11n
55
(K11n
55
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 11 10 4 5 6 9 8
Solving Sequence
1,4
2
5,8
9 3 11 6 10 7
c
1
c
4
c
8
c
3
c
11
c
5
c
10
c
6
c
2
, c
7
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−43u
35
− 186u
34
+ ··· + 32b + 285, 41u
35
+ 278u
34
+ ··· + 4a + 100, u
36
+ 7u
35
+ ··· + 7u + 1i
I
u
2
= hb
6
− b
5
− b
4
+ 2b
3
− b + 1, a, u − 1i
* 2 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
= h−43u
35
− 186u
34
+ · · · + 32b + 285, 41u
35
+ 278u
34
+ · · · + 4a +
100, u
36
+ 7u
35
+ · · · + 7u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
8
=
−
41
4
u
35
−
139
2
u
34
+ ··· − 126u − 25
1.34375u
35
+ 5.81250u
34
+ ··· − 28.4375u − 8.90625
a
9
=
−12.5938u
35
− 82.0625u
34
+ ··· − 128.313u − 23.8438
−0.906250u
35
− 9.43750u
34
+ ··· − 50.6875u − 13.9063
a
3
=
−u
2
+ 1
u
2
a
11
=
u
0.0312500u
35
+ 0.187500u
34
+ ··· + 1.18750u + 0.0312500
a
6
=
−0.0312500u
35
− 0.187500u
34
+ ··· − 1.18750u − 0.0312500
−0.906250u
35
− 5.50000u
34
+ ··· − 4.75000u − 0.968750
a
10
=
−0.625000u
35
− 3.81250u
34
+ ··· − 4.06250u + 0.312500
0.968750u
35
+ 5.87500u
34
+ ··· + 6.12500u + 1.03125
a
7
=
41
4
u
35
+
139
2
u
34
+ ··· + 126u + 25
−2.84375u
35
− 13.8125u
34
+ ··· + 33.9375u + 11.1563
a
7
=
41
4
u
35
+
139
2
u
34
+ ··· + 126u + 25
−2.84375u
35
− 13.8125u
34
+ ··· + 33.9375u + 11.1563
(ii) Obstruction class = −1
(iii) Cusp Shapes =
161
16
u
35
+
897
16
u
34
+ ··· −
87
16
u −
49
8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
36
− 7u
35
+ ··· − 7u + 1
c
2
u
36
+ 9u
35
+ ··· + 11u + 1
c
3
, c
7
u
36
− u
35
+ ··· − 128u + 64
c
5
u
36
− 6u
35
+ ··· − 74u + 17
c
6
, c
9
u
36
− 2u
35
+ ··· − 2u + 1
c
8
u
36
+ 2u
35
+ ··· − 56u + 49
c
10
u
36
− 18u
35
+ ··· − 2u + 1
c
11
u
36
− 2u
35
+ ··· − 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
36
− 9y
35
+ ··· − 11y + 1
c
2
y
36
+ 43y
35
+ ··· + 57y + 1
c
3
, c
7
y
36
− 39y
35
+ ··· − 61440y + 4096
c
5
y
36
+ 14y
35
+ ··· + 2990y + 289
c
6
, c
9
y
36
− 18y
35
+ ··· − 2y + 1
c
8
y
36
+ 6y
35
+ ··· + 490y + 2401
c
10
y
36
+ 2y
35
+ ··· + 18y + 1
c
11
y
36
+ 42y
35
+ ··· − 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.061290 + 0.326287I
a = −0.642038 − 0.460220I
b = 0.303700 + 0.290509I
−0.00265 + 2.23213I −0.68244 − 4.15610I
u = 1.061290 − 0.326287I
a = −0.642038 + 0.460220I
b = 0.303700 − 0.290509I
−0.00265 − 2.23213I −0.68244 + 4.15610I
u = 0.515673 + 0.710660I
a = −0.862097 − 0.946791I
b = 0.155712 + 0.653329I
2.02771 − 6.40530I 1.88436 + 7.11312I
u = 0.515673 − 0.710660I
a = −0.862097 + 0.946791I
b = 0.155712 − 0.653329I
2.02771 + 6.40530I 1.88436 − 7.11312I
u = 0.785644 + 0.339119I
a = 0.531433 + 0.754793I
b = −0.166697 − 0.401361I
−1.48377 − 1.33270I −5.42230 + 4.02694I
u = 0.785644 − 0.339119I
a = 0.531433 − 0.754793I
b = −0.166697 + 0.401361I
−1.48377 + 1.33270I −5.42230 − 4.02694I
u = 1.177000 + 0.085808I
a = 0.589499 + 0.125039I
b = −0.316874 − 0.078348I
−2.50865 − 0.37469I −1.90153 − 1.63609I
u = 1.177000 − 0.085808I
a = 0.589499 − 0.125039I
b = −0.316874 + 0.078348I
−2.50865 + 0.37469I −1.90153 + 1.63609I
u = −0.888941 + 0.845189I
a = 0.755005 + 0.931742I
b = 0.44408 − 1.66748I
3.30506 + 0.68404I −1.00000 + 0.503330I
u = −0.888941 − 0.845189I
a = 0.755005 − 0.931742I
b = 0.44408 + 1.66748I
3.30506 − 0.68404I −1.00000 − 0.503330I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.512682 + 0.569033I
a = 0.780897 + 0.985416I
b = −0.113983 − 0.584856I
−0.43379 − 1.94575I −1.80433 + 3.98828I
u = 0.512682 − 0.569033I
a = 0.780897 − 0.985416I
b = −0.113983 + 0.584856I
−0.43379 + 1.94575I −1.80433 − 3.98828I
u = 1.231010 + 0.192137I
a = −0.699294 − 0.225880I
b = 0.379504 + 0.152175I
−0.47071 − 4.51088I 0.51157 + 3.50709I
u = 1.231010 − 0.192137I
a = −0.699294 + 0.225880I
b = 0.379504 − 0.152175I
−0.47071 + 4.51088I 0.51157 − 3.50709I
u = −0.977681 + 0.835405I
a = −0.710165 − 0.943837I
b = −0.44152 + 1.76315I
3.02757 + 5.62134I −1.94819 − 5.64508I
u = −0.977681 − 0.835405I
a = −0.710165 + 0.943837I
b = −0.44152 − 1.76315I
3.02757 − 5.62134I −1.94819 + 5.64508I
u = 0.268102 + 0.646439I
a = −0.923789 − 1.064280I
b = 0.005979 + 0.692881I
3.06250 + 1.09495I 4.73244 − 0.17091I
u = 0.268102 − 0.646439I
a = −0.923789 + 1.064280I
b = 0.005979 − 0.692881I
3.06250 − 1.09495I 4.73244 + 0.17091I
u = −0.813768 + 1.033210I
a = 0.813317 + 1.002390I
b = 0.26375 − 1.59543I
6.28852 − 0.72428I 0
u = −0.813768 − 1.033210I
a = 0.813317 − 1.002390I
b = 0.26375 + 1.59543I
6.28852 + 0.72428I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.791993 + 1.084540I
a = −0.827522 − 1.018650I
b = −0.21954 + 1.58084I
9.06514 − 5.65458I 3.21416 + 3.51542I
u = −0.791993 − 1.084540I
a = −0.827522 + 1.018650I
b = −0.21954 − 1.58084I
9.06514 + 5.65458I 3.21416 − 3.51542I
u = −0.890160 + 1.056150I
a = −0.787704 − 1.021790I
b = −0.24615 + 1.66009I
10.62450 + 2.78646I 4.96714 + 0.I
u = −0.890160 − 1.056150I
a = −0.787704 + 1.021790I
b = −0.24615 − 1.66009I
10.62450 − 2.78646I 4.96714 + 0.I
u = −0.584281 + 0.166417I
a = 1.042350 + 0.296118I
b = 1.36445 − 0.54728I
−0.74873 + 6.02926I 3.12323 − 6.76386I
u = −0.584281 − 0.166417I
a = 1.042350 − 0.296118I
b = 1.36445 + 0.54728I
−0.74873 − 6.02926I 3.12323 + 6.76386I
u = −1.094200 + 0.888442I
a = −0.670282 − 0.996339I
b = −0.35959 + 1.85999I
5.38321 + 7.74752I 0
u = −1.094200 − 0.888442I
a = −0.670282 + 0.996339I
b = −0.35959 − 1.85999I
5.38321 − 7.74752I 0
u = −1.07312 + 0.94681I
a = 0.693971 + 1.015250I
b = 0.31569 − 1.82618I
10.02090 + 4.50426I 0
u = −1.07312 − 0.94681I
a = 0.693971 − 1.015250I
b = 0.31569 + 1.82618I
10.02090 − 4.50426I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.12915 + 0.89488I
a = 0.657336 + 1.008600I
b = 0.34290 − 1.88751I
7.9678 + 12.8462I 0
u = −1.12915 − 0.89488I
a = 0.657336 − 1.008600I
b = 0.34290 + 1.88751I
7.9678 − 12.8462I 0
u = −0.540727 + 0.094568I
a = −1.134090 − 0.180057I
b = −1.312240 + 0.291727I
−2.55842 + 1.10908I −0.054318 − 1.238814I
u = −0.540727 − 0.094568I
a = −1.134090 + 0.180057I
b = −1.312240 − 0.291727I
−2.55842 − 1.10908I −0.054318 + 1.238814I
u = −0.267380 + 0.307929I
a = 1.39318 + 0.77500I
b = 0.600824 − 0.555255I
1.71662 − 0.40164I 5.72630 + 0.15643I
u = −0.267380 − 0.307929I
a = 1.39318 − 0.77500I
b = 0.600824 + 0.555255I
1.71662 + 0.40164I 5.72630 − 0.15643I
8
II. I
u
2
= hb
6
− b
5
− b
4
+ 2b
3
− b + 1, a, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
8
=
0
b
a
9
=
−b
b
a
3
=
0
1
a
11
=
1
−b
2
a
6
=
b
2
− 1
−b
4
a
10
=
b
4
− b
2
+ 1
−b
4
a
7
=
0
b
a
7
=
0
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = b
5
+ 4b
4
− 2b
3
− 4b
2
+ 6b − 3
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
6
c
2
, c
4
(u + 1)
6
c
3
, c
7
u
6
c
5
, c
10
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
c
6
, c
8
, c
11
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
9
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
7
y
6
c
5
, c
10
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
c
6
, c
8
, c
9
c
11
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = −1.002190 + 0.295542I
−3.53554 + 0.92430I −9.40317 − 0.69886I
u = 1.00000
a = 0
b = −1.002190 − 0.295542I
−3.53554 − 0.92430I −9.40317 + 0.69886I
u = 1.00000
a = 0
b = 0.428243 + 0.664531I
0.245672 + 0.924305I 0.635956 + 0.093695I
u = 1.00000
a = 0
b = 0.428243 − 0.664531I
0.245672 − 0.924305I 0.635956 − 0.093695I
u = 1.00000
a = 0
b = 1.073950 + 0.558752I
−1.64493 − 5.69302I −5.23279 + 4.86918I
u = 1.00000
a = 0
b = 1.073950 − 0.558752I
−1.64493 + 5.69302I −5.23279 − 4.86918I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
36
− 7u
35
+ ··· − 7u + 1)
c
2
((u + 1)
6
)(u
36
+ 9u
35
+ ··· + 11u + 1)
c
3
, c
7
u
6
(u
36
− u
35
+ ··· − 128u + 64)
c
4
((u + 1)
6
)(u
36
− 7u
35
+ ··· − 7u + 1)
c
5
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)(u
36
− 6u
35
+ ··· − 74u + 17)
c
6
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
36
− 2u
35
+ ··· − 2u + 1)
c
8
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
36
+ 2u
35
+ ··· − 56u + 49)
c
9
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
36
− 2u
35
+ ··· − 2u + 1)
c
10
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)(u
36
− 18u
35
+ ··· − 2u + 1)
c
11
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
36
− 2u
35
+ ··· − 2u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
6
)(y
36
− 9y
35
+ ··· − 11y + 1)
c
2
((y −1)
6
)(y
36
+ 43y
35
+ ··· + 57y + 1)
c
3
, c
7
y
6
(y
36
− 39y
35
+ ··· − 61440y + 4096)
c
5
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
36
+ 14y
35
+ ··· + 2990y + 289)
c
6
, c
9
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
36
− 18y
35
+ ··· − 2y + 1)
c
8
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
36
+ 6y
35
+ ··· + 490y + 2401)
c
10
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
36
+ 2y
35
+ ··· + 18y + 1)
c
11
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
36
+ 42y
35
+ ··· − 2y + 1)
14
