12a
1027
(K12a
1027
)
A knot diagram
1
Linearized knot diagam
4 7 8 10 11 2 3 12 1 5 6 9
Solving Sequence
8,12
9
1,4
2 10 5 3 7 6 11
c
8
c
12
c
1
c
9
c
4
c
3
c
7
c
6
c
11
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−5.36041 × 10
31
u
46
− 1.13554 × 10
32
u
45
+ ··· + 1.29302 × 10
32
b − 1.94404 × 10
32
,
2.24156 × 10
31
u
46
+ 7.53273 × 10
31
u
45
+ ··· + 3.23255 × 10
31
a + 5.49837 × 10
32
, u
47
+ 3u
46
+ ··· + 11u + 1i
I
u
2
= hb
2
− b − 1, a, u + 1i
I
u
3
= hb
2
+ b − 1, a
2
− 2, u − 1i
* 3 irreducible components of dim
C
= 0, with total 53 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−5.36 × 10
31
u
46
− 1.14 × 10
32
u
45
+ · · · + 1.29 × 10
32
b − 1.94 ×
10
32
, 2.24 × 10
31
u
46
+ 7.53 × 10
31
u
45
+ · · · + 3.23 × 10
31
a + 5.50 ×
10
32
, u
47
+ 3u
46
+ · · · + 11u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
−0.693436u
46
− 2.33028u
45
+ ··· + 51.8733u − 17.0094
0.414565u
46
+ 0.878210u
45
+ ··· − 2.42558u + 1.50349
a
2
=
1.27062u
46
+ 4.66656u
45
+ ··· − 92.6821u + 25.3900
−1.34352u
46
− 2.38910u
45
+ ··· − 1.23584u − 2.87972
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
5
=
−0.269337u
46
− 1.48602u
45
+ ··· + 49.2076u − 15.7275
0.532338u
46
+ 1.05855u
45
+ ··· − 1.02870u + 1.61606
a
3
=
−0.278871u
46
− 1.45207u
45
+ ··· + 49.4477u − 15.5059
0.414565u
46
+ 0.878210u
45
+ ··· − 2.42558u + 1.50349
a
7
=
1.09240u
46
+ 4.31714u
45
+ ··· − 94.1022u + 24.9244
−1.00057u
46
− 1.67329u
45
+ ··· − 1.56609u − 2.90262
a
6
=
−0.741274u
46
− 2.48932u
45
+ ··· + 48.4395u − 16.5010
1.01218u
46
+ 1.69492u
45
+ ··· + 0.702410u + 1.76957
a
11
=
2.45199u
46
+ 6.38937u
45
+ ··· − 77.9752u + 22.6332
0.951398u
46
+ 1.42569u
45
+ ··· + 12.6443u − 1.14744
(ii) Obstruction class = −1
(iii) Cusp Shapes = 0.0277644u
46
− 0.160999u
45
+ ··· − 5.16637u − 2.38003
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
47
− 10u
46
+ ··· − 1252u − 41
c
2
, c
3
, c
6
c
7
u
47
− 2u
46
+ ··· − 10u + 1
c
4
, c
5
, c
10
c
11
u
47
− u
46
+ ··· − 4u + 4
c
8
, c
9
, c
12
u
47
− 3u
46
+ ··· + 11u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
47
+ 18y
46
+ ··· + 1361766y −1681
c
2
, c
3
, c
6
c
7
y
47
− 54y
46
+ ··· + 74y −1
c
4
, c
5
, c
10
c
11
y
47
− 57y
46
+ ··· + 304y −16
c
8
, c
9
, c
12
y
47
− 47y
46
+ ··· + 211y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.384168 + 0.933735I
a = −1.12092 − 1.02399I
b = −1.55340 + 0.17162I
1.07008 + 7.34464I 2.44287 − 4.66692I
u = 0.384168 − 0.933735I
a = −1.12092 + 1.02399I
b = −1.55340 − 0.17162I
1.07008 − 7.34464I 2.44287 + 4.66692I
u = 0.655220 + 0.738260I
a = −0.024512 − 0.284636I
b = 0.394578 + 0.582858I
8.68282 + 0.69167I 7.69697 + 0.39730I
u = 0.655220 − 0.738260I
a = −0.024512 + 0.284636I
b = 0.394578 − 0.582858I
8.68282 − 0.69167I 7.69697 − 0.39730I
u = 0.500977 + 0.840491I
a = 0.619681 + 0.697367I
b = 0.573333 − 0.570047I
8.15872 + 4.64474I 5.91649 − 5.94308I
u = 0.500977 − 0.840491I
a = 0.619681 − 0.697367I
b = 0.573333 + 0.570047I
8.15872 − 4.64474I 5.91649 + 5.94308I
u = 1.05786
a = 1.39495
b = −0.132955
6.53482 14.2750
u = −0.808561 + 0.480294I
a = 0.244914 − 0.563300I
b = 1.53534 − 0.04664I
−5.40245 + 0.60387I 2.86876 − 0.80089I
u = −0.808561 − 0.480294I
a = 0.244914 + 0.563300I
b = 1.53534 + 0.04664I
−5.40245 − 0.60387I 2.86876 + 0.80089I
u = 0.897365 + 0.697135I
a = −0.520554 − 0.385254I
b = −1.47865 − 0.13246I
2.62151 − 1.72568I 4.31042 + 0.20195I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.897365 − 0.697135I
a = −0.520554 + 0.385254I
b = −1.47865 + 0.13246I
2.62151 + 1.72568I 4.31042 − 0.20195I
u = −0.860635
a = −0.263406
b = −0.380700
1.06395 15.0300
u = −0.347438 + 0.773858I
a = 1.05026 − 1.26332I
b = 1.55717 + 0.11542I
−6.84691 − 5.11929I −0.37879 + 6.36237I
u = −0.347438 − 0.773858I
a = 1.05026 + 1.26332I
b = 1.55717 − 0.11542I
−6.84691 + 5.11929I −0.37879 − 6.36237I
u = 1.17129
a = −0.134391
b = −1.65236
−6.66969 8.57260
u = −0.375308 + 0.614572I
a = −0.633040 + 0.852370I
b = −0.568261 − 0.421058I
0.32221 − 3.19990I 3.06029 + 9.19456I
u = −0.375308 − 0.614572I
a = −0.633040 − 0.852370I
b = −0.568261 + 0.421058I
0.32221 + 3.19990I 3.06029 − 9.19456I
u = 1.28718
a = −0.273773
b = 0.949055
2.25442 5.01370
u = 0.337492 + 0.539565I
a = −0.60781 − 1.74002I
b = −1.56840 + 0.05663I
−8.40867 + 1.57482I −4.99229 − 1.00293I
u = 0.337492 − 0.539565I
a = −0.60781 + 1.74002I
b = −1.56840 − 0.05663I
−8.40867 − 1.57482I −4.99229 + 1.00293I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.39429
a = 0.234243
b = 1.70918
0.706458 0
u = −0.480504 + 0.364841I
a = 0.028159 − 0.293507I
b = −0.310058 + 0.354613I
1.036960 − 0.314856I 8.23983 + 0.59404I
u = −0.480504 − 0.364841I
a = 0.028159 + 0.293507I
b = −0.310058 − 0.354613I
1.036960 + 0.314856I 8.23983 − 0.59404I
u = −1.398200 + 0.068490I
a = −0.31034 − 1.47254I
b = 0.490370 + 0.539803I
3.97392 − 1.86699I 0
u = −1.398200 − 0.068490I
a = −0.31034 + 1.47254I
b = 0.490370 − 0.539803I
3.97392 + 1.86699I 0
u = 1.42950 + 0.07034I
a = −1.27663 + 0.73949I
b = 1.44285 − 0.09561I
1.56798 + 0.38836I 0
u = 1.42950 − 0.07034I
a = −1.27663 − 0.73949I
b = 1.44285 + 0.09561I
1.56798 − 0.38836I 0
u = −1.42981 + 0.19455I
a = 0.91947 + 1.50523I
b = −1.52603 − 0.14762I
−2.72276 − 4.28492I 0
u = −1.42981 − 0.19455I
a = 0.91947 − 1.50523I
b = −1.52603 + 0.14762I
−2.72276 + 4.28492I 0
u = 1.44250 + 0.10539I
a = 0.421487 + 1.174390I
b = −0.340123 − 0.621543I
7.07024 + 2.02020I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.44250 − 0.10539I
a = 0.421487 − 1.174390I
b = −0.340123 + 0.621543I
7.07024 − 2.02020I 0
u = 1.45842 + 0.19966I
a = 0.02427 − 1.46022I
b = −0.613332 + 0.585418I
6.27479 + 6.12368I 0
u = 1.45842 − 0.19966I
a = 0.02427 + 1.46022I
b = −0.613332 − 0.585418I
6.27479 − 6.12368I 0
u = 1.46180 + 0.28357I
a = −0.45847 + 1.62265I
b = 1.56761 − 0.18045I
−1.00497 + 8.94411I 0
u = 1.46180 − 0.28357I
a = −0.45847 − 1.62265I
b = 1.56761 + 0.18045I
−1.00497 − 8.94411I 0
u = −1.50841 + 0.35834I
a = 0.14634 + 1.54270I
b = −1.60053 − 0.20896I
7.16899 − 12.03310I 0
u = −1.50841 − 0.35834I
a = 0.14634 − 1.54270I
b = −1.60053 + 0.20896I
7.16899 + 12.03310I 0
u = −1.55108 + 0.09583I
a = 0.399380 − 0.446871I
b = −1.235980 + 0.286169I
11.18770 − 0.40908I 0
u = −1.55108 − 0.09583I
a = 0.399380 + 0.446871I
b = −1.235980 − 0.286169I
11.18770 + 0.40908I 0
u = −1.54096 + 0.29431I
a = 0.118024 − 1.335240I
b = 0.690898 + 0.651096I
14.8277 − 8.8002I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.54096 − 0.29431I
a = 0.118024 + 1.335240I
b = 0.690898 − 0.651096I
14.8277 + 8.8002I 0
u = −1.56276 + 0.21755I
a = −0.336855 + 1.023130I
b = 0.291108 − 0.752857I
16.0174 − 4.1262I 0
u = −1.56276 − 0.21755I
a = −0.336855 − 1.023130I
b = 0.291108 + 0.752857I
16.0174 + 4.1262I 0
u = 0.168164 + 0.305680I
a = 0.47692 + 1.60928I
b = 0.579305 − 0.209601I
−1.053100 + 0.603352I −5.28744 − 1.90691I
u = 0.168164 − 0.305680I
a = 0.47692 − 1.60928I
b = 0.579305 + 0.209601I
−1.053100 − 0.603352I −5.28744 + 1.90691I
u = 0.343456
a = −3.57272
b = −0.759828
4.20368 −1.73630
u = −0.0700122
a = −20.7044
b = 1.61200
−3.93845 −2.00240
9
II. I
u
2
= hb
2
− b − 1, a, u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
−1
a
9
=
1
−1
a
1
=
−1
0
a
4
=
0
b
a
2
=
−1
−b − 1
a
10
=
0
−1
a
5
=
0
b
a
3
=
b
b
a
7
=
−b
−b − 1
a
6
=
0
b
a
11
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
u
2
+ u − 1
c
4
, c
5
, c
10
c
11
u
2
c
6
, c
7
u
2
− u − 1
c
8
, c
9
(u + 1)
2
c
12
(u − 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
6
, c
7
y
2
− 3y + 1
c
4
, c
5
, c
10
c
11
y
2
c
8
, c
9
, c
12
(y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = −0.618034
0.657974 −6.00000
u = −1.00000
a = 0
b = 1.61803
−7.23771 −6.00000
13
III. I
u
3
= hb
2
+ b − 1, a
2
− 2, u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
1
a
9
=
1
−1
a
1
=
1
0
a
4
=
a
b
a
2
=
ba + 1
−b + 1
a
10
=
0
−1
a
5
=
a
b − a
a
3
=
b + a
b
a
7
=
−ba + b
b − 1
a
6
=
−a
−b
a
11
=
−2
−ba + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
(u
2
+ u − 1)
2
c
2
, c
3
(u
2
− u − 1)
2
c
4
, c
5
, c
10
c
11
(u
2
− 2)
2
c
8
, c
9
(u − 1)
4
c
12
(u + 1)
4
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
6
, c
7
(y
2
− 3y + 1)
2
c
4
, c
5
, c
10
c
11
(y −2)
4
c
8
, c
9
, c
12
(y −1)
4
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.41421
b = 0.618034
5.59278 4.00000
u = 1.00000
a = 1.41421
b = −1.61803
−2.30291 4.00000
u = 1.00000
a = −1.41421
b = 0.618034
5.59278 4.00000
u = 1.00000
a = −1.41421
b = −1.61803
−2.30291 4.00000
17
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u − 1)
3
)(u
47
− 10u
46
+ ··· − 1252u − 41)
c
2
, c
3
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
47
− 2u
46
+ ··· − 10u + 1)
c
4
, c
5
, c
10
c
11
u
2
(u
2
− 2)
2
(u
47
− u
46
+ ··· − 4u + 4)
c
6
, c
7
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
47
− 2u
46
+ ··· − 10u + 1)
c
8
, c
9
((u − 1)
4
)(u + 1)
2
(u
47
− 3u
46
+ ··· + 11u − 1)
c
12
((u − 1)
2
)(u + 1)
4
(u
47
− 3u
46
+ ··· + 11u − 1)
18
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
− 3y + 1)
3
)(y
47
+ 18y
46
+ ··· + 1361766y −1681)
c
2
, c
3
, c
6
c
7
((y
2
− 3y + 1)
3
)(y
47
− 54y
46
+ ··· + 74y −1)
c
4
, c
5
, c
10
c
11
y
2
(y −2)
4
(y
47
− 57y
46
+ ··· + 304y −16)
c
8
, c
9
, c
12
((y −1)
6
)(y
47
− 47y
46
+ ··· + 211y −1)
19
