11n
16
(K11n
16
)
A knot diagram
1
Linearized knot diagam
5 1 8 2 3 10 11 4 1 8 7
Solving Sequence
7,11 3,8
4 1 2 10 6 5 9
c
7
c
3
c
11
c
2
c
10
c
6
c
5
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
23
+ 7u
22
+ ··· + 2b − 5u, −3u
23
− 11u
22
+ ··· + 2a + 8, u
24
+ 3u
23
+ ··· − 5u − 1i
I
u
2
= h−u
2
a + b, u
2
a + a
2
+ u
2
+ a − u + 2, u
3
− u
2
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 30 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
=
h2u
23
+7u
22
+· · ·+2b−5u, −3u
23
−11u
22
+· · ·+2a+8, u
24
+3u
23
+· · ·−5u−1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
3
2
u
23
+
11
2
u
22
+ ··· −
33
2
u − 4
−u
23
−
7
2
u
22
+ ··· +
21
2
u
2
+
5
2
u
a
8
=
1
u
2
a
4
=
3
2
u
23
+
9
2
u
22
+ ··· −
25
2
u − 3
−
1
2
u
22
− u
21
+ ··· −
5
2
u − 1
a
1
=
−u
u
a
2
=
2u
23
+ 6u
22
+ ··· −
27
2
u −
7
2
−
3
2
u
23
− 4u
22
+ ··· −
1
2
u −
1
2
a
10
=
u
u
3
+ u
a
6
=
−u
4
− u
2
+ 1
−u
6
− 2u
4
− u
2
a
5
=
1
2
u
23
+
3
2
u
22
+ ··· −
15
2
u + 1
−
1
2
u
22
− u
21
+ ··· +
7
2
u
2
−
1
2
u
a
9
=
u
5
+ 2u
3
+ u
−u
5
− u
3
+ u
a
9
=
u
5
+ 2u
3
+ u
−u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
23
−
9
2
u
22
−
45
2
u
21
−40u
20
−105u
19
−
307
2
u
18
−
527
2
u
17
−325u
16
−
757
2
u
15
−
803
2
u
14
−
589
2
u
13
−
531
2
u
12
−90u
11
−
75
2
u
10
+3u
9
+
129
2
u
8
−35u
7
+
15
2
u
6
−
89
2
u
5
−57u
4
−11u
3
−
67
2
u
2
−
3
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
24
+ 4u
23
+ ··· + 8u + 1
c
2
u
24
+ 16u
23
+ ··· − 16u + 1
c
3
, c
8
u
24
− u
23
+ ··· − 96u − 64
c
5
u
24
− 4u
23
+ ··· + 2u + 1
c
6
u
24
+ 3u
23
+ ··· + u − 1
c
7
, c
10
, c
11
u
24
− 3u
23
+ ··· + 5u − 1
c
9
u
24
− 13u
23
+ ··· − 995u + 563
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
24
+ 16y
23
+ ··· − 16y + 1
c
2
y
24
− 12y
23
+ ··· − 612y + 1
c
3
, c
8
y
24
− 35y
23
+ ··· − 13312y + 4096
c
5
y
24
− 40y
23
+ ··· − 16y + 1
c
6
y
24
− 33y
23
+ ··· − 11y + 1
c
7
, c
10
, c
11
y
24
+ 19y
23
+ ··· − 11y + 1
c
9
y
24
− 53y
23
+ ··· + 17132945y + 316969
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.977245 + 0.071776I
a = 0.150461 + 0.477664I
b = −1.86833 + 0.85025I
−13.8937 + 5.6522I −11.88129 − 3.05170I
u = −0.977245 − 0.071776I
a = 0.150461 − 0.477664I
b = −1.86833 − 0.85025I
−13.8937 − 5.6522I −11.88129 + 3.05170I
u = −0.944342
a = −0.374599
b = 1.57977
−9.62200 −9.45700
u = −0.132356 + 1.101640I
a = 0.224192 − 1.262680I
b = −1.38202 + 0.85732I
2.09684 + 3.39237I −6.49952 − 2.22048I
u = −0.132356 − 1.101640I
a = 0.224192 + 1.262680I
b = −1.38202 − 0.85732I
2.09684 − 3.39237I −6.49952 + 2.22048I
u = 0.369901 + 1.056050I
a = 0.19239 − 1.75435I
b = 0.509437 + 0.688724I
−1.33599 − 3.11324I −9.44737 + 3.66544I
u = 0.369901 − 1.056050I
a = 0.19239 + 1.75435I
b = 0.509437 − 0.688724I
−1.33599 + 3.11324I −9.44737 − 3.66544I
u = 0.023030 + 1.170740I
a = 0.00332 + 1.52505I
b = 0.53462 − 1.45060I
3.34786 − 1.42933I −3.02808 + 3.24576I
u = 0.023030 − 1.170740I
a = 0.00332 − 1.52505I
b = 0.53462 + 1.45060I
3.34786 + 1.42933I −3.02808 − 3.24576I
u = 0.736962 + 0.245534I
a = 0.482708 + 0.172356I
b = 1.141730 − 0.517295I
−3.71809 − 1.00013I −12.92204 + 1.61108I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.736962 − 0.245534I
a = 0.482708 − 0.172356I
b = 1.141730 + 0.517295I
−3.71809 + 1.00013I −12.92204 − 1.61108I
u = 0.149866 + 1.301080I
a = 0.468754 + 1.001100I
b = −0.632329 − 1.109160I
3.26690 − 2.11293I −4.50894 + 4.47286I
u = 0.149866 − 1.301080I
a = 0.468754 − 1.001100I
b = −0.632329 + 1.109160I
3.26690 + 2.11293I −4.50894 − 4.47286I
u = −0.521270 + 1.255460I
a = 1.48874 − 0.97432I
b = −1.105310 − 0.534231I
−10.25040 − 0.34153I −9.34191 − 0.16934I
u = −0.521270 − 1.255460I
a = 1.48874 + 0.97432I
b = −1.105310 + 0.534231I
−10.25040 + 0.34153I −9.34191 + 0.16934I
u = −0.461477 + 1.300210I
a = −0.98110 + 1.48411I
b = 1.37119 − 0.77637I
−5.57948 + 5.01306I −6.18016 − 2.85769I
u = −0.461477 − 1.300210I
a = −0.98110 − 1.48411I
b = 1.37119 + 0.77637I
−5.57948 − 5.01306I −6.18016 + 2.85769I
u = 0.24692 + 1.39353I
a = −1.57305 + 0.00181I
b = 2.02058 − 0.67462I
1.55911 − 4.48321I −9.00126 + 3.05253I
u = 0.24692 − 1.39353I
a = −1.57305 − 0.00181I
b = 2.02058 + 0.67462I
1.55911 + 4.48321I −9.00126 − 3.05253I
u = −0.45937 + 1.35651I
a = 1.05212 − 2.06998I
b = −2.35956 + 1.45257I
−9.4212 + 10.7764I −8.42021 − 5.67335I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.45937 − 1.35651I
a = 1.05212 + 2.06998I
b = −2.35956 − 1.45257I
−9.4212 − 10.7764I −8.42021 + 5.67335I
u = 0.450904
a = 0.305459
b = −0.393799
−0.785516 −12.5270
u = −0.228245 + 0.158994I
a = 0.02603 − 2.87156I
b = −0.322979 − 0.586238I
−0.34648 − 1.75564I −2.27719 + 2.42480I
u = −0.228245 − 0.158994I
a = 0.02603 + 2.87156I
b = −0.322979 + 0.586238I
−0.34648 + 1.75564I −2.27719 − 2.42480I
7
II. I
u
2
= h−u
2
a + b, u
2
a + a
2
+ u
2
+ a − u + 2, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
3
=
a
u
2
a
a
8
=
1
u
2
a
4
=
a
u
2
a
a
1
=
−u
u
a
2
=
−au + 2a
u
2
a + au − a
a
10
=
u
u
2
− u + 1
a
6
=
u
−u
a
5
=
u
2
+ a + u + 1
u
2
a − 2u + 1
a
9
=
1
u
2
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
2
a + 3au − 5u
2
− 4a + 5u − 16
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
3
c
3
, c
8
u
6
c
4
(u
2
− u + 1)
3
c
6
, c
9
(u
3
+ u
2
− 1)
2
c
7
(u
3
− u
2
+ 2u − 1)
2
c
10
, c
11
(u
3
+ u
2
+ 2u + 1)
2
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
3
c
3
, c
8
y
6
c
6
, c
9
(y
3
− y
2
+ 2y −1)
2
c
7
, c
10
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = 0.818128 + 0.292480I
b = −1.52448 − 0.02619I
3.02413 − 0.79824I −6.43615 − 0.68567I
u = 0.215080 + 1.307140I
a = −0.155769 − 0.854759I
b = 0.73956 + 1.33333I
3.02413 − 4.85801I −2.88198 + 6.08229I
u = 0.215080 − 1.307140I
a = 0.818128 − 0.292480I
b = −1.52448 + 0.02619I
3.02413 + 0.79824I −6.43615 + 0.68567I
u = 0.215080 − 1.307140I
a = −0.155769 + 0.854759I
b = 0.73956 − 1.33333I
3.02413 + 4.85801I −2.88198 − 6.08229I
u = 0.569840
a = −0.662359 + 1.147240I
b = −0.215080 + 0.372529I
−1.11345 + 2.02988I −12.18187 − 4.49037I
u = 0.569840
a = −0.662359 − 1.147240I
b = −0.215080 − 0.372529I
−1.11345 − 2.02988I −12.18187 + 4.49037I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
3
)(u
24
+ 4u
23
+ ··· + 8u + 1)
c
2
((u
2
+ u + 1)
3
)(u
24
+ 16u
23
+ ··· − 16u + 1)
c
3
, c
8
u
6
(u
24
− u
23
+ ··· − 96u − 64)
c
4
((u
2
− u + 1)
3
)(u
24
+ 4u
23
+ ··· + 8u + 1)
c
5
((u
2
+ u + 1)
3
)(u
24
− 4u
23
+ ··· + 2u + 1)
c
6
((u
3
+ u
2
− 1)
2
)(u
24
+ 3u
23
+ ··· + u − 1)
c
7
((u
3
− u
2
+ 2u − 1)
2
)(u
24
− 3u
23
+ ··· + 5u − 1)
c
9
((u
3
+ u
2
− 1)
2
)(u
24
− 13u
23
+ ··· − 995u + 563)
c
10
, c
11
((u
3
+ u
2
+ 2u + 1)
2
)(u
24
− 3u
23
+ ··· + 5u − 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
3
)(y
24
+ 16y
23
+ ··· − 16y + 1)
c
2
((y
2
+ y + 1)
3
)(y
24
− 12y
23
+ ··· − 612y + 1)
c
3
, c
8
y
6
(y
24
− 35y
23
+ ··· − 13312y + 4096)
c
5
((y
2
+ y + 1)
3
)(y
24
− 40y
23
+ ··· − 16y + 1)
c
6
((y
3
− y
2
+ 2y −1)
2
)(y
24
− 33y
23
+ ··· − 11y + 1)
c
7
, c
10
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
24
+ 19y
23
+ ··· − 11y + 1)
c
9
((y
3
− y
2
+ 2y −1)
2
)(y
24
− 53y
23
+ ··· + 1.71329 × 10
7
y + 316969)
13
