11n
140
(K11n
140
)
A knot diagram
1
Linearized knot diagam
8 1 9 8 11 10 2 4 7 6 9
Solving Sequence
2,7 8,9
10 1 3 4 5 6 11
c
7
c
9
c
1
c
2
c
3
c
4
c
6
c
11
c
5
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
12
+ u
11
− u
10
+ 2u
9
− 6u
8
+ 5u
7
− u
6
+ 4u
5
− 6u
4
+ 4u
3
+ 6u
2
+ 4b + 1,
− u
12
+ u
11
− u
10
+ 2u
9
− 6u
8
+ 5u
7
− u
6
+ 4u
5
− 10u
4
+ 4u
3
+ 2u
2
+ 4a − 3,
u
13
+ 2u
11
− u
10
+ 6u
9
− u
8
+ 6u
7
− 3u
6
+ 8u
5
+ 4u
3
+ u + 1i
I
u
2
= h2336u
15
− 694u
14
+ ··· + 13139b − 13544, −7442u
15
+ 7948u
14
+ ··· + 65695a + 191727,
u
16
+ u
15
+ ··· + 4u + 5i
I
u
3
= hb − a − 1, a
2
− au + 2a − u + 2, u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 33 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−u
12
+u
11
+· · ·+4b+1, −u
12
+u
11
+· · ·+4a−3, u
13
+2u
11
+· · ·+u+1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
−u
2
a
9
=
1
4
u
12
−
1
4
u
11
+ ··· −
1
2
u
2
+
3
4
1
4
u
12
−
1
4
u
11
+ ··· −
3
2
u
2
−
1
4
a
10
=
1
2
u
12
−
1
2
u
11
+ ··· − 2u
2
+
1
2
1
4
u
12
−
1
4
u
11
+ ··· −
3
2
u
2
−
1
4
a
1
=
−u
u
3
+ u
a
3
=
−u
3
u
5
+ u
3
+ u
a
4
=
−
1
4
u
12
−
1
4
u
11
+ ··· +
1
2
u −
1
4
−
1
4
u
12
−
1
4
u
11
+ ··· +
1
2
u −
1
4
a
5
=
−
1
4
u
12
−
1
4
u
11
+ ··· +
3
2
u −
1
4
−
1
4
u
12
−
1
4
u
11
+ ··· +
1
2
u −
1
4
a
6
=
1
4
u
12
−
5
4
u
11
+ ··· − u −
5
4
−
1
2
u
11
+
1
2
u
10
+ ··· −
1
2
u − 1
a
11
=
−
5
4
u
12
−
1
4
u
11
+ ··· −
5
2
u −
3
4
−u
12
−
3
2
u
10
+ ··· − u −
1
2
a
11
=
−
5
4
u
12
−
1
4
u
11
+ ··· −
5
2
u −
3
4
−u
12
−
3
2
u
10
+ ··· − u −
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 3u
12
− 3u
11
+ 6u
10
− 9u
9
+ 20u
8
− 20u
7
+ 19u
6
− 24u
5
+ 29u
4
− 20u
3
+ 7u
2
− 8u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
u
13
+ 2u
11
+ u
10
+ 6u
9
+ u
8
+ 6u
7
+ 3u
6
+ 8u
5
+ 4u
3
+ u − 1
c
2
u
13
+ 4u
12
+ ··· + u − 1
c
5
, c
6
, c
9
c
10
u
13
+ 3u
12
+ ··· − 3u − 2
c
11
u
13
− 3u
12
+ ··· + 41u − 24
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
y
13
+ 4y
12
+ ··· + y −1
c
2
y
13
+ 16y
12
+ ··· + 17y −1
c
5
, c
6
, c
9
c
10
y
13
+ 15y
12
+ ··· + 17y −4
c
11
y
13
+ 3y
12
+ ··· + 6385y −576
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.849803 + 0.688633I
a = 0.100600 + 0.370738I
b = 0.161020 − 1.380070I
−2.33364 + 0.79324I 4.73185 − 2.01069I
u = 0.849803 − 0.688633I
a = 0.100600 − 0.370738I
b = 0.161020 + 1.380070I
−2.33364 − 0.79324I 4.73185 + 2.01069I
u = −0.301931 + 0.795374I
a = 0.53836 + 1.69819I
b = 0.01733 + 1.65836I
−11.18850 − 1.28224I 2.05046 + 5.61257I
u = −0.301931 − 0.795374I
a = 0.53836 − 1.69819I
b = 0.01733 − 1.65836I
−11.18850 + 1.28224I 2.05046 − 5.61257I
u = −0.793875 + 0.936102I
a = −0.703154 − 0.416062I
b = 0.691430 + 0.338829I
3.03190 − 3.86102I 7.31704 + 2.47395I
u = −0.793875 − 0.936102I
a = −0.703154 + 0.416062I
b = 0.691430 − 0.338829I
3.03190 + 3.86102I 7.31704 − 2.47395I
u = 0.426416 + 0.596732I
a = 0.581070 − 0.572912I
b = −0.016047 − 0.904459I
−2.28202 + 1.46619I 3.02331 − 4.77758I
u = 0.426416 − 0.596732I
a = 0.581070 + 0.572912I
b = −0.016047 + 0.904459I
−2.28202 − 1.46619I 3.02331 + 4.77758I
u = 0.760740 + 1.064260I
a = −1.237650 + 0.471225I
b = 0.631392 + 0.645827I
2.12458 + 8.30943I 4.97203 − 7.67433I
u = 0.760740 − 1.064260I
a = −1.237650 − 0.471225I
b = 0.631392 − 0.645827I
2.12458 − 8.30943I 4.97203 + 7.67433I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.717554 + 1.160670I
a = −1.71256 − 0.47703I
b = 0.20153 − 1.58392I
−5.31862 − 11.41160I 1.59544 + 6.78413I
u = −0.717554 − 1.160670I
a = −1.71256 + 0.47703I
b = 0.20153 + 1.58392I
−5.31862 + 11.41160I 1.59544 − 6.78413I
u = −0.447199
a = 0.866672
b = −0.373309
0.678852 14.6200
6
II. I
u
2
= h2336u
15
− 694u
14
+ · · · + 13139b − 13544, −7442u
15
+ 7948u
14
+
· · · + 65695a + 191727, u
16
+ u
15
+ · · · + 4u + 5i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
−u
2
a
9
=
0.113281u
15
− 0.120983u
14
+ ··· − 0.144562u − 2.91844
−0.177791u
15
+ 0.0528198u
14
+ ··· − 0.306035u + 1.03082
a
10
=
−0.0645102u
15
− 0.0681635u
14
+ ··· − 0.450597u − 1.88762
−0.177791u
15
+ 0.0528198u
14
+ ··· − 0.306035u + 1.03082
a
1
=
−u
u
3
+ u
a
3
=
−u
3
u
5
+ u
3
+ u
a
4
=
0.440429u
15
+ 0.374290u
14
+ ··· + 5.23398u + 1.69710
−0.246594u
15
− 0.358246u
14
+ ··· − 0.496385u − 1.22780
a
5
=
1
5
u
15
+
1
5
u
14
+ ··· +
14
5
u +
4
5
−0.240429u
15
− 0.174290u
14
+ ··· − 1.43398u − 0.897100
a
6
=
0.0488926u
15
− 0.264525u
14
+ ··· − 0.240840u − 0.483477
−0.287236u
15
− 0.200624u
14
+ ··· − 0.500419u − 0.889413
a
11
=
0.206165u
15
+ 0.383956u
14
+ ··· + 0.862410u + 1.13069
−0.0159830u
15
− 0.106553u
14
+ ··· + 1.24560u − 0.338839
a
11
=
0.206165u
15
+ 0.383956u
14
+ ··· + 0.862410u + 1.13069
−0.0159830u
15
− 0.106553u
14
+ ··· + 1.24560u − 0.338839
(ii) Obstruction class = −1
(iii) Cusp Shapes =
488
1877
u
15
−
1752
1877
u
14
+ ··· −
16008
1877
u −
9926
1877
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
u
16
− u
15
+ ··· − 4u + 5
c
2
u
16
+ 7u
15
+ ··· + 124u + 25
c
5
, c
6
, c
9
c
10
, c
11
(u
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
y
16
+ 7y
15
+ ··· + 124y + 25
c
2
y
16
+ 3y
15
+ ··· + 824y + 625
c
5
, c
6
, c
9
c
10
, c
11
(y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 39y
4
+ 22y
3
+ 18y
2
+ 4y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.548614 + 0.832668I
a = −1.83759 + 1.08227I
b = 0.06382 − 1.51723I
−9.89946 − 2.18536I 0.41681 + 3.14055I
u = −0.548614 − 0.832668I
a = −1.83759 − 1.08227I
b = 0.06382 + 1.51723I
−9.89946 + 2.18536I 0.41681 − 3.14055I
u = −0.969644 + 0.496042I
a = 0.353364 + 0.413115I
b = −0.19980 − 1.51366I
−3.28987 + 5.23868I 4.00000 − 3.04258I
u = −0.969644 − 0.496042I
a = 0.353364 − 0.413115I
b = −0.19980 + 1.51366I
−3.28987 − 5.23868I 4.00000 + 3.04258I
u = 0.886697 + 0.673651I
a = 0.835367 − 0.338086I
b = −0.647085 + 0.502738I
3.31972 − 2.18536I 7.58319 + 3.14055I
u = 0.886697 − 0.673651I
a = 0.835367 + 0.338086I
b = −0.647085 − 0.502738I
3.31972 + 2.18536I 7.58319 − 3.14055I
u = −0.822874 + 0.843581I
a = 1.216240 + 0.256877I
b = −0.647085 + 0.502738I
3.31972 − 2.18536I 7.58319 + 3.14055I
u = −0.822874 − 0.843581I
a = 1.216240 − 0.256877I
b = −0.647085 − 0.502738I
3.31972 + 2.18536I 7.58319 − 3.14055I
u = 0.043421 + 1.182100I
a = −0.104055 − 0.579236I
b = 0.283060 − 0.443755I
−3.28987 − 1.04600I 4.00000 + 6.68545I
u = 0.043421 − 1.182100I
a = −0.104055 + 0.579236I
b = 0.283060 + 0.443755I
−3.28987 + 1.04600I 4.00000 − 6.68545I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.239639 + 0.738346I
a = −1.82664 − 0.62597I
b = 0.283060 + 0.443755I
−3.28987 + 1.04600I 4.00000 − 6.68545I
u = 0.239639 − 0.738346I
a = −1.82664 + 0.62597I
b = 0.283060 − 0.443755I
−3.28987 − 1.04600I 4.00000 + 6.68545I
u = 0.769845 + 1.017620I
a = 1.54571 − 0.12141I
b = −0.19980 − 1.51366I
−3.28987 + 5.23868I 4.00000 − 3.04258I
u = 0.769845 − 1.017620I
a = 1.54571 + 0.12141I
b = −0.19980 + 1.51366I
−3.28987 − 5.23868I 4.00000 + 3.04258I
u = −0.098471 + 1.335410I
a = 0.017611 + 1.367960I
b = 0.06382 + 1.51723I
−9.89946 + 2.18536I 0.41681 − 3.14055I
u = −0.098471 − 1.335410I
a = 0.017611 − 1.367960I
b = 0.06382 − 1.51723I
−9.89946 − 2.18536I 0.41681 + 3.14055I
11
III. I
u
3
= hb − a − 1, a
2
− au + 2a − u + 2, u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
1
a
9
=
a
a + 1
a
10
=
2a + 1
a + 1
a
1
=
−u
0
a
3
=
u
u
a
4
=
−au + u
−au
a
5
=
−au
−au − u
a
6
=
2au − a + 2u − 2
au + u − 1
a
11
=
−au − a − 3u − 1
−a − u − 1
a
11
=
−au − a − 3u − 1
−a − u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
2
c
2
(u + 1)
4
c
5
, c
6
, c
9
c
10
u
4
+ 3u
2
+ 1
c
11
(u
2
− u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(y + 1)
4
c
2
(y −1)
4
c
5
, c
6
, c
9
c
10
(y
2
+ 3y + 1)
2
c
11
(y
2
− 3y + 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −1.000000 − 0.618034I
b = − 0.618034I
−4.27683 −4.00000
u = 1.000000I
a = −1.00000 + 1.61803I
b = 1.61803I
−12.1725 −4.00000
u = − 1.000000I
a = −1.000000 + 0.618034I
b = 0.618034I
−4.27683 −4.00000
u = − 1.000000I
a = −1.00000 − 1.61803I
b = − 1.61803I
−12.1725 −4.00000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
2
(u
13
+ 2u
11
+ u
10
+ 6u
9
+ u
8
+ 6u
7
+ 3u
6
+ 8u
5
+ 4u
3
+ u − 1)
· (u
16
− u
15
+ ··· − 4u + 5)
c
2
((u + 1)
4
)(u
13
+ 4u
12
+ ··· + u − 1)(u
16
+ 7u
15
+ ··· + 124u + 25)
c
5
, c
6
, c
9
c
10
(u
4
+ 3u
2
+ 1)(u
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ 1)
2
· (u
13
+ 3u
12
+ ··· − 3u − 2)
c
11
(u
2
− u − 1)
2
(u
8
− u
7
+ 5u
6
− 4u
5
+ 7u
4
− 4u
3
+ 2u
2
+ 1)
2
· (u
13
− 3u
12
+ ··· + 41u − 24)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
((y + 1)
4
)(y
13
+ 4y
12
+ ··· + y −1)(y
16
+ 7y
15
+ ··· + 124y + 25)
c
2
((y −1)
4
)(y
13
+ 16y
12
+ ··· + 17y −1)(y
16
+ 3y
15
+ ··· + 824y + 625)
c
5
, c
6
, c
9
c
10
(y
2
+ 3y + 1)
2
· (y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 39y
4
+ 22y
3
+ 18y
2
+ 4y + 1)
2
· (y
13
+ 15y
12
+ ··· + 17y −4)
c
11
(y
2
− 3y + 1)
2
· (y
8
+ 9y
7
+ 31y
6
+ 50y
5
+ 39y
4
+ 22y
3
+ 18y
2
+ 4y + 1)
2
· (y
13
+ 3y
12
+ ··· + 6385y −576)
17
