11n
67
(K11n
67
)
A knot diagram
1
Linearized knot diagam
4 1 9 2 10 9 11 3 6 1 7
Solving Sequence
5,10 2,6
4 1 9 7 3 11 8
c
5
c
4
c
1
c
9
c
6
c
3
c
11
c
7
c
2
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−53523809u
13
+ 113678375u
12
+ ··· + 2227279840b + 1256772587,
− 7347336541u
13
+ 13427745311u
12
+ ··· + 75727514560a + 1617396731,
u
14
− 2u
13
+ ··· + 24u + 17i
I
u
2
= hb + 1, −u
3
+ 2u
2
+ 2a − 3u + 5, u
4
− u
3
+ 3u
2
− 2u + 1i
I
u
3
= h−a
2
u − 2a
2
+ 4au + 5b + 3a − 5, a
3
− 3a
2
u − 2a
2
+ au − a + u − 2, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 24 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−5.35 × 10
7
u
13
+ 1.14 × 10
8
u
12
+ · · · + 2.23 × 10
9
b + 1.26 × 10
9
, −7.35 ×
10
9
u
13
+1.34×10
10
u
12
+· · ·+7.57×10
10
a+1.62×10
9
, u
14
−2u
13
+· · ·+24u+17i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
0.0970233u
13
− 0.177317u
12
+ ··· + 5.72525u − 0.0213581
0.0240310u
13
− 0.0510391u
12
+ ··· + 0.391365u − 0.564263
a
6
=
1
u
2
a
4
=
0.0860286u
13
− 0.162052u
12
+ ··· + 4.32173u + 0.0613420
0.0297616u
13
− 0.0719847u
12
+ ··· + 0.0943418u − 0.681013
a
1
=
0.0390836u
13
− 0.0710568u
12
+ ··· + 3.02865u + 0.931294
0.00745046u
13
− 0.0211156u
12
+ ··· + 0.794129u + 0.282743
a
9
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
3
=
0.0815510u
13
− 0.158251u
12
+ ··· + 3.40648u − 0.102474
0.0311547u
13
− 0.0816865u
12
+ ··· − 0.621089u − 0.757212
a
11
=
0.0118356u
13
− 0.0182548u
12
+ ··· + 1.04576u + 0.329948
−0.00412141u
13
+ 0.0230782u
12
+ ··· + 1.46293u + 0.190664
a
8
=
−0.0299569u
13
+ 0.0755900u
12
+ ··· + 0.453086u + 0.932725
0.00941903u
13
+ 0.0187909u
12
+ ··· + 0.243684u + 0.271269
a
8
=
−0.0299569u
13
+ 0.0755900u
12
+ ··· + 0.453086u + 0.932725
0.00941903u
13
+ 0.0187909u
12
+ ··· + 0.243684u + 0.271269
(ii) Obstruction class = −1
(iii) Cusp Shapes =
159749339
1781823872
u
13
−
841106653
8909119360
u
12
+ ··· −
20311908351
8909119360
u −
30534615889
8909119360
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
14
− 7u
13
+ ··· + 3u + 4
c
2
u
14
− 3u
13
+ ··· − 127u + 16
c
3
, c
8
u
14
+ 8u
13
+ ··· + 80u + 64
c
5
, c
6
, c
9
u
14
− 2u
13
+ ··· + 24u + 17
c
7
, c
11
u
14
− 2u
13
+ ··· − 12u + 17
c
10
u
14
− 4u
13
+ ··· + 2066u + 289
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
14
+ 3y
13
+ ··· + 127y + 16
c
2
y
14
+ 47y
13
+ ··· + 32223y + 256
c
3
, c
8
y
14
− 42y
13
+ ··· + 4864y + 4096
c
5
, c
6
, c
9
y
14
+ 28y
13
+ ··· + 2994y + 289
c
7
, c
11
y
14
− 4y
13
+ ··· + 2066y + 289
c
10
y
14
+ 52y
13
+ ··· + 189758y + 83521
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.739038 + 0.298276I
a = 0.673213 − 0.315821I
b = 0.313142 − 0.702457I
0.02319 + 2.21939I −1.77809 − 3.53992I
u = −0.739038 − 0.298276I
a = 0.673213 + 0.315821I
b = 0.313142 + 0.702457I
0.02319 − 2.21939I −1.77809 + 3.53992I
u = −0.267566 + 0.668739I
a = 0.619498 − 0.223590I
b = −0.172651 + 0.268532I
0.212568 + 1.285480I 1.55268 − 6.08941I
u = −0.267566 − 0.668739I
a = 0.619498 + 0.223590I
b = −0.172651 − 0.268532I
0.212568 − 1.285480I 1.55268 + 6.08941I
u = 0.01822 + 1.41811I
a = 0.361569 + 0.414725I
b = 0.983382 − 0.463084I
5.01039 + 4.24504I 1.99936 − 6.80413I
u = 0.01822 − 1.41811I
a = 0.361569 − 0.414725I
b = 0.983382 + 0.463084I
5.01039 − 4.24504I 1.99936 + 6.80413I
u = 0.120536 + 0.452712I
a = −3.28623 + 1.15613I
b = −1.024040 − 0.163148I
−2.12302 − 0.75753I −7.75042 − 3.06748I
u = 0.120536 − 0.452712I
a = −3.28623 − 1.15613I
b = −1.024040 + 0.163148I
−2.12302 + 0.75753I −7.75042 + 3.06748I
u = 0.60560 + 1.93212I
a = 0.342835 − 1.047120I
b = 1.50068 + 1.04479I
−19.4276 − 10.6503I −1.06301 + 4.03963I
u = 0.60560 − 1.93212I
a = 0.342835 + 1.047120I
b = 1.50068 − 1.04479I
−19.4276 + 10.6503I −1.06301 − 4.03963I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.05779 + 1.83805I
a = 0.405077 − 0.680123I
b = 0.63249 + 1.72109I
9.20003 − 2.06852I 0.364251 + 1.127832I
u = 1.05779 − 1.83805I
a = 0.405077 + 0.680123I
b = 0.63249 − 1.72109I
9.20003 + 2.06852I 0.364251 − 1.127832I
u = 0.20446 + 2.50927I
a = −0.130671 + 0.682293I
b = 1.26699 − 1.74372I
−17.5696 − 0.3825I −0.199766 + 0.045547I
u = 0.20446 − 2.50927I
a = −0.130671 − 0.682293I
b = 1.26699 + 1.74372I
−17.5696 + 0.3825I −0.199766 − 0.045547I
6
II. I
u
2
= hb + 1, −u
3
+ 2u
2
+ 2a − 3u + 5, u
4
− u
3
+ 3u
2
− 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
1
2
u
3
− u
2
+
3
2
u −
5
2
−1
a
6
=
1
u
2
a
4
=
1
2
u
3
− u
2
+
3
2
u −
3
2
−1
a
1
=
−1
0
a
9
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
3
− u
2
+ 2u − 1
a
3
=
1
2
u
3
− u
2
+
3
2
u −
3
2
−1
a
11
=
−u
u
a
8
=
u
u
3
+ u
a
8
=
u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes =
23
4
u
3
−
11
2
u
2
+
59
4
u −
33
4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
c
2
, c
4
(u + 1)
4
c
3
, c
8
u
4
c
5
, c
6
, c
10
u
4
− u
3
+ 3u
2
− 2u + 1
c
7
u
4
− u
3
+ u
2
+ 1
c
9
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
11
u
4
+ u
3
+ u
2
+ 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
8
y
4
c
5
, c
6
, c
9
c
10
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
7
, c
11
y
4
+ y
3
+ 3y
2
+ 2y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.395123 + 0.506844I
a = −1.92796 + 0.41333I
b = −1.00000
−1.85594 − 1.41510I −3.26394 + 5.88934I
u = 0.395123 − 0.506844I
a = −1.92796 − 0.41333I
b = −1.00000
−1.85594 + 1.41510I −3.26394 − 5.88934I
u = 0.10488 + 1.55249I
a = −0.322042 + 0.157780I
b = −1.00000
5.14581 − 3.16396I 2.13894 − 0.11292I
u = 0.10488 − 1.55249I
a = −0.322042 − 0.157780I
b = −1.00000
5.14581 + 3.16396I 2.13894 + 0.11292I
10
III.
I
u
3
= h−a
2
u − 2a
2
+ 4au + 5b + 3a −5, a
3
− 3a
2
u − 2a
2
+ au − a + u −2, u
2
+ 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
a
1
5
a
2
u +
2
5
a
2
−
4
5
au −
3
5
a + 1
a
6
=
1
−1
a
4
=
−
4
5
a
2
u +
2
5
a
2
+
1
5
au −
8
5
a
1
5
a
2
u −
3
5
a
2
+
6
5
au +
7
5
a
a
1
=
u
2
5
a
2
u −
1
5
a
2
+
2
5
au +
4
5
a + 2
a
9
=
u
0
a
7
=
0
−1
a
3
=
−
3
5
a
2
u −
1
5
a
2
+
7
5
au −
1
5
a
1
5
a
2
u −
3
5
a
2
+
6
5
au +
7
5
a
a
11
=
u
2
5
a
2
u +
2
5
au + ··· +
4
5
a + 2
a
8
=
1
1
5
a
2
u −
4
5
au + ··· +
2
5
a
2
+
2
5
a
a
8
=
1
1
5
a
2
u −
4
5
au + ··· +
2
5
a
2
+
2
5
a
(ii) Obstruction class = 1
(iii) Cusp Shapes =
4
5
a
2
u +
8
5
a
2
−
16
5
au −
12
5
a
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
3
+ u
2
− 1)
2
c
2
(u
3
+ u
2
+ 2u + 1)
2
c
3
, c
8
u
6
− 3u
4
+ 2u
2
+ 1
c
4
(u
3
− u
2
+ 1)
2
c
5
, c
6
, c
7
c
9
, c
11
(u
2
+ 1)
3
c
10
(u − 1)
6
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
3
− y
2
+ 2y −1)
2
c
2
(y
3
+ 3y
2
+ 2y −1)
2
c
3
, c
8
(y
3
− 3y
2
+ 2y + 1)
2
c
5
, c
6
, c
7
c
9
, c
11
(y + 1)
6
c
10
(y −1)
6
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = 0.684841 + 1.082500I
b = 0.877439 − 0.744862I
3.02413 + 2.82812I −0.49024 − 2.97945I
u = 1.000000I
a = −0.439718 − 0.407221I
b = 0.877439 + 0.744862I
3.02413 − 2.82812I −0.49024 + 2.97945I
u = 1.000000I
a = 1.75488 + 2.32472I
b = −0.754878
−1.11345 −7.01951 + 0.I
u = − 1.000000I
a = 0.684841 − 1.082500I
b = 0.877439 + 0.744862I
3.02413 − 2.82812I −0.49024 + 2.97945I
u = − 1.000000I
a = −0.439718 + 0.407221I
b = 0.877439 − 0.744862I
3.02413 + 2.82812I −0.49024 − 2.97945I
u = − 1.000000I
a = 1.75488 − 2.32472I
b = −0.754878
−1.11345 −7.01951 + 0.I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
3
+ u
2
− 1)
2
(u
14
− 7u
13
+ ··· + 3u + 4)
c
2
((u + 1)
4
)(u
3
+ u
2
+ 2u + 1)
2
(u
14
− 3u
13
+ ··· − 127u + 16)
c
3
, c
8
u
4
(u
6
− 3u
4
+ 2u
2
+ 1)(u
14
+ 8u
13
+ ··· + 80u + 64)
c
4
((u + 1)
4
)(u
3
− u
2
+ 1)
2
(u
14
− 7u
13
+ ··· + 3u + 4)
c
5
, c
6
((u
2
+ 1)
3
)(u
4
− u
3
+ 3u
2
− 2u + 1)(u
14
− 2u
13
+ ··· + 24u + 17)
c
7
((u
2
+ 1)
3
)(u
4
− u
3
+ u
2
+ 1)(u
14
− 2u
13
+ ··· − 12u + 17)
c
9
((u
2
+ 1)
3
)(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
14
− 2u
13
+ ··· + 24u + 17)
c
10
((u − 1)
6
)(u
4
− u
3
+ 3u
2
− 2u + 1)(u
14
− 4u
13
+ ··· + 2066u + 289)
c
11
((u
2
+ 1)
3
)(u
4
+ u
3
+ u
2
+ 1)(u
14
− 2u
13
+ ··· − 12u + 17)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
3
− y
2
+ 2y −1)
2
(y
14
+ 3y
13
+ ··· + 127y + 16)
c
2
((y −1)
4
)(y
3
+ 3y
2
+ 2y −1)
2
(y
14
+ 47y
13
+ ··· + 32223y + 256)
c
3
, c
8
y
4
(y
3
− 3y
2
+ 2y + 1)
2
(y
14
− 42y
13
+ ··· + 4864y + 4096)
c
5
, c
6
, c
9
((y + 1)
6
)(y
4
+ 5y
3
+ ··· + 2y + 1)(y
14
+ 28y
13
+ ··· + 2994y + 289)
c
7
, c
11
((y + 1)
6
)(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
14
− 4y
13
+ ··· + 2066y + 289)
c
10
(y −1)
6
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
· (y
14
+ 52y
13
+ ··· + 189758y + 83521)
16
