11n
80
(K11n
80
)
A knot diagram
1
Linearized knot diagam
5 1 7 2 4 10 3 11 1 7 9
Solving Sequence
2,4
5
6,10
7 1 3 8 9 11
c
4
c
5
c
6
c
1
c
3
c
7
c
9
c
11
c
2
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h35u
14
− 160u
13
+ ··· + 2098b − 1482, −1803u
14
− 7193u
13
+ ··· + 2098a − 16837,
u
15
+ 4u
14
+ ··· + 12u + 1i
I
u
2
= h−u
2
+ b − u − 1, −u
3
− u
2
+ a − u + 1, u
4
+ u
3
+ u
2
+ 1i
I
u
3
= hb
2
− bu + b + u, a + u, u
2
− u + 1i
* 3 irreducible components of dim
C
= 0, with total 23 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
= h35u
14
− 160u
13
+ · · · + 2098b − 1482, −1803u
14
− 7193u
13
+ · · · +
2098a − 16837, u
15
+ 4u
14
+ · · · + 12u + 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
−u
2
a
6
=
u
2
+ 1
−u
2
a
10
=
0.859390u
14
+ 3.42850u
13
+ ··· + 8.95853u + 8.02526
−0.0166826u
14
+ 0.0762631u
13
+ ··· + 2.44423u + 0.706387
a
7
=
−0.477121u
14
− 1.81888u
13
+ ··· − 5.69495u − 3.69733
0.0319352u
14
+ 0.211153u
13
+ ··· − 0.407531u − 0.337941
a
1
=
−u
u
3
+ u
a
3
=
−u
3
u
5
+ u
3
+ u
a
8
=
−0.277407u
14
− 0.946139u
13
+ ··· − 3.19876u − 3.48236
0.167779u
14
+ 0.661582u
13
+ ··· − 1.09628u − 0.447092
a
9
=
0.715443u
14
+ 2.87226u
13
+ ··· + 8.20591u + 7.89180
0.00905624u
14
+ 0.0300286u
13
+ ··· + 3.28742u + 0.859390
a
11
=
0.782173u
14
+ 3.06721u
13
+ ··· + 7.92898u + 6.06625
−0.0619638u
14
− 0.0738799u
13
+ ··· + 1.00715u + 0.409438
a
11
=
0.782173u
14
+ 3.06721u
13
+ ··· + 7.92898u + 6.06625
−0.0619638u
14
− 0.0738799u
13
+ ··· + 1.00715u + 0.409438
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
713
2098
u
14
−
2585
2098
u
13
+ ··· +
20215
2098
u +
10090
1049
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
15
+ 4u
14
+ ··· + 12u + 1
c
2
, c
5
u
15
+ 10u
14
+ ··· + 112u − 1
c
3
, c
7
u
15
− 2u
14
+ ··· + 16u − 16
c
6
, c
10
u
15
− 3u
14
+ ··· − 24u − 16
c
8
, c
9
, c
11
u
15
+ 7u
14
+ ··· − 16u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
15
+ 10y
14
+ ··· + 112y −1
c
2
, c
5
y
15
− 6y
14
+ ··· + 13488y −1
c
3
, c
7
y
15
− 20y
14
+ ··· + 128y −256
c
6
, c
10
y
15
+ 21y
14
+ ··· − 1984y −256
c
8
, c
9
, c
11
y
15
− 3y
14
+ ··· + 134y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.443471 + 0.899923I
a = −2.22456 − 0.69076I
b = −0.00537 + 2.93789I
1.31612 + 1.82919I 23.9935 − 13.4254I
u = 0.443471 − 0.899923I
a = −2.22456 + 0.69076I
b = −0.00537 − 2.93789I
1.31612 − 1.82919I 23.9935 + 13.4254I
u = −1.154120 + 0.257445I
a = −0.114455 − 1.265520I
b = 0.21776 + 1.65198I
−5.23991 + 4.29122I 5.74651 − 1.92061I
u = −1.154120 − 0.257445I
a = −0.114455 + 1.265520I
b = 0.21776 − 1.65198I
−5.23991 − 4.29122I 5.74651 + 1.92061I
u = −0.707815 + 0.947595I
a = 0.362571 − 0.587039I
b = −0.068426 − 0.205683I
9.44393 − 2.71266I 11.43593 + 3.34052I
u = −0.707815 − 0.947595I
a = 0.362571 + 0.587039I
b = −0.068426 + 0.205683I
9.44393 + 2.71266I 11.43593 − 3.34052I
u = 0.416218 + 0.666363I
a = −0.168507 − 0.746645I
b = −0.225041 + 0.497206I
−0.075833 + 1.377120I −0.42484 − 4.74084I
u = 0.416218 − 0.666363I
a = −0.168507 + 0.746645I
b = −0.225041 − 0.497206I
−0.075833 − 1.377120I −0.42484 + 4.74084I
u = 0.136912 + 1.276840I
a = −1.156530 − 0.039261I
b = 0.148725 + 0.753403I
−2.05262 + 0.52363I 2.28909 − 0.30141I
u = 0.136912 − 1.276840I
a = −1.156530 + 0.039261I
b = 0.148725 − 0.753403I
−2.05262 − 0.52363I 2.28909 + 0.30141I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.68964 + 1.30605I
a = 1.367780 − 0.217281I
b = −0.35945 + 1.80414I
−8.47013 − 10.83430I 4.46568 + 4.98924I
u = −0.68964 − 1.30605I
a = 1.367780 + 0.217281I
b = −0.35945 − 1.80414I
−8.47013 + 10.83430I 4.46568 − 4.98924I
u = −0.39863 + 1.51864I
a = −0.749856 + 0.006232I
b = 0.03559 − 1.70713I
−11.10030 − 1.26356I 2.58190 + 0.63912I
u = −0.39863 − 1.51864I
a = −0.749856 − 0.006232I
b = 0.03559 + 1.70713I
−11.10030 + 1.26356I 2.58190 − 0.63912I
u = −0.0927870
a = 7.36713
b = 0.512405
1.10369 8.82440
6
II. I
u
2
= h−u
2
+ b − u − 1, −u
3
− u
2
+ a − u + 1, u
4
+ u
3
+ u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
−u
2
a
6
=
u
2
+ 1
−u
2
a
10
=
u
3
+ u
2
+ u − 1
u
2
+ u + 1
a
7
=
u
2
+ 1
−u
2
a
1
=
−u
u
3
+ u
a
3
=
−u
3
u
3
+ u
2
+ 1
a
8
=
u
−u
3
− u
a
9
=
u
3
+ u
2
+ 2u − 1
−u
3
+ u
2
+ 1
a
11
=
u
3
+ u
2
+ u − 1
u
2
+ u + 1
a
11
=
u
3
+ u
2
+ u − 1
u
2
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
3
− 5u
2
+ 8
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
4
− u
3
+ u
2
+ 1
c
2
, c
5
, c
7
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
3
u
4
− u
3
+ 3u
2
− 2u + 1
c
4
u
4
+ u
3
+ u
2
+ 1
c
6
, c
10
u
4
c
8
, c
9
(u + 1)
4
c
11
(u − 1)
4
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
2
, c
3
, c
5
c
7
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
6
, c
10
y
4
c
8
, c
9
, c
11
(y −1)
4
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351808 + 0.720342I
a = −1.54742 + 1.12087I
b = 0.95668 + 1.22719I
1.43393 + 1.41510I 11.48794 − 2.21528I
u = 0.351808 − 0.720342I
a = −1.54742 − 1.12087I
b = 0.95668 − 1.22719I
1.43393 − 1.41510I 11.48794 + 2.21528I
u = −0.851808 + 0.911292I
a = −0.452576 + 0.585652I
b = 0.043315 − 0.641200I
8.43568 − 3.16396I 4.01206 + 4.08190I
u = −0.851808 − 0.911292I
a = −0.452576 − 0.585652I
b = 0.043315 + 0.641200I
8.43568 + 3.16396I 4.01206 − 4.08190I
10
III. I
u
3
= hb
2
− bu + b + u, a + u, u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
4
=
1
0
a
5
=
1
−u + 1
a
6
=
u
−u + 1
a
10
=
−u
b
a
7
=
bu − b + 2u
0
a
1
=
−u
u − 1
a
3
=
1
0
a
8
=
bu − b + 2u
0
a
9
=
−bu + b − 2u
u − 1
a
11
=
2bu − 2b + 2u
b
a
11
=
2bu − 2b + 2u
b
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3bu − 6b − u + 5
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
(u
2
+ u + 1)
2
c
3
, c
7
u
4
c
4
(u
2
− u + 1)
2
c
6
, c
8
, c
9
(u
2
− u − 1)
2
c
10
, c
11
(u
2
+ u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
2
c
3
, c
7
y
4
c
6
, c
8
, c
9
c
10
, c
11
(y
2
− 3y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = −0.500000 − 0.866025I
b = 0.309017 − 0.535233I
8.88264 + 2.02988I 4.50000 + 2.34537I
u = 0.500000 + 0.866025I
a = −0.500000 − 0.866025I
b = −0.80902 + 1.40126I
0.98696 + 2.02988I 4.50000 − 9.27358I
u = 0.500000 − 0.866025I
a = −0.500000 + 0.866025I
b = 0.309017 + 0.535233I
8.88264 − 2.02988I 4.50000 − 2.34537I
u = 0.500000 − 0.866025I
a = −0.500000 + 0.866025I
b = −0.80902 − 1.40126I
0.98696 − 2.02988I 4.50000 + 9.27358I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
+ u + 1)
2
)(u
4
− u
3
+ u
2
+ 1)(u
15
+ 4u
14
+ ··· + 12u + 1)
c
2
, c
5
((u
2
+ u + 1)
2
)(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
15
+ 10u
14
+ ··· + 112u − 1)
c
3
u
4
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
15
− 2u
14
+ ··· + 16u − 16)
c
4
((u
2
− u + 1)
2
)(u
4
+ u
3
+ u
2
+ 1)(u
15
+ 4u
14
+ ··· + 12u + 1)
c
6
u
4
(u
2
− u − 1)
2
(u
15
− 3u
14
+ ··· − 24u − 16)
c
7
u
4
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
15
− 2u
14
+ ··· + 16u − 16)
c
8
, c
9
((u + 1)
4
)(u
2
− u − 1)
2
(u
15
+ 7u
14
+ ··· − 16u − 1)
c
10
u
4
(u
2
+ u − 1)
2
(u
15
− 3u
14
+ ··· − 24u − 16)
c
11
((u − 1)
4
)(u
2
+ u − 1)
2
(u
15
+ 7u
14
+ ··· − 16u − 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
2
)(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
15
+ 10y
14
+ ··· + 112y −1)
c
2
, c
5
((y
2
+ y + 1)
2
)(y
4
+ 5y
3
+ ··· + 2y + 1)(y
15
− 6y
14
+ ··· + 13488y −1)
c
3
, c
7
y
4
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
15
− 20y
14
+ ··· + 128y −256)
c
6
, c
10
y
4
(y
2
− 3y + 1)
2
(y
15
+ 21y
14
+ ··· − 1984y −256)
c
8
, c
9
, c
11
((y −1)
4
)(y
2
− 3y + 1)
2
(y
15
− 3y
14
+ ··· + 134y −1)
16
