12n
0110
(K12n
0110
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 9 4 12 11 5 7 8 10
Solving Sequence
7,12 4,8
3 6 11 9 5 2 10 1
c
7
c
3
c
6
c
11
c
8
c
5
c
2
c
10
c
12
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h133471u
25
+ 669325u
24
+ ··· + 3665628b − 2077739,
− 1424696u
25
− 7258133u
24
+ ··· + 3665628a + 94549, u
26
+ 5u
25
+ ··· − u + 1i
I
u
2
= hb + u, u
2
+ a − u + 3, u
3
− u
2
+ 2u − 1i
I
u
3
= hb, 2u
2
+ a + u + 4, u
3
+ 2u − 1i
I
u
4
= h−2u
2
a − au − u
2
+ 5b − 3a − 3u + 1, a
2
+ 2u
2
+ a + 2, u
3
− u
2
+ 2u − 1i
I
u
5
= hb, −u
2
+ a − u − 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
* 5 irreducible components of dim
C
= 0, with total 42 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
= h1.33 × 10
5
u
25
+ 6.69 × 10
5
u
24
+ · · · + 3.67 × 10
6
b − 2.08 × 10
6
, −1.42 ×
10
6
u
25
− 7.26 × 10
6
u
24
+ · · · + 3.67 × 10
6
a + 9.45 × 10
4
, u
26
+ 5u
25
+ · · · − u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
0.388664u
25
+ 1.98005u
24
+ ··· + 4.43142u − 0.0257934
−0.0364115u
25
− 0.182595u
24
+ ··· + 0.168887u + 0.566817
a
8
=
1
u
2
a
3
=
0.352252u
25
+ 1.79746u
24
+ ··· + 4.60030u + 0.541023
−0.0364115u
25
− 0.182595u
24
+ ··· + 0.168887u + 0.566817
a
6
=
0.00205667u
25
− 0.185867u
24
+ ··· − 3.15088u + 1.35901
0.207590u
25
+ 1.02161u
24
+ ··· − 0.929535u − 0.00263065
a
11
=
u
u
3
+ u
a
9
=
u
2
+ 1
u
4
+ 2u
2
a
5
=
0.0668167u
25
+ 0.120495u
24
+ ··· − 3.91725u + 1.01430
0.213589u
25
+ 1.06741u
24
+ ··· − 1.08111u + 0.0668167
a
2
=
−0.281381u
25
− 1.19353u
24
+ ··· + 2.43481u + 0.262430
−0.213589u
25
− 1.06741u
24
+ ··· + 1.08111u − 0.0668167
a
10
=
u
3
+ 2u
u
3
+ u
a
1
=
−u
7
− 4u
5
− 4u
3
−u
7
− 3u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
112969
610938
u
25
−
563647
610938
u
24
+ ··· −
59207
1221876
u −
10043435
1221876
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
26
− 3u
25
+ ··· + 284u + 1
c
2
, c
4
u
26
− 11u
25
+ ··· + 6u + 1
c
3
, c
6
u
26
− 4u
25
+ ··· − 64u − 128
c
5
, c
9
u
26
+ 2u
25
+ ··· − 2048u − 512
c
7
, c
8
, c
11
u
26
− 5u
25
+ ··· + u + 1
c
10
u
26
+ 5u
25
+ ··· + 1376u + 292
c
12
u
26
+ u
25
+ ··· − 1131u + 99
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
26
+ 75y
25
+ ··· − 58160y + 1
c
2
, c
4
y
26
+ 3y
25
+ ··· − 284y + 1
c
3
, c
6
y
26
+ 54y
25
+ ··· − 421888y + 16384
c
5
, c
9
y
26
+ 56y
25
+ ··· − 5636096y + 262144
c
7
, c
8
, c
11
y
26
+ 29y
25
+ ··· + 19y + 1
c
10
y
26
+ 33y
25
+ ··· + 2440488y + 85264
c
12
y
26
+ 65y
25
+ ··· + 226431y + 9801
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.922033 + 0.504258I
a = −1.58448 − 0.11744I
b = −0.82029 + 2.30398I
15.8271 + 7.8059I −9.96367 − 4.06326I
u = −0.922033 − 0.504258I
a = −1.58448 + 0.11744I
b = −0.82029 − 2.30398I
15.8271 − 7.8059I −9.96367 + 4.06326I
u = −0.876316 + 0.685407I
a = 0.907139 − 0.269953I
b = −0.52680 − 2.57055I
16.3557 − 1.8854I −9.21556 − 0.32820I
u = −0.876316 − 0.685407I
a = 0.907139 + 0.269953I
b = −0.52680 + 2.57055I
16.3557 + 1.8854I −9.21556 + 0.32820I
u = −0.310950 + 0.788647I
a = −1.030360 − 0.414279I
b = −0.383630 + 1.328970I
3.78741 − 0.69574I −7.19922 + 0.53889I
u = −0.310950 − 0.788647I
a = −1.030360 + 0.414279I
b = −0.383630 − 1.328970I
3.78741 + 0.69574I −7.19922 − 0.53889I
u = 0.120825 + 1.259610I
a = −0.370078 − 0.116328I
b = 0.006697 + 0.465321I
3.03417 − 1.95544I −4.88432 + 3.72797I
u = 0.120825 − 1.259610I
a = −0.370078 + 0.116328I
b = 0.006697 − 0.465321I
3.03417 + 1.95544I −4.88432 − 3.72797I
u = 0.208874 + 1.332040I
a = −2.13409 + 1.95191I
b = −0.471849 − 0.213919I
1.74705 − 2.61908I −28.5726 + 3.9663I
u = 0.208874 − 1.332040I
a = −2.13409 − 1.95191I
b = −0.471849 + 0.213919I
1.74705 + 2.61908I −28.5726 − 3.9663I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.525198 + 0.265240I
a = 1.16923 − 0.91232I
b = 0.181981 − 1.063290I
2.08935 + 3.66090I −6.80246 − 8.91631I
u = −0.525198 − 0.265240I
a = 1.16923 + 0.91232I
b = 0.181981 + 1.063290I
2.08935 − 3.66090I −6.80246 + 8.91631I
u = −0.18444 + 1.43942I
a = 0.520303 + 0.741442I
b = 0.441502 − 0.900551I
7.67139 + 6.20240I −4.64456 − 7.21742I
u = −0.18444 − 1.43942I
a = 0.520303 − 0.741442I
b = 0.441502 + 0.900551I
7.67139 − 6.20240I −4.64456 + 7.21742I
u = 0.521911
a = −6.33214
b = −0.265813
−2.54481 −86.8730
u = 0.14645 + 1.51159I
a = −1.051750 + 0.162879I
b = 1.46948 − 0.48937I
5.01908 − 1.34239I −6.85851 + 0.18617I
u = 0.14645 − 1.51159I
a = −1.051750 − 0.162879I
b = 1.46948 + 0.48937I
5.01908 + 1.34239I −6.85851 − 0.18617I
u = −0.35319 + 1.54442I
a = −1.05184 − 1.89806I
b = −1.06767 + 2.09935I
−17.0555 + 12.4841I −7.58939 − 4.89118I
u = −0.35319 − 1.54442I
a = −1.05184 + 1.89806I
b = −1.06767 − 2.09935I
−17.0555 − 12.4841I −7.58939 + 4.89118I
u = 0.406135
a = −0.606983
b = 0.253631
−0.735338 −13.2750
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.06274 + 1.65934I
a = 0.46673 − 1.82293I
b = −1.39474 + 2.17520I
12.38470 + 0.60705I −6.03823 + 0.I
u = −0.06274 − 1.65934I
a = 0.46673 + 1.82293I
b = −1.39474 − 2.17520I
12.38470 − 0.60705I −6.03823 + 0.I
u = −0.30091 + 1.65456I
a = 1.14317 + 1.77349I
b = −0.09116 − 3.00698I
−15.3762 + 2.6062I −6.76990 − 0.82332I
u = −0.30091 − 1.65456I
a = 1.14317 − 1.77349I
b = −0.09116 + 3.00698I
−15.3762 − 2.6062I −6.76990 + 0.82332I
u = 0.095602 + 0.202458I
a = −0.51442 + 1.84308I
b = 0.662563 + 0.037560I
−0.945468 + 0.077764I −9.88736 + 1.05285I
u = 0.095602 − 0.202458I
a = −0.51442 − 1.84308I
b = 0.662563 − 0.037560I
−0.945468 − 0.077764I −9.88736 − 1.05285I
7
II. I
u
2
= hb + u, u
2
+ a − u + 3, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−u
2
+ u − 3
−u
a
8
=
1
u
2
a
3
=
−u
2
− 3
−u
a
6
=
−u
−u
2
a
11
=
u
u
2
− u + 1
a
9
=
u
2
+ 1
u
2
− u + 1
a
5
=
−u
−u
2
a
2
=
−u
2
− u − 2
−u
2
a
10
=
u
2
+ 1
u
2
− u + 1
a
1
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
+ 5u − 16
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
8
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
4
, c
10
, c
12
u
3
− u
2
+ 1
c
5
, c
9
u
3
c
6
, c
11
u
3
+ u
2
+ 2u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
11
y
3
+ 3y
2
+ 2y −1
c
2
, c
4
, c
10
c
12
y
3
− y
2
+ 2y −1
c
5
, c
9
y
3
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −1.122560 + 0.744862I
b = −0.215080 − 1.307140I
6.04826 − 5.65624I −8.27516 + 4.28659I
u = 0.215080 − 1.307140I
a = −1.122560 − 0.744862I
b = −0.215080 + 1.307140I
6.04826 + 5.65624I −8.27516 − 4.28659I
u = 0.569840
a = −2.75488
b = −0.569840
−2.22691 −14.4500
11
III. I
u
3
= hb, 2u
2
+ a + u + 4, u
3
+ 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
−2u
2
− u − 4
0
a
8
=
1
u
2
a
3
=
−2u
2
− u − 4
0
a
6
=
1
0
a
11
=
u
−u + 1
a
9
=
u
2
+ 1
u
a
5
=
u
−u
2
a
2
=
−2u
2
− 2u − 4
u
2
a
10
=
1
−u + 1
a
1
=
−u
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
2
+ u + 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
6
u
3
c
4
(u + 1)
3
c
5
, c
7
, c
8
u
3
+ 2u − 1
c
9
, c
11
, c
12
u
3
+ 2u + 1
c
10
u
3
+ 3u
2
+ 5u + 2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
6
y
3
c
5
, c
7
, c
8
c
9
, c
11
, c
12
y
3
+ 4y
2
+ 4y −1
c
10
y
3
+ y
2
+ 13y −4
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.22670 + 1.46771I
a = 0.432268 − 0.136798I
b = 0
7.79580 + 5.13794I −4.53505 − 0.52866I
u = −0.22670 − 1.46771I
a = 0.432268 + 0.136798I
b = 0
7.79580 − 5.13794I −4.53505 + 0.52866I
u = 0.453398
a = −4.86454
b = 0
−2.43213 3.07010
15
IV.
I
u
4
= h−2u
2
a − au − u
2
+ 5b − 3a − 3u + 1, a
2
+ 2u
2
+ a + 2, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
a
2
5
u
2
a +
1
5
u
2
+ ··· +
3
5
a −
1
5
a
8
=
1
u
2
a
3
=
2
5
u
2
a +
1
5
u
2
+ ··· +
8
5
a −
1
5
2
5
u
2
a +
1
5
u
2
+ ··· +
3
5
a −
1
5
a
6
=
1
5
u
2
a +
8
5
u
2
+ ··· +
4
5
a +
17
5
−
1
5
u
2
a +
2
5
u
2
+ ··· +
1
5
a +
8
5
a
11
=
u
u
2
− u + 1
a
9
=
u
2
+ 1
u
2
− u + 1
a
5
=
1
5
u
2
a +
8
5
u
2
+ ··· +
4
5
a +
17
5
−
1
5
u
2
a +
2
5
u
2
+ ··· +
1
5
a +
8
5
a
2
=
2u
2
+ a − u + 3
−
1
5
u
2
a +
2
5
u
2
+ ··· +
1
5
a +
8
5
a
10
=
u
2
+ 1
u
2
− u + 1
a
1
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
7
5
u
2
a −
16
5
au −
11
5
u
2
+
12
5
a +
22
5
u −
54
5
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
7
c
8
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
, c
10
, c
12
(u
3
− u
2
+ 1)
2
c
5
, c
9
u
6
c
6
, c
11
(u
3
+ u
2
+ 2u + 1)
2
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
, c
10
c
12
(y
3
− y
2
+ 2y −1)
2
c
5
, c
9
y
6
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = 0.824718 − 0.424452I
b = −0.215080 + 1.307140I
6.04826 −4.97493 − 1.29886I
u = 0.215080 + 1.307140I
a = −1.82472 + 0.42445I
b = −0.569840
1.91067 − 2.82812I −11.4570 + 15.2977I
u = 0.215080 − 1.307140I
a = 0.824718 + 0.424452I
b = −0.215080 − 1.307140I
6.04826 −4.97493 + 1.29886I
u = 0.215080 − 1.307140I
a = −1.82472 − 0.42445I
b = −0.569840
1.91067 + 2.82812I −11.4570 − 15.2977I
u = 0.569840
a = −0.50000 + 1.54901I
b = −0.215080 + 1.307140I
1.91067 + 2.82812I −9.06804 + 0.18883I
u = 0.569840
a = −0.50000 − 1.54901I
b = −0.215080 − 1.307140I
1.91067 − 2.82812I −9.06804 − 0.18883I
19
V. I
u
5
= hb, −u
2
+ a − u − 1, u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
4
=
u
2
+ u + 1
0
a
8
=
1
u
2
a
3
=
u
2
+ u + 1
0
a
6
=
1
0
a
11
=
u
u
3
+ u
a
9
=
u
2
+ 1
−u
3
− 2u − 1
a
5
=
2u
3
+ u
2
+ 3u + 3
u
3
+ u
2
+ u + 2
a
2
=
−2u
3
− 2u − 2
−u
3
− u
2
− u − 2
a
10
=
u
3
+ 2u
u
3
+ u
a
1
=
−2u
3
− u
2
− 3u − 3
−u
3
− u
2
− u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
3
+ 3u
2
− u − 10
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
4
c
3
, c
6
u
4
c
4
(u + 1)
4
c
5
, c
7
, c
8
u
4
+ u
3
+ 2u
2
+ 2u + 1
c
9
, c
11
, c
12
u
4
− u
3
+ 2u
2
− 2u + 1
c
10
(u
2
− u + 1)
2
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
6
y
4
c
5
, c
7
, c
8
c
9
, c
11
, c
12
y
4
+ 3y
3
+ 2y
2
+ 1
c
10
(y
2
+ y + 1)
2
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.621744 + 0.440597I
a = 0.570696 − 0.107280I
b = 0
1.64493 + 2.02988I −8.92268 − 2.50966I
u = −0.621744 − 0.440597I
a = 0.570696 + 0.107280I
b = 0
1.64493 − 2.02988I −8.92268 + 2.50966I
u = 0.121744 + 1.306620I
a = −0.57070 + 1.62477I
b = 0
1.64493 − 2.02988I −14.5773 + 1.8205I
u = 0.121744 − 1.306620I
a = −0.57070 − 1.62477I
b = 0
1.64493 + 2.02988I −14.5773 − 1.8205I
23
VI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
7
)(u
3
− u
2
+ 2u − 1)
3
(u
26
− 3u
25
+ ··· + 284u + 1)
c
2
((u − 1)
7
)(u
3
+ u
2
− 1)
3
(u
26
− 11u
25
+ ··· + 6u + 1)
c
3
u
7
(u
3
− u
2
+ 2u − 1)
3
(u
26
− 4u
25
+ ··· − 64u − 128)
c
4
((u + 1)
7
)(u
3
− u
2
+ 1)
3
(u
26
− 11u
25
+ ··· + 6u + 1)
c
5
u
9
(u
3
+ 2u − 1)(u
4
+ u
3
+ ··· + 2u + 1)(u
26
+ 2u
25
+ ··· − 2048u − 512)
c
6
u
7
(u
3
+ u
2
+ 2u + 1)
3
(u
26
− 4u
25
+ ··· − 64u − 128)
c
7
, c
8
(u
3
+ 2u − 1)(u
3
− u
2
+ 2u − 1)
3
(u
4
+ u
3
+ 2u
2
+ 2u + 1)
· (u
26
− 5u
25
+ ··· + u + 1)
c
9
u
9
(u
3
+ 2u + 1)(u
4
− u
3
+ ··· − 2u + 1)(u
26
+ 2u
25
+ ··· − 2048u − 512)
c
10
(u
2
− u + 1)
2
(u
3
− u
2
+ 1)
3
(u
3
+ 3u
2
+ 5u + 2)
· (u
26
+ 5u
25
+ ··· + 1376u + 292)
c
11
(u
3
+ 2u + 1)(u
3
+ u
2
+ 2u + 1)
3
(u
4
− u
3
+ 2u
2
− 2u + 1)
· (u
26
− 5u
25
+ ··· + u + 1)
c
12
(u
3
+ 2u + 1)(u
3
− u
2
+ 1)
3
(u
4
− u
3
+ 2u
2
− 2u + 1)
· (u
26
+ u
25
+ ··· − 1131u + 99)
24
VII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
7
)(y
3
+ 3y
2
+ 2y −1)
3
(y
26
+ 75y
25
+ ··· − 58160y + 1)
c
2
, c
4
((y −1)
7
)(y
3
− y
2
+ 2y −1)
3
(y
26
+ 3y
25
+ ··· − 284y + 1)
c
3
, c
6
y
7
(y
3
+ 3y
2
+ 2y −1)
3
(y
26
+ 54y
25
+ ··· − 421888y + 16384)
c
5
, c
9
y
9
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
26
+ 56y
25
+ ··· − 5636096y + 262144)
c
7
, c
8
, c
11
(y
3
+ 3y
2
+ 2y −1)
3
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
26
+ 29y
25
+ ··· + 19y + 1)
c
10
(y
2
+ y + 1)
2
(y
3
− y
2
+ 2y −1)
3
(y
3
+ y
2
+ 13y −4)
· (y
26
+ 33y
25
+ ··· + 2440488y + 85264)
c
12
(y
3
− y
2
+ 2y −1)
3
(y
3
+ 4y
2
+ 4y −1)(y
4
+ 3y
3
+ 2y
2
+ 1)
· (y
26
+ 65y
25
+ ··· + 226431y + 9801)
25
