − 0.729255I
b = 0.02169 + 1.76519I
5.49041 − 1.38297I 2.93425 + 0.71622I
u = −0.485725 − 1.232220I
a = 0.968192 + 0.729255I
b = 0.02169 − 1.76519I
5.49041 + 1.38297I 2.93425 − 0.71622I
u = 0.527181 + 1.230800I
a = −0.968929 − 0.539477I
b = 0.25689 + 1.55234I
1.07459 − 3.30324I −0.83610 + 2.39821I
u = 0.527181 − 1.230800I
a = −0.968929 + 0.539477I
b = 0.25689 − 1.55234I
1.07459 + 3.30324I −0.83610 − 2.39821I
u = 0.101397 + 1.371440I
a = 0.241803 − 0.465532I
b = −0.032142 − 0.650070I
−3.89003 + 2.36301I 2.56487 − 4.19898I
u = 0.101397 − 1.371440I
a = 0.241803 + 0.465532I
b = −0.032142 + 0.650070I
−3.89003 − 2.36301I 2.56487 + 4.19898I
u = 0.408597 + 1.339370I
a = 1.32284 + 0.50744I
b = 0.25689 − 1.55234I
1.07459 + 3.30324I −0.83610 − 2.39821I
u = 0.408597 − 1.339370I
a = 1.32284 − 0.50744I
b = 0.25689 + 1.55234I
1.07459 − 3.30324I −0.83610 + 2.39821I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.43667 + 1.34910I
a = −1.26249 + 0.83530I
b = −0.50699 − 1.66583I
4.58598 − 8.60203I 1.58542 + 5.32797I
u = −0.43667 − 1.34910I
a = −1.26249 − 0.83530I
b = −0.50699 + 1.66583I
4.58598 + 8.60203I 1.58542 − 5.32797I
u = −0.517741 + 0.054555I
a = 1.276610 − 0.533825I
b = −0.997974 − 0.288600I
−1.68175 − 2.52293I 2.35428 + 4.38707I
u = −0.517741 − 0.054555I
a = 1.276610 + 0.533825I
b = −0.997974 + 0.288600I
−1.68175 + 2.52293I 2.35428 − 4.38707I
u = 0.229089 + 0.294081I
a = 0.08363 − 4.21290I
b = 0.452299 + 0.637242I
−4.25522 + 0.99909I 0.456384 + 0.581912I
u = 0.229089 − 0.294081I
a = 0.08363 + 4.21290I
b = 0.452299 − 0.637242I
−4.25522 − 0.99909I 0.456384 − 0.581912I
12
III. I
u
3
= hb, −u
2
+ 2a − u − 3, u
3
+ 2u − 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
−u
−u + 1
a
12
=
u
2
+ 1
−u
a
3
=
1
2
u
2
+
1
2
u +
3
2
0
a
5
=
−1
−u + 1
a
2
=
1
2
u
2
+
1
2
u +
5
2
u − 1
a
1
=
1
u − 1
a
4
=
1
2
u
2
+
1
2
u +
3
2
0
a
9
=
1
0
a
8
=
u
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =
25
4
u
2
+
11
4
u +
23
4
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
9
u
3
c
4
(u + 1)
3
c
5
u
3
+ 3u
2
+ 5u + 2
c
6
, c
7
, c
8
u
3
+ 2u + 1
c
10
, c
11
, c
12
u
3
+ 2u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
9
y
3
c
5
y
3
+ y
2
+ 13y −4
c
6
, c
7
, c
8
c
10
, c
11
, c
12
y
3
+ 4y
2
+ 4y −1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.22670 + 1.46771I
a = 0.335258 + 0.401127I
b = 0
−11.08570 − 5.13794I −8.01583 − 0.12290I
u = −0.22670 − 1.46771I
a = 0.335258 − 0.401127I
b = 0
−11.08570 + 5.13794I −8.01583 + 0.12290I
u = 0.453398
a = 1.82948
b = 0
−0.857735 8.28170
16
IV. I
u
4
= hb, u
3
+ a + u + 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
−u
u
3
+ u
a
12
=
u
2
+ 1
u
3
+ 2u + 1
a
3
=
−u
3
− u − 1
0
a
5
=
−u
3
− 2u
u
3
+ u
a
2
=
u − 1
−u
3
− u
a
1
=
u
3
+ 2u
−u
3
− u
a
4
=
−u
3
− u − 1
0
a
9
=
1
0
a
8
=
2u
3
+ u
2
+ 3u + 3
−u
3
− u
2
− u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
3
+ 4u − 3
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
9
u
4
c
4
(u + 1)
4
c
5
(u
2
− u + 1)
2
c
6
, c
7
, c
8
u
4
− u
3
+ 2u
2
− 2u + 1
c
10
, c
11
, c
12
u
4
+ u
3
+ 2u
2
+ 2u + 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
9
y
4
c
5
(y
2
+ y + 1)
2
c
6
, c
7
, c
8
c
10
, c
11
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.621744 + 0.440597I
a = −0.500000 − 0.866025I
b = 0
−4.93480 − 2.02988I −5.00000 + 3.46410I
u = −0.621744 − 0.440597I
a = −0.500000 + 0.866025I
b = 0
−4.93480 + 2.02988I −5.00000 − 3.46410I
u = 0.121744 + 1.306620I
a = −0.500000 + 0.866025I
b = 0
−4.93480 + 2.02988I −5.00000 − 3.46410I
u = 0.121744 − 1.306620I
a = −0.500000 − 0.866025I
b = 0
−4.93480 − 2.02988I −5.00000 + 3.46410I
20
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
7
)(u
13
+ 3u
12
+ ··· + 8u + 1)
2
(u
15
+ 4u
14
+ ··· − 127u + 16)
c
2
((u − 1)
7
)(u
13
− 3u
12
+ ··· − 2u + 1)
2
(u
15
− 2u
14
+ ··· − 11u + 4)
c
3
, c
9
u
7
(u
13
+ u
12
+ ··· + 4u − 4)
2
(u
15
− 3u
14
+ ··· + 8u + 32)
c
4
((u + 1)
7
)(u
13
− 3u
12
+ ··· − 2u + 1)
2
(u
15
− 2u
14
+ ··· − 11u + 4)
c
5
((u
2
− u + 1)
2
)(u
3
+ 3u
2
+ 5u + 2)(u
13
− 2u
12
+ ··· + 3u − 1)
2
· (u
15
+ 6u
14
+ ··· + 16u + 4)
c
6
, c
7
, c
8
(u
3
+ 2u + 1)(u
4
− u
3
+ 2u
2
− 2u + 1)(u
15
+ 7u
13
+ ··· + 3u + 1)
· (u
26
− 2u
25
+ ··· − 2u + 1)
c
10
, c
11
, c
12
(u
3
+ 2u − 1)(u
4
+ u
3
+ 2u
2
+ 2u + 1)(u
15
+ 7u
13
+ ··· + 3u + 1)
· (u
26
− 2u
25
+ ··· − 2u + 1)
21
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
7
)(y
13
+ 17y
12
+ ··· + 8y −1)
2
· (y
15
+ 16y
14
+ ··· + 15841y −256)
c
2
, c
4
((y −1)
7
)(y
13
− 3y
12
+ ··· + 8y −1)
2
(y
15
− 4y
14
+ ··· − 127y −16)
c
3
, c
9
y
7
(y
13
+ 15y
12
+ ··· − 56y −16)
2
(y
15
+ 15y
14
+ ··· + 320y −1024)
c
5
((y
2
+ y + 1)
2
)(y
3
+ y
2
+ 13y −4)(y
13
− 16y
12
+ ··· + 5y −1)
2
· (y
15
− 16y
14
+ ··· + 408y −16)
c
6
, c
7
, c
8
c
10
, c
11
, c
12
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)(y
15
+ 14y
14
+ ··· + 27y −1)
· (y
26
+ 18y
25
+ ··· + 30y
2
+ 1)
22
