12n
0329
(K12n
0329
)
A knot diagram
1
Linearized knot diagam
3 5 7 10 2 4 3 12 11 5 8 9
Solving Sequence
3,5
2 6
1,11
10 4 7 8 9 12
c
2
c
5
c
1
c
10
c
4
c
6
c
7
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h47u
14
+ 142u
13
+ ··· + 256b + 305, 49u
14
+ 88u
13
+ ··· + 64a + 45, u
15
+ u
14
+ ··· + 4u + 1i
I
u
2
= h−a
4
u − a
4
+ a
3
u + a
3
− 2a
2
u − 2a
2
+ au + b − u − 1, a
5
− a
4
+ 2a
3
− a
2
+ a − 1, u
2
+ 1i
I
u
3
= h10u
5
+ 9u
4
− 4u
3
+ 144u
2
+ 107b − 160u + 346,
92u
5
− 174u
4
+ 648u
3
− 965u
2
+ 1819a + 1310u − 733, u
6
− 3u
5
+ 10u
4
− 14u
3
+ 22u
2
− 10u + 17i
I
u
4
= h2b − 1, a, u + 1i
* 4 irreducible components of dim
C
= 0, with total 32 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
= h47u
14
+ 142u
13
+ · · · + 256b + 305, 49u
14
+ 88u
13
+ · · · + 64a +
45, u
15
+ u
14
+ · · · + 4u + 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
11
=
−0.765625u
14
− 1.37500u
13
+ ··· − 8.10938u − 0.703125
−0.183594u
14
− 0.554688u
13
+ ··· − 6.01953u − 1.19141
a
10
=
−0.765625u
14
− 1.37500u
13
+ ··· − 8.10938u − 0.703125
0.238281u
14
− 0.0546875u
13
+ ··· − 2.81641u − 0.582031
a
4
=
−
1
32
u
14
−
1
32
u
13
+ ··· −
1
32
u + 1
−u
2
a
7
=
1
32
u
13
+
1
32
u
12
+ ··· +
17
8
u +
1
32
u
a
8
=
1
32
u
13
+
1
32
u
12
+ ··· +
25
8
u +
1
32
u
a
9
=
−1.42188u
14
− 1.21875u
13
+ ··· − 2.07813u + 1.29688
−0.230469u
14
− 0.210938u
13
+ ··· − 2.34766u + 0.0429688
a
12
=
−0.0312500u
14
− 0.500000u
13
+ ··· + 0.593750u + 1.15625
−0.0351563u
14
− 0.320313u
13
+ ··· − 3.38672u − 0.589844
(ii) Obstruction class = −1
(iii) Cusp Shapes =
1851
512
u
14
+
499
256
u
13
+ ··· +
2553
512
u +
5749
512
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
+ 27u
14
+ ··· − 10u − 1
c
2
, c
3
, c
5
c
6
, c
7
u
15
− u
14
+ ··· + 4u − 1
c
4
, c
10
u
15
− 3u
14
+ ··· + 18u − 8
c
8
, c
11
, c
12
u
15
+ 2u
14
+ ··· + 13u − 4
c
9
u
15
− 3u
14
+ ··· + 244u − 64
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
− 105y
14
+ ··· + 150y −1
c
2
, c
3
, c
5
c
6
, c
7
y
15
+ 27y
14
+ ··· − 10y −1
c
4
, c
10
y
15
− 3y
14
+ ··· + 244y −64
c
8
, c
11
, c
12
y
15
− 12y
14
+ ··· + 209y −16
c
9
y
15
+ 65y
14
+ ··· + 27664y −4096
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.085190 + 0.639113I
a = −0.187470 − 0.679349I
b = 0.244487 − 0.095273I
3.76428 + 0.87599I 12.27688 − 4.04131I
u = −1.085190 − 0.639113I
a = −0.187470 + 0.679349I
b = 0.244487 + 0.095273I
3.76428 − 0.87599I 12.27688 + 4.04131I
u = −0.171837 + 0.650095I
a = 1.10255 + 1.42436I
b = 0.682015 + 0.663583I
3.90578 − 5.04152I 15.1629 + 7.2560I
u = −0.171837 − 0.650095I
a = 1.10255 − 1.42436I
b = 0.682015 − 0.663583I
3.90578 + 5.04152I 15.1629 − 7.2560I
u = −0.113165 + 0.510319I
a = −1.23811 − 1.27126I
b = −0.714447 − 0.300419I
−1.07843 − 2.01114I 7.99911 + 6.03699I
u = −0.113165 − 0.510319I
a = −1.23811 + 1.27126I
b = −0.714447 + 0.300419I
−1.07843 + 2.01114I 7.99911 − 6.03699I
u = 0.139618 + 0.358203I
a = 1.15825 + 1.37391I
b = 0.824625 − 0.322737I
1.56037 + 0.76584I 8.88156 − 1.45117I
u = 0.139618 − 0.358203I
a = 1.15825 − 1.37391I
b = 0.824625 + 0.322737I
1.56037 − 0.76584I 8.88156 + 1.45117I
u = −0.301931
a = 1.66090
b = 0.268883
0.626145 16.4510
u = 0.61910 + 1.98975I
a = −0.734091 + 0.400986I
b = −2.19546 + 0.22693I
−16.5790 + 11.3907I 8.72673 − 4.72209I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.61910 − 1.98975I
a = −0.734091 − 0.400986I
b = −2.19546 − 0.22693I
−16.5790 − 11.3907I 8.72673 + 4.72209I
u = −0.10073 + 2.21378I
a = −0.558378 + 0.607390I
b = −1.76407 + 0.10317I
−16.6861 − 2.0181I 8.39824 + 0.80099I
u = −0.10073 − 2.21378I
a = −0.558378 − 0.607390I
b = −1.76407 − 0.10317I
−16.6861 + 2.0181I 8.39824 − 0.80099I
u = 0.36316 + 2.28705I
a = 0.626797 − 0.473627I
b = 2.03841 − 0.08740I
18.2203 + 4.7904I 6.45412 − 1.91248I
u = 0.36316 − 2.28705I
a = 0.626797 + 0.473627I
b = 2.03841 + 0.08740I
18.2203 − 4.7904I 6.45412 + 1.91248I
6
II. I
u
2
= h−a
4
u + a
3
u + · · · − 2a
2
− 1, a
5
− a
4
+ 2a
3
− a
2
+ a − 1, u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
−1
a
6
=
u
0
a
1
=
0
−1
a
11
=
a
a
4
u + a
4
− a
3
u − a
3
+ 2a
2
u + 2a
2
− au + u + 1
a
10
=
a
a
4
u + a
4
− a
3
u − a
3
+ 2a
2
u + 2a
2
− au − a + u + 1
a
4
=
−a
2
u
1
a
7
=
−a
2
+ u
−u
a
8
=
−a
2
−u
a
9
=
a
3
+ a
a
4
u + a
4
− a
3
u + 2a
2
u + 2a
2
+ u + 1
a
12
=
a
4
− a
3
+ a
2
+ 1
a
4
u + a
4
− a
3
+ 2a
2
u + 2a
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4a
3
+ 4a
2
− 4a + 8
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
10
c
2
, c
3
, c
5
c
6
, c
7
(u
2
+ 1)
5
c
4
, c
10
u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1
c
8
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
9
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
c
11
, c
12
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y −1)
10
c
2
, c
3
, c
5
c
6
, c
7
(y + 1)
10
c
4
, c
10
(y
5
− 3y
4
+ 4y
3
− y
2
− y + 1)
2
c
8
, c
11
, c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
9
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −0.339110 + 0.822375I
b = 0.271616 − 0.645450I
−2.96077 + 1.53058I 4.51511 − 4.43065I
u = 1.000000I
a = −0.339110 − 0.822375I
b = −1.80694 − 0.21165I
−2.96077 − 1.53058I 4.51511 + 4.43065I
u = 1.000000I
a = 0.766826
b = 2.07090 + 1.30408I
−0.888787 5.48110
u = 1.000000I
a = 0.455697 + 1.200150I
b = 1.46044 + 0.74843I
2.58269 − 4.40083I 8.74431 + 3.49859I
u = 1.000000I
a = 0.455697 − 1.200150I
b = 0.003972 − 0.195404I
2.58269 + 4.40083I 8.74431 − 3.49859I
u = − 1.000000I
a = −0.339110 + 0.822375I
b = −1.80694 + 0.21165I
−2.96077 + 1.53058I 4.51511 − 4.43065I
u = − 1.000000I
a = −0.339110 − 0.822375I
b = 0.271616 + 0.645450I
−2.96077 − 1.53058I 4.51511 + 4.43065I
u = − 1.000000I
a = 0.766826
b = 2.07090 − 1.30408I
−0.888787 5.48110
u = − 1.000000I
a = 0.455697 + 1.200150I
b = 0.003972 + 0.195404I
2.58269 − 4.40083I 8.74431 + 3.49859I
u = − 1.000000I
a = 0.455697 − 1.200150I
b = 1.46044 − 0.74843I
2.58269 + 4.40083I 8.74431 − 3.49859I
10
III. I
u
3
= h10u
5
+ 9u
4
+ · · · + 107b + 346, 92u
5
− 174u
4
+ · · · + 1819a −
733, u
6
− 3u
5
+ 10u
4
− 14u
3
+ 22u
2
− 10u + 17i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
6
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
11
=
−0.0505772u
5
+ 0.0956570u
4
+ ··· − 0.720176u + 0.402969
−0.0934579u
5
− 0.0841121u
4
+ ··· + 1.49533u − 3.23364
a
10
=
−0.0505772u
5
+ 0.0956570u
4
+ ··· − 0.720176u + 0.402969
−0.112150u
5
+ 0.299065u
4
+ ··· + 1.79439u − 2.28037
a
4
=
−0.0170423u
5
− 0.0329852u
4
+ ··· + 0.213854u − 0.483782
−0.0280374u
5
+ 0.0747664u
4
+ ··· − 0.551402u + 2.42991
a
7
=
0.0588235u
5
− 0.176471u
4
+ ··· + 1.29412u − 0.588235
−0.0841121u
5
+ 0.224299u
4
+ ··· − 1.65421u + 0.289720
a
8
=
−0.0252886u
5
+ 0.0478285u
4
+ ··· − 0.360088u − 0.298516
−0.0841121u
5
+ 0.224299u
4
+ ··· − 1.65421u + 0.289720
a
9
=
−0.109951u
5
+ 0.0775151u
4
+ ··· − 0.652556u + 0.136888
−0.149533u
5
+ 0.0654206u
4
+ ··· + 1.39252u − 2.37383
a
12
=
0.0340847u
5
+ 0.0659703u
4
+ ··· − 0.427708u + 0.967565
u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes =
60
107
u
5
−
160
107
u
4
+
404
107
u
3
−
420
107
u
2
+
324
107
u +
1006
107
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
+ 11u
5
+ 60u
4
+ 218u
3
+ 544u
2
+ 648u + 289
c
2
, c
3
, c
5
c
6
, c
7
u
6
+ 3u
5
+ 10u
4
+ 14u
3
+ 22u
2
+ 10u + 17
c
4
, c
10
(u
3
+ u
2
+ 2u + 1)
2
c
8
, c
11
, c
12
(u
3
+ u
2
− 1)
2
c
9
(u
3
+ 3u
2
+ 2u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− y
5
− 108y
4
+ 4078y
3
+ 48088y
2
− 105472y + 83521
c
2
, c
3
, c
5
c
6
, c
7
y
6
+ 11y
5
+ 60y
4
+ 218y
3
+ 544y
2
+ 648y + 289
c
4
, c
10
(y
3
+ 3y
2
+ 2y −1)
2
c
8
, c
11
, c
12
(y
3
− y
2
+ 2y −1)
2
c
9
(y
3
− 5y
2
+ 10y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.162359 + 1.038790I
a = −0.083694 − 0.535481I
b = −2.03980 + 1.82295I
−2.17641 13.01951 + 0.I
u = −0.162359 − 1.038790I
a = −0.083694 + 0.535481I
b = −2.03980 − 1.82295I
−2.17641 13.01951 + 0.I
u = 1.23597 + 1.45071I
a = 0.595267 + 0.358893I
b = 1.109500 + 0.002038I
−6.31400 + 2.82812I 6.49024 − 2.97945I
u = 1.23597 − 1.45071I
a = 0.595267 − 0.358893I
b = 1.109500 − 0.002038I
−6.31400 − 2.82812I 6.49024 + 2.97945I
u = 0.42639 + 2.01299I
a = −0.599808 − 0.233897I
b = −1.56970 − 0.18054I
−6.31400 − 2.82812I 6.49024 + 2.97945I
u = 0.42639 − 2.01299I
a = −0.599808 + 0.233897I
b = −1.56970 + 0.18054I
−6.31400 + 2.82812I 6.49024 − 2.97945I
14
IV. I
u
4
= h2b − 1, a, u + 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
−1
a
2
=
1
1
a
6
=
−1
−2
a
1
=
2
1
a
11
=
0
0.5
a
10
=
0
0.5
a
4
=
0
−1
a
7
=
−1
−1
a
8
=
−2
−1
a
9
=
0
0.5
a
12
=
2
1.5
(ii) Obstruction class = 1
(iii) Cusp Shapes = 9.75
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
8
u + 1
c
4
, c
9
, c
10
u
c
5
, c
6
, c
7
c
11
, c
12
u − 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
6
, c
7
c
8
, c
11
, c
12
y −1
c
4
, c
9
, c
10
y
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0
b = 0.500000
3.28987 9.75000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
10
)(u + 1)(u
6
+ 11u
5
+ ··· + 648u + 289)
· (u
15
+ 27u
14
+ ··· − 10u − 1)
c
2
, c
3
(u + 1)(u
2
+ 1)
5
(u
6
+ 3u
5
+ 10u
4
+ 14u
3
+ 22u
2
+ 10u + 17)
· (u
15
− u
14
+ ··· + 4u − 1)
c
4
, c
10
u(u
3
+ u
2
+ 2u + 1)
2
(u
10
− 3u
8
+ 4u
6
− u
4
− u
2
+ 1)
· (u
15
− 3u
14
+ ··· + 18u − 8)
c
5
, c
6
, c
7
(u − 1)(u
2
+ 1)
5
(u
6
+ 3u
5
+ 10u
4
+ 14u
3
+ 22u
2
+ 10u + 17)
· (u
15
− u
14
+ ··· + 4u − 1)
c
8
(u + 1)(u
3
+ u
2
− 1)
2
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
· (u
15
+ 2u
14
+ ··· + 13u − 4)
c
9
u(u
3
+ 3u
2
+ 2u − 1)
2
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
· (u
15
− 3u
14
+ ··· + 244u − 64)
c
11
, c
12
(u − 1)(u
3
+ u
2
− 1)
2
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
· (u
15
+ 2u
14
+ ··· + 13u − 4)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
11
)(y
6
− y
5
+ ··· − 105472y + 83521)
· (y
15
− 105y
14
+ ··· + 150y −1)
c
2
, c
3
, c
5
c
6
, c
7
(y −1)(y + 1)
10
(y
6
+ 11y
5
+ ··· + 648y + 289)
· (y
15
+ 27y
14
+ ··· − 10y −1)
c
4
, c
10
y(y
3
+ 3y
2
+ 2y −1)
2
(y
5
− 3y
4
+ 4y
3
− y
2
− y + 1)
2
· (y
15
− 3y
14
+ ··· + 244y −64)
c
8
, c
11
, c
12
(y −1)(y
3
− y
2
+ 2y −1)
2
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
· (y
15
− 12y
14
+ ··· + 209y −16)
c
9
y(y
3
− 5y
2
+ 10y −1)
2
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
· (y
15
+ 65y
14
+ ··· + 27664y −4096)
20
