12n
0592
(K12n
0592
)
A knot diagram
1
Linearized knot diagam
3 7 10 7 11 2 4 12 1 6 4 9
Solving Sequence
2,7 3,10
4 1 6 11 5 9 12 8
c
2
c
3
c
1
c
6
c
10
c
5
c
9
c
12
c
8
c
4
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−13u
16
54u
15
+ ··· + 4b 36, 9u
16
28u
15
+ ··· + 8a + 20, u
17
+ 6u
16
+ ··· + 56u + 8i
I
u
2
= h3u
10
8u
9
+ 3u
8
+ 16u
7
11u
6
21u
5
+ 21u
4
+ 16u
3
17u
2
+ b 6u + 6,
6u
10
15u
9
+ 4u
8
+ 33u
7
20u
6
41u
5
+ 39u
4
+ 33u
3
32u
2
+ a 11u + 12,
u
11
3u
10
+ 2u
9
+ 5u
8
6u
7
5u
6
+ 10u
5
+ 2u
4
8u
3
+ u
2
+ 3u 1i
I
u
3
= h−37a
5
u
2
+ 123a
4
u
2
+ ··· 100a + 168, a
5
u
2
3a
4
u
2
+ ··· + 13a + 20, u
3
u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 46 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−13u
16
54u
15
+ · · · + 4b 36, 9u
16
28u
15
+ · · · + 8a +
20, u
17
+ 6u
16
+ · · · + 56u + 8i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
10
=
9
8
u
16
+
7
2
u
15
+ ···
29
2
u
5
2
13
4
u
16
+
27
2
u
15
+ ··· +
131
2
u + 9
a
4
=
1
4
u
16
+ u
15
+ ··· +
3
2
u +
1
2
1
2
u
16
+
5
2
u
15
+ ··· +
29
2
u + 2
a
1
=
u
2
+ 1
u
4
a
6
=
u
u
a
11
=
33
8
u
16
+
39
2
u
15
+ ··· +
245
2
u +
39
2
25
4
u
16
+
59
2
u
15
+ ··· +
405
2
u + 31
a
5
=
1
4
u
16
+ u
15
+ ··· +
3
2
u +
1
2
1
2
u
15
2u
14
+ ···
23
2
u 2
a
9
=
9
8
u
16
4u
15
+ ··· + 12u +
9
2
15
4
u
16
29
2
u
15
+ ···
29
2
u + 1
a
12
=
9
4
u
16
+ 12u
15
+ ··· +
201
2
u +
35
2
2u
16
+
25
2
u
15
+ ··· +
311
2
u + 26
a
8
=
11
4
u
16
59
4
u
15
+ ···
429
4
u 16
9
4
u
16
14u
15
+ ··· 141u 22
(ii) Obstruction class = 1
(iii) Cusp Shapes = 9u
16
52u
15
116u
14
68u
13
+ 190u
12
+ 347u
11
140u
10
1029u
9
1025u
8
+ 472u
7
+ 1947u
6
+ 1504u
5
415u
4
1702u
3
1433u
2
576u 82
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
+ 6u
16
+ ··· + 480u + 64
c
2
, c
6
u
17
6u
16
+ ··· + 56u 8
c
3
, c
5
, c
10
u
17
u
15
+ ··· + 3u 1
c
4
, c
7
u
17
+ 4u
16
+ ··· + 8u 1
c
8
, c
9
, c
12
u
17
7u
16
+ ··· + 20u 8
c
11
u
17
2u
16
+ ··· + u 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
+ 22y
16
+ ··· + 66048y 4096
c
2
, c
6
y
17
6y
16
+ ··· + 480y 64
c
3
, c
5
, c
10
y
17
2y
16
+ ··· + 3y 1
c
4
, c
7
y
17
36y
16
+ ··· + 102y 1
c
8
, c
9
, c
12
y
17
21y
16
+ ··· + 272y 64
c
11
y
17
44y
16
+ ··· + 33y 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.582631 + 0.962982I
a = 0.764828 + 0.649431I
b = 0.179778 + 1.114890I
6.08323 3.07842I 10.91849 + 3.36026I
u = 0.582631 0.962982I
a = 0.764828 0.649431I
b = 0.179778 1.114890I
6.08323 + 3.07842I 10.91849 3.36026I
u = 0.998116 + 0.574654I
a = 1.30029 + 0.75173I
b = 0.86585 + 1.49753I
0.04390 + 4.58131I 7.64977 6.92778I
u = 0.998116 0.574654I
a = 1.30029 0.75173I
b = 0.86585 1.49753I
0.04390 4.58131I 7.64977 + 6.92778I
u = 1.139290 + 0.263587I
a = 0.072389 + 0.255610I
b = 0.015097 0.310294I
2.02646 1.69138I 0.73300 + 3.24851I
u = 1.139290 0.263587I
a = 0.072389 0.255610I
b = 0.015097 + 0.310294I
2.02646 + 1.69138I 0.73300 3.24851I
u = 0.612038 + 0.494740I
a = 1.36351 0.44192I
b = 0.615883 0.945055I
1.133960 0.080801I 10.62347 + 1.71983I
u = 0.612038 0.494740I
a = 1.36351 + 0.44192I
b = 0.615883 + 0.945055I
1.133960 + 0.080801I 10.62347 1.71983I
u = 0.710496 + 1.098260I
a = 0.517258 0.878567I
b = 0.597387 1.192300I
16.3616 5.2854I 10.20532 + 2.53472I
u = 0.710496 1.098260I
a = 0.517258 + 0.878567I
b = 0.597387 + 1.192300I
16.3616 + 5.2854I 10.20532 2.53472I
5
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.105090 + 0.729938I
a = 1.30956 0.56888I
b = 1.03194 1.58456I
4.45386 + 9.24930I 8.34005 7.37887I
u = 1.105090 0.729938I
a = 1.30956 + 0.56888I
b = 1.03194 + 1.58456I
4.45386 9.24930I 8.34005 + 7.37887I
u = 1.126030 + 0.831805I
a = 1.43522 + 0.37456I
b = 1.30455 + 1.61559I
14.9808 + 12.2193I 8.76311 5.87051I
u = 1.126030 0.831805I
a = 1.43522 0.37456I
b = 1.30455 1.61559I
14.9808 12.2193I 8.76311 + 5.87051I
u = 1.22314 + 0.77434I
a = 0.144243 0.159249I
b = 0.053117 + 0.306476I
4.66481 3.72406I 7.31637 0.66089I
u = 1.22314 0.77434I
a = 0.144243 + 0.159249I
b = 0.053117 0.306476I
4.66481 + 3.72406I 7.31637 + 0.66089I
u = 0.456042
a = 1.76372
b = 0.804333
0.900511 10.9010
6
II.
I
u
2
= h3u
10
8u
9
+· · ·+b+6, 6u
10
15u
9
+· · ·+a+12, u
11
3u
10
+· · ·+3u1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
10
=
6u
10
+ 15u
9
+ ··· + 11u 12
3u
10
+ 8u
9
+ ··· + 6u 6
a
4
=
3u
10
+ 8u
9
+ ··· + 5u 9
u
10
+ 3u
9
2u
8
5u
7
+ 6u
6
+ 5u
5
10u
4
2u
3
+ 9u
2
u 3
a
1
=
u
2
+ 1
u
4
a
6
=
u
u
a
11
=
5u
10
+ 13u
9
+ ··· + 8u 10
2u
10
+ 6u
9
4u
8
9u
7
+ 9u
6
+ 12u
5
16u
4
8u
3
+ 12u
2
+ 3u 4
a
5
=
3u
10
+ 8u
9
+ ··· + 5u 9
u
10
+ 3u
9
2u
8
5u
7
+ 6u
6
+ 5u
5
10u
4
2u
3
+ 8u
2
u 2
a
9
=
4u
10
+ 10u
9
+ ··· + 7u 8
3u
10
+ 8u
9
+ ··· + 6u 6
a
12
=
4u
10
10u
9
+ ··· 4u + 11
3u
10
7u
9
+ u
8
+ 16u
7
8u
6
19u
5
+ 17u
4
+ 14u
3
13u
2
3u + 6
a
8
=
8u
10
21u
9
+ ··· 13u + 19
3u
10
8u
9
+ ··· 5u + 9
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 7u
10
19u
9
+ 13u
8
+ 28u
7
27u
6
29u
5
+ 51u
4
+ 18u
3
28u
2
+ u + 16
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
11
5u
10
+ ··· + 11u 1
c
2
u
11
3u
10
+ 2u
9
+ 5u
8
6u
7
5u
6
+ 10u
5
+ 2u
4
8u
3
+ u
2
+ 3u 1
c
3
, c
10
u
11
+ 4u
9
+ u
8
+ 4u
7
+ 4u
6
u
5
+ 5u
4
2u
3
+ 3u
2
u + 1
c
4
u
11
+ 3u
9
+ 6u
8
+ u
7
+ 20u
6
4u
5
+ 26u
4
6u
3
+ 15u
2
4u + 3
c
5
u
11
+ 4u
9
u
8
+ 4u
7
4u
6
u
5
5u
4
2u
3
3u
2
u 1
c
6
u
11
+ 3u
10
+ 2u
9
5u
8
6u
7
+ 5u
6
+ 10u
5
2u
4
8u
3
u
2
+ 3u + 1
c
7
u
11
+ 3u
9
6u
8
+ u
7
20u
6
4u
5
26u
4
6u
3
15u
2
4u 3
c
8
, c
9
u
11
8u
9
u
8
+ 23u
7
+ 5u
6
28u
5
7u
4
+ 13u
3
+ 2u
2
2u 1
c
11
u
11
+ 7u
9
22u
8
+ 7u
7
56u
6
3u
5
45u
4
6u
3
12u
2
u 1
c
12
u
11
8u
9
+ u
8
+ 23u
7
5u
6
28u
5
+ 7u
4
+ 13u
3
2u
2
2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
11
+ 19y
10
+ ··· + 31y 1
c
2
, c
6
y
11
5y
10
+ ··· + 11y 1
c
3
, c
5
, c
10
y
11
+ 8y
10
+ ··· 5y 1
c
4
, c
7
y
11
+ 6y
10
+ ··· 74y 9
c
8
, c
9
, c
12
y
11
16y
10
+ ··· + 8y 1
c
11
y
11
+ 14y
10
+ ··· 23y 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
1(vol +
1CS) Cusp shape
u = 0.856562 + 0.586236I
a = 0.588949 + 0.335331I
b = 0.307889 0.632495I
2.23625 + 2.32410I 6.78884 2.44901I
u = 0.856562 0.586236I
a = 0.588949 0.335331I
b = 0.307889 + 0.632495I
2.23625 2.32410I 6.78884 + 2.44901I
u = 0.855558 + 0.209235I
a = 0.357696 0.809834I
b = 0.136583 + 0.767702I
5.34223 + 0.90366I 2.99552 7.97302I
u = 0.855558 0.209235I
a = 0.357696 + 0.809834I
b = 0.136583 0.767702I
5.34223 0.90366I 2.99552 + 7.97302I
u = 0.736045 + 0.353997I
a = 1.99508 0.41979I
b = 1.61707 + 0.39727I
0.857139 1.116710I 8.63274 + 6.10960I
u = 0.736045 0.353997I
a = 1.99508 + 0.41979I
b = 1.61707 0.39727I
0.857139 + 1.116710I 8.63274 6.10960I
u = 1.011880 + 0.753500I
a = 0.801625 + 0.134986I
b = 0.912860 0.467435I
1.90477 3.17083I 2.15194 + 8.40607I
u = 1.011880 0.753500I
a = 0.801625 0.134986I
b = 0.912860 + 0.467435I
1.90477 + 3.17083I 2.15194 8.40607I
u = 0.419548
a = 4.82997
b = 2.02640
12.4935 13.9460
u = 1.25442 + 1.05470I
a = 0.452785 0.066086I
b = 0.637684 + 0.394653I
4.48658 4.37367I 3.76171 + 10.00302I
10
Solutions to I
u
2
1(vol +
1CS) Cusp shape
u = 1.25442 1.05470I
a = 0.452785 + 0.066086I
b = 0.637684 0.394653I
4.48658 + 4.37367I 3.76171 10.00302I
11
III. I
u
3
=
h−37a
5
u
2
+123a
4
u
2
+· · ·100a+168, a
5
u
2
3a
4
u
2
+· · ·+13a+20, u
3
u
2
+1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
10
=
a
0.110778a
5
u
2
0.368263a
4
u
2
+ ··· + 0.299401a 0.502994
a
4
=
0.182635a
5
u
2
0.323353a
4
u
2
+ ··· 0.371257a + 0.143713
0.365269a
5
u
2
0.646707a
4
u
2
+ ··· 0.742515a + 0.287425
a
1
=
u
2
+ 1
u
2
+ u + 1
a
6
=
u
u
a
11
=
0.434132a
5
u
2
+ 0.0838323a
4
u
2
+ ··· 0.718563a 0.592814
0.544910a
5
u
2
0.284431a
4
u
2
+ ··· 1.41916a 1.09581
a
5
=
0.182635a
5
u
2
0.323353a
4
u
2
+ ··· 0.371257a + 0.143713
0.703593a
5
u
2
1.12275a
4
u
2
+ ··· 0.733533a + 0.832335
a
9
=
0.799401a
5
u
2
0.0628743a
4
u
2
+ ··· 0.461078a 0.305389
0.868263a
5
u
2
+ 0.167665a
4
u
2
+ ··· 1.43713a 1.18563
a
12
=
1.00898a
5
u
2
+ 1.55689a
4
u
2
+ ··· + 0.583832a 0.580838
0.853293a
5
u
2
+ 1.40419a
4
u
2
+ ··· + 0.964072a 0.179641
a
8
=
0.407186a
5
u
2
+ 1.24551a
4
u
2
+ ··· + 0.467066a 0.664671
0.736527a
5
u
2
+ 1.66467a
4
u
2
+ ··· + 0.874251a 0.628743
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u + 6
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
3
+ u
2
+ 2u + 1)
6
c
2
, c
6
(u
3
+ u
2
1)
6
c
3
, c
5
, c
10
u
18
+ u
17
+ ··· + 4u 8
c
4
, c
7
u
18
+ u
17
+ ··· 684u 216
c
8
, c
9
, c
12
(u
3
+ u
2
2u 1)
6
c
11
u
18
u
17
+ ··· 196u 392
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
3
+ 3y
2
+ 2y 1)
6
c
2
, c
6
(y
3
y
2
+ 2y 1)
6
c
3
, c
5
, c
10
y
18
+ 3y
17
+ ··· 80y + 64
c
4
, c
7
y
18
21y
17
+ ··· + 55728y + 46656
c
8
, c
9
, c
12
(y
3
5y
2
+ 6y 1)
6
c
11
y
18
33y
17
+ ··· 16464y + 153664
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
1(vol +
1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.961970 0.199906I
b = 0.842708 + 0.290892I
1.20570 2.82812I 9.50976 + 2.97945I
u = 0.877439 + 0.744862I
a = 0.641730 0.729183I
b = 0.62835 2.13101I
15.7136 2.8281I 9.50976 + 2.97945I
u = 0.877439 + 0.744862I
a = 0.721738 + 0.281163I
b = 0.992973 0.541129I
1.20570 2.82812I 9.50976 + 2.97945I
u = 0.877439 + 0.744862I
a = 1.189520 0.508099I
b = 0.244232 0.630700I
4.43407 2.82812I 9.50976 + 2.97945I
u = 0.877439 + 0.744862I
a = 0.516399 + 0.280423I
b = 0.66526 + 1.33185I
4.43407 2.82812I 9.50976 + 2.97945I
u = 0.877439 + 0.744862I
a = 1.61441 + 1.05819I
b = 0.019938 + 1.117810I
15.7136 2.8281I 9.50976 + 2.97945I
u = 0.877439 0.744862I
a = 0.961970 + 0.199906I
b = 0.842708 0.290892I
1.20570 + 2.82812I 9.50976 2.97945I
u = 0.877439 0.744862I
a = 0.641730 + 0.729183I
b = 0.62835 + 2.13101I
15.7136 + 2.8281I 9.50976 2.97945I
u = 0.877439 0.744862I
a = 0.721738 0.281163I
b = 0.992973 + 0.541129I
1.20570 + 2.82812I 9.50976 2.97945I
u = 0.877439 0.744862I
a = 1.189520 + 0.508099I
b = 0.244232 + 0.630700I
4.43407 + 2.82812I 9.50976 2.97945I
15
Solutions to I
u
3
1(vol +
1CS) Cusp shape
u = 0.877439 0.744862I
a = 0.516399 0.280423I
b = 0.66526 1.33185I
4.43407 + 2.82812I 9.50976 2.97945I
u = 0.877439 0.744862I
a = 1.61441 1.05819I
b = 0.019938 1.117810I
15.7136 + 2.8281I 9.50976 2.97945I
u = 0.754878
a = 0.685274 + 1.096670I
b = 0.517298 0.827854I
5.34329 2.98050
u = 0.754878
a = 0.685274 1.096670I
b = 0.517298 + 0.827854I
5.34329 2.98050
u = 0.754878
a = 1.95258
b = 2.71504
11.5760 2.98050
u = 0.754878
a = 1.92010 + 0.60556I
b = 1.44944 0.45713I
0.296489 2.98050
u = 0.754878
a = 1.92010 0.60556I
b = 1.44944 + 0.45713I
0.296489 2.98050
u = 0.754878
a = 3.59666
b = 1.47396
11.5760 2.98050
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
+ u
2
+ 2u + 1)
6
)(u
11
5u
10
+ ··· + 11u 1)
· (u
17
+ 6u
16
+ ··· + 480u + 64)
c
2
(u
3
+ u
2
1)
6
· (u
11
3u
10
+ 2u
9
+ 5u
8
6u
7
5u
6
+ 10u
5
+ 2u
4
8u
3
+ u
2
+ 3u 1)
· (u
17
6u
16
+ ··· + 56u 8)
c
3
, c
10
(u
11
+ 4u
9
+ u
8
+ 4u
7
+ 4u
6
u
5
+ 5u
4
2u
3
+ 3u
2
u + 1)
· (u
17
u
15
+ ··· + 3u 1)(u
18
+ u
17
+ ··· + 4u 8)
c
4
(u
11
+ 3u
9
+ 6u
8
+ u
7
+ 20u
6
4u
5
+ 26u
4
6u
3
+ 15u
2
4u + 3)
· (u
17
+ 4u
16
+ ··· + 8u 1)(u
18
+ u
17
+ ··· 684u 216)
c
5
(u
11
+ 4u
9
u
8
+ 4u
7
4u
6
u
5
5u
4
2u
3
3u
2
u 1)
· (u
17
u
15
+ ··· + 3u 1)(u
18
+ u
17
+ ··· + 4u 8)
c
6
(u
3
+ u
2
1)
6
· (u
11
+ 3u
10
+ 2u
9
5u
8
6u
7
+ 5u
6
+ 10u
5
2u
4
8u
3
u
2
+ 3u + 1)
· (u
17
6u
16
+ ··· + 56u 8)
c
7
(u
11
+ 3u
9
6u
8
+ u
7
20u
6
4u
5
26u
4
6u
3
15u
2
4u 3)
· (u
17
+ 4u
16
+ ··· + 8u 1)(u
18
+ u
17
+ ··· 684u 216)
c
8
, c
9
(u
3
+ u
2
2u 1)
6
· (u
11
8u
9
u
8
+ 23u
7
+ 5u
6
28u
5
7u
4
+ 13u
3
+ 2u
2
2u 1)
· (u
17
7u
16
+ ··· + 20u 8)
c
11
(u
11
+ 7u
9
22u
8
+ 7u
7
56u
6
3u
5
45u
4
6u
3
12u
2
u 1)
· (u
17
2u
16
+ ··· + u 1)(u
18
u
17
+ ··· 196u 392)
c
12
(u
3
+ u
2
2u 1)
6
· (u
11
8u
9
+ u
8
+ 23u
7
5u
6
28u
5
+ 7u
4
+ 13u
3
2u
2
2u + 1)
· (u
17
7u
16
+ ··· + 20u 8)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
+ 3y
2
+ 2y 1)
6
)(y
11
+ 19y
10
+ ··· + 31y 1)
· (y
17
+ 22y
16
+ ··· + 66048y 4096)
c
2
, c
6
((y
3
y
2
+ 2y 1)
6
)(y
11
5y
10
+ ··· + 11y 1)
· (y
17
6y
16
+ ··· + 480y 64)
c
3
, c
5
, c
10
(y
11
+ 8y
10
+ ··· 5y 1)(y
17
2y
16
+ ··· + 3y 1)
· (y
18
+ 3y
17
+ ··· 80y + 64)
c
4
, c
7
(y
11
+ 6y
10
+ ··· 74y 9)(y
17
36y
16
+ ··· + 102y 1)
· (y
18
21y
17
+ ··· + 55728y + 46656)
c
8
, c
9
, c
12
((y
3
5y
2
+ 6y 1)
6
)(y
11
16y
10
+ ··· + 8y 1)
· (y
17
21y
16
+ ··· + 272y 64)
c
11
(y
11
+ 14y
10
+ ··· 23y 1)(y
17
44y
16
+ ··· + 33y 1)
· (y
18
33y
17
+ ··· 16464y + 153664)
18