11n
68
(K11n
68
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 9 11 4 6 5 7 10
Solving Sequence
6,11 4,7
8 9 3 5 10 1 2
c
6
c
7
c
8
c
3
c
5
c
10
c
11
c
2
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h77815u
25
− 80433u
24
+ ··· + 101496b + 29131,
2049367u
25
+ 17170719u
24
+ ··· + 12687000a + 47575387, u
26
− 2u
25
+ ··· − 2u + 1i
I
u
2
= h−u
3
+ 2b + u + 1, −u
3
− 2u
2
+ 2a − 3u − 1, u
4
+ u
3
+ u
2
+ 1i
I
u
3
= hu
8
− u
7
+ 2u
6
− u
4
+ u
3
− u
2
+ b − u, u
7
+ u
6
+ 2u
5
+ 4u
4
+ 3u
3
+ 3u
2
+ a + 3u + 1,
u
9
+ 3u
7
+ 3u
6
+ 3u
5
+ 6u
4
+ 3u
3
+ 3u
2
+ 2u − 1i
* 3 irreducible components of dim
C
= 0, with total 39 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
= h77815u
25
− 80433u
24
+ · · · + 101496b + 29131, 2.05 × 10
6
u
25
+
1.72 × 10
7
u
24
+ · · · + 1.27 × 10
7
a + 4.76 × 10
7
, u
26
− 2u
25
+ · · · − 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
4
=
−0.161533u
25
− 1.35341u
24
+ ··· + 4.59237u − 3.74993
−0.766680u
25
+ 0.792475u
24
+ ··· − 0.209299u − 0.287016
a
7
=
1
u
2
a
8
=
−4.34995u
25
+ 8.30824u
24
+ ··· − 11.2562u + 4.44452
0.111927u
25
+ 1.17301u
24
+ ··· − 2.00774u + 1.43931
a
9
=
−4.23802u
25
+ 9.48125u
24
+ ··· − 13.2639u + 5.88383
0.111927u
25
+ 1.17301u
24
+ ··· − 2.00774u + 1.43931
a
3
=
0.236517u
25
− 2.54906u
24
+ ··· + 7.57449u − 5.71342
−1.07634u
25
+ 1.55091u
24
+ ··· − 1.40645u + 0.112537
a
5
=
−2.44452u
25
+ 1.53910u
24
+ ··· + 0.243519u − 3.36715
−1.39686u
25
+ 1.36224u
24
+ ··· − 1.66316u − 0.888073
a
10
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
2
=
−0.889032u
25
− 0.919202u
24
+ ··· + 5.34807u − 4.99096
−1.17523u
25
+ 0.972343u
24
+ ··· + 0.156712u − 0.961125
a
2
=
−0.889032u
25
− 0.919202u
24
+ ··· + 5.34807u − 4.99096
−1.17523u
25
+ 0.972343u
24
+ ··· + 0.156712u − 0.961125
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
48418177
8458000
u
25
+
48940111
8458000
u
24
+ ··· −
705789
8458000
u −
61013797
8458000
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
26
− 2u
25
+ ··· − 35u + 4
c
2
u
26
+ 10u
25
+ ··· + 481u + 16
c
3
, c
7
u
26
− 2u
25
+ ··· − 112u + 64
c
5
, c
8
, c
9
u
26
+ 2u
25
+ ··· + 2u + 1
c
6
, c
10
u
26
+ 2u
25
+ ··· + 2u + 1
c
11
u
26
− 14u
25
+ ··· − 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
26
− 10y
25
+ ··· − 481y + 16
c
2
y
26
+ 14y
25
+ ··· − 80993y + 256
c
3
, c
7
y
26
+ 18y
25
+ ··· + 70400y + 4096
c
5
, c
8
, c
9
y
26
+ 22y
25
+ ··· + 4y + 1
c
6
, c
10
y
26
+ 14y
25
+ ··· + 4y + 1
c
11
y
26
− 2y
25
+ ··· + 20y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.011190 + 0.136706I
a = 0.035883 + 0.146636I
b = −0.62158 + 1.42798I
−1.86313 + 7.71246I −6.86228 − 5.25734I
u = 1.011190 − 0.136706I
a = 0.035883 − 0.146636I
b = −0.62158 − 1.42798I
−1.86313 − 7.71246I −6.86228 + 5.25734I
u = −0.370532 + 0.998437I
a = 1.109240 − 0.521057I
b = 0.369770 − 0.293138I
−2.47557 + 4.95345I −6.39722 − 7.47760I
u = −0.370532 − 0.998437I
a = 1.109240 + 0.521057I
b = 0.369770 + 0.293138I
−2.47557 − 4.95345I −6.39722 + 7.47760I
u = 0.269068 + 1.038770I
a = −0.127266 − 0.719999I
b = −0.463650 + 0.532995I
1.31071 − 2.42285I 0.84038 + 4.76679I
u = 0.269068 − 1.038770I
a = −0.127266 + 0.719999I
b = −0.463650 − 0.532995I
1.31071 + 2.42285I 0.84038 − 4.76679I
u = −0.132101 + 0.846386I
a = −0.69055 + 2.07222I
b = 1.30633 − 0.71384I
−0.911584 + 0.890121I 0.87423 + 1.36491I
u = −0.132101 − 0.846386I
a = −0.69055 − 2.07222I
b = 1.30633 + 0.71384I
−0.911584 − 0.890121I 0.87423 − 1.36491I
u = 0.330433 + 0.724477I
a = −0.95370 + 2.08665I
b = −1.142600 + 0.109328I
−5.04252 − 1.61304I −10.18355 + 3.58696I
u = 0.330433 − 0.724477I
a = −0.95370 − 2.08665I
b = −1.142600 − 0.109328I
−5.04252 + 1.61304I −10.18355 − 3.58696I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.774839 + 0.143637I
a = 0.175331 − 0.241236I
b = 0.487210 + 1.045470I
0.05851 − 2.13854I −4.58802 + 1.91237I
u = 0.774839 − 0.143637I
a = 0.175331 + 0.241236I
b = 0.487210 − 1.045470I
0.05851 + 2.13854I −4.58802 − 1.91237I
u = 0.419572 + 0.612728I
a = −0.290716 − 0.333902I
b = 0.237389 + 0.425546I
−0.10620 − 1.46904I −0.77851 + 4.66825I
u = 0.419572 − 0.612728I
a = −0.290716 + 0.333902I
b = 0.237389 − 0.425546I
−0.10620 + 1.46904I −0.77851 − 4.66825I
u = −0.914066 + 0.917616I
a = 0.250251 + 0.130322I
b = 0.155646 + 0.140338I
−7.98517 + 3.33888I 1.72089 − 5.46783I
u = −0.914066 − 0.917616I
a = 0.250251 − 0.130322I
b = 0.155646 − 0.140338I
−7.98517 − 3.33888I 1.72089 + 5.46783I
u = 0.402493 + 1.239940I
a = 0.37826 − 1.94903I
b = 0.80677 + 1.32805I
4.09462 − 6.26991I −1.96309 + 5.01662I
u = 0.402493 − 1.239940I
a = 0.37826 + 1.94903I
b = 0.80677 − 1.32805I
4.09462 + 6.26991I −1.96309 − 5.01662I
u = −0.387462 + 1.292200I
a = −0.67996 − 1.61333I
b = −0.42179 + 1.55871I
7.58035 + 1.64459I 1.58550 − 0.59315I
u = −0.387462 − 1.292200I
a = −0.67996 + 1.61333I
b = −0.42179 − 1.55871I
7.58035 − 1.64459I 1.58550 + 0.59315I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.553190 + 1.252930I
a = 0.89905 + 1.50339I
b = 0.43042 − 1.66549I
6.35723 + 8.21738I −0.54202 − 5.78684I
u = −0.553190 − 1.252930I
a = 0.89905 − 1.50339I
b = 0.43042 + 1.66549I
6.35723 − 8.21738I −0.54202 + 5.78684I
u = 0.560294 + 1.283400I
a = −0.58734 + 1.77218I
b = −0.96323 − 1.71069I
1.68898 − 13.33640I −4.27120 + 7.69267I
u = 0.560294 − 1.283400I
a = −0.58734 − 1.77218I
b = −0.96323 + 1.71069I
1.68898 + 13.33640I −4.27120 − 7.69267I
u = −0.410537 + 0.270705I
a = 0.73153 + 2.35210I
b = −0.430697 + 0.672525I
−4.35117 − 1.59149I −10.56012 + 0.81365I
u = −0.410537 − 0.270705I
a = 0.73153 − 2.35210I
b = −0.430697 − 0.672525I
−4.35117 + 1.59149I −10.56012 − 0.81365I
7
II. I
u
2
= h−u
3
+ 2b + u + 1, −u
3
− 2u
2
+ 2a − 3u − 1, u
4
+ u
3
+ u
2
+ 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
4
=
1
2
u
3
+ u
2
+
3
2
u +
1
2
1
2
u
3
−
1
2
u −
1
2
a
7
=
1
u
2
a
8
=
1
u
2
a
9
=
u
2
+ 1
u
2
a
3
=
1
2
u
3
+ u
2
+
3
2
u +
1
2
1
2
u
3
−
1
2
u −
1
2
a
5
=
−u
3
−u
3
− u
2
− 1
a
10
=
u
u
3
+ u
a
1
=
u
3
u
3
+ u
2
+ 1
a
2
=
3
2
u
3
+ u
2
+
3
2
u +
1
2
3
2
u
3
+ u
2
−
1
2
u +
1
2
a
2
=
3
2
u
3
+ u
2
+
3
2
u +
1
2
3
2
u
3
+ u
2
−
1
2
u +
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1
4
u
3
+
7
2
u
2
+
23
4
u −
37
4
8
(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
7
u
4
c
5
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
6
u
4
+ u
3
+ u
2
+ 1
c
8
, c
9
, c
11
u
4
− u
3
+ 3u
2
− 2u + 1
c
10
u
4
− u
3
+ u
2
+ 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
7
y
4
c
5
, c
8
, c
9
c
11
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
6
, c
10
y
4
+ y
3
+ 3y
2
+ 2y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351808 + 0.720342I
a = 0.38053 + 1.53420I
b = −0.927958 − 0.413327I
−1.43393 − 1.41510I −8.73606 + 5.88934I
u = 0.351808 − 0.720342I
a = 0.38053 − 1.53420I
b = −0.927958 + 0.413327I
−1.43393 + 1.41510I −8.73606 − 5.88934I
u = −0.851808 + 0.911292I
a = −0.130534 + 0.427872I
b = 0.677958 + 0.157780I
−8.43568 + 3.16396I −14.13894 + 0.11292I
u = −0.851808 − 0.911292I
a = −0.130534 − 0.427872I
b = 0.677958 − 0.157780I
−8.43568 − 3.16396I −14.13894 − 0.11292I
11
III. I
u
3
= hu
8
− u
7
+ 2u
6
− u
4
+ u
3
− u
2
+ b − u, u
7
+ u
6
+ 2u
5
+ 4u
4
+ 3u
3
+
3u
2
+ a + 3u + 1, u
9
+ 3u
7
+ · · · + 2u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
4
=
−u
7
− u
6
− 2u
5
− 4u
4
− 3u
3
− 3u
2
− 3u − 1
−u
8
+ u
7
− 2u
6
+ u
4
− u
3
+ u
2
+ u
a
7
=
1
u
2
a
8
=
−u
u
a
9
=
0
u
a
3
=
−u
7
− 2u
6
− 2u
5
− 6u
4
− 4u
3
− 4u
2
− 4u
−2u
8
+ u
7
− 4u
6
− u
5
− 2u
3
+ 2u
2
+ u
a
5
=
1
u
2
a
10
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
2
=
−u
4
− u
2
− 2u − 1
u
4
+ 2u
a
2
=
−u
4
− u
2
− 2u − 1
u
4
+ 2u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
6
− 8u
4
− 8u
3
− 4u
2
− 8u − 6
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
(u
3
− u
2
+ 1)
3
c
2
, c
3
, c
7
(u
3
+ u
2
+ 2u + 1)
3
c
5
, c
6
, c
8
c
9
, c
10
u
9
+ 3u
7
− 3u
6
+ 3u
5
− 6u
4
+ 3u
3
− 3u
2
+ 2u + 1
c
11
u
9
− 6u
8
+ 15u
7
− 15u
6
− 5u
5
+ 24u
4
− 9u
3
− 15u
2
+ 10u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
3
− y
2
+ 2y −1)
3
c
2
, c
3
, c
7
(y
3
+ 3y
2
+ 2y −1)
3
c
5
, c
6
, c
8
c
9
, c
10
y
9
+ 6y
8
+ 15y
7
+ 15y
6
− 5y
5
− 24y
4
− 9y
3
+ 15y
2
+ 10y −1
c
11
y
9
− 6y
8
+ ··· + 130y −1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.149100 + 1.032810I
a = −1.49322 − 1.81245I
b = 1.57125 + 2.35293I
−1.11345 −9.01951 + 0.I
u = −0.149100 − 1.032810I
a = −1.49322 + 1.81245I
b = 1.57125 − 2.35293I
−1.11345 −9.01951 + 0.I
u = −0.929255 + 0.157692I
a = −0.1261290 + 0.0333681I
b = 0.119081 + 1.372090I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = −0.929255 − 0.157692I
a = −0.1261290 − 0.0333681I
b = 0.119081 − 1.372090I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = 0.550542 + 1.200360I
a = −1.08414 + 1.00782I
b = 0.116542 − 1.272430I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = 0.550542 − 1.200360I
a = −1.08414 − 1.00782I
b = 0.116542 + 1.272430I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = 0.378713 + 1.358050I
a = 0.84258 − 1.19340I
b = 0.00950 + 1.58939I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = 0.378713 − 1.358050I
a = 0.84258 + 1.19340I
b = 0.00950 − 1.58939I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = 0.298201
a = −2.27818
b = 0.367256
−1.11345 −9.01950
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
3
− u
2
+ 1)
3
(u
26
− 2u
25
+ ··· − 35u + 4)
c
2
((u + 1)
4
)(u
3
+ u
2
+ 2u + 1)
3
(u
26
+ 10u
25
+ ··· + 481u + 16)
c
3
, c
7
u
4
(u
3
+ u
2
+ 2u + 1)
3
(u
26
− 2u
25
+ ··· − 112u + 64)
c
4
((u + 1)
4
)(u
3
− u
2
+ 1)
3
(u
26
− 2u
25
+ ··· − 35u + 4)
c
5
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
· (u
9
+ 3u
7
− 3u
6
+ 3u
5
− 6u
4
+ 3u
3
− 3u
2
+ 2u + 1)
· (u
26
+ 2u
25
+ ··· + 2u + 1)
c
6
(u
4
+ u
3
+ u
2
+ 1)(u
9
+ 3u
7
− 3u
6
+ 3u
5
− 6u
4
+ 3u
3
− 3u
2
+ 2u + 1)
· (u
26
+ 2u
25
+ ··· + 2u + 1)
c
8
, c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)
· (u
9
+ 3u
7
− 3u
6
+ 3u
5
− 6u
4
+ 3u
3
− 3u
2
+ 2u + 1)
· (u
26
+ 2u
25
+ ··· + 2u + 1)
c
10
(u
4
− u
3
+ u
2
+ 1)(u
9
+ 3u
7
− 3u
6
+ 3u
5
− 6u
4
+ 3u
3
− 3u
2
+ 2u + 1)
· (u
26
+ 2u
25
+ ··· + 2u + 1)
c
11
(u
4
− u
3
+ 3u
2
− 2u + 1)
· (u
9
− 6u
8
+ 15u
7
− 15u
6
− 5u
5
+ 24u
4
− 9u
3
− 15u
2
+ 10u + 1)
· (u
26
− 14u
25
+ ··· − 4u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
3
− y
2
+ 2y −1)
3
(y
26
− 10y
25
+ ··· − 481y + 16)
c
2
((y −1)
4
)(y
3
+ 3y
2
+ 2y −1)
3
(y
26
+ 14y
25
+ ··· − 80993y + 256)
c
3
, c
7
y
4
(y
3
+ 3y
2
+ 2y −1)
3
(y
26
+ 18y
25
+ ··· + 70400y + 4096)
c
5
, c
8
, c
9
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
· (y
9
+ 6y
8
+ 15y
7
+ 15y
6
− 5y
5
− 24y
4
− 9y
3
+ 15y
2
+ 10y −1)
· (y
26
+ 22y
25
+ ··· + 4y + 1)
c
6
, c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
· (y
9
+ 6y
8
+ 15y
7
+ 15y
6
− 5y
5
− 24y
4
− 9y
3
+ 15y
2
+ 10y −1)
· (y
26
+ 14y
25
+ ··· + 4y + 1)
c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
9
− 6y
8
+ ··· + 130y −1)
· (y
26
− 2y
25
+ ··· + 20y + 1)
17
