11n
34
(K11n
34
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 9 10 3 11 1 8 6
Solving Sequence
8,11 4,9
3 7 10 6 1 2 5
c
8
c
3
c
7
c
10
c
6
c
11
c
2
c
4
c
1
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
10
+ 32u
9
− 129u
8
+ 203u
7
+ 53u
6
− 484u
5
+ 234u
4
+ 343u
3
− 48u
2
+ 32b − 158u − 11,
− 9u
10
+ 101u
9
− 444u
8
+ 857u
7
− 238u
6
− 1654u
5
+ 1824u
4
+ 793u
3
− 1185u
2
+ 16a − 713u + 124,
u
11
− 11u
10
+ 47u
9
− 86u
8
+ 12u
7
+ 181u
6
− 170u
5
− 107u
4
+ 111u
3
+ 86u
2
− u + 1i
I
u
2
= h3a
5
− 13a
4
+ 7a
3
+ 17a
2
+ 13b + 21a − 7, a
6
− 6a
5
+ 11a
4
− 4a
3
− a
2
− a + 1, u + 1i
I
u
3
= hb, −u
4
+ 2u
3
+ u
2
+ a − 2u − 1, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 22 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−3u
10
+ 32u
9
+ · · · + 32b − 11, −9u
10
+ 101u
9
+ · · · + 16a +
124, u
11
− 11u
10
+ · · · − u + 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
4
=
0.562500u
10
− 6.31250u
9
+ ··· + 44.5625u − 7.75000
3
32
u
10
− u
9
+ ··· +
79
16
u +
11
32
a
9
=
1
−u
2
a
3
=
0.468750u
10
− 5.31250u
9
+ ··· + 39.6250u − 8.09375
3
32
u
10
− u
9
+ ··· +
79
16
u +
11
32
a
7
=
−0.437500u
10
+ 4.31250u
9
+ ··· − 25.5625u − 4.37500
7
32
u
10
−
33
16
u
9
+ ··· +
31
8
u −
7
32
a
10
=
−u
u
a
6
=
−0.218750u
10
+ 2.31250u
9
+ ··· − 25.6250u − 4.53125
−0.0625000u
9
+ 0.562500u
8
+ ··· + 3.93750u − 0.0625000
a
1
=
0.218750u
10
− 2.43750u
9
+ ··· + 16.5000u − 2.46875
1
32
u
10
−
5
16
u
9
+ ··· +
9
4
u +
7
32
a
2
=
0.718750u
10
− 7.87500u
9
+ ··· + 45.5625u − 11.2813
−0.0937500u
10
+ 0.812500u
9
+ ··· + 7.87500u + 0.593750
a
5
=
0.312500u
10
− 3.06250u
9
+ ··· + 21.5625u + 4.50000
−0.593750u
10
+ 5.56250u
9
+ ··· − 4.12500u + 0.343750
a
5
=
0.312500u
10
− 3.06250u
9
+ ··· + 21.5625u + 4.50000
−0.593750u
10
+ 5.56250u
9
+ ··· − 4.12500u + 0.343750
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
1
16
u
10
−
13
16
u
9
+
35
8
u
8
−
193
16
u
7
+
119
8
u
6
+
19
4
u
5
−
275
8
u
4
+
389
16
u
3
+
223
16
u
2
−
147
16
u −
39
8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
11
− 10u
10
+ ··· + 10u − 1
c
2
u
11
+ 24u
10
+ ··· + 182u + 1
c
3
, c
7
u
11
+ u
10
+ ··· + 96u − 32
c
5
u
11
+ 13u
9
+ ··· + 66u − 101
c
6
u
11
− 2u
10
+ ··· + 136u − 1357
c
8
, c
10
u
11
+ 11u
10
+ ··· − u − 1
c
9
u
11
− u
10
+ ··· − 192u − 64
c
11
u
11
+ 2u
10
+ 2u
9
+ 6u
7
+ 12u
6
+ 12u
5
+ u
3
+ 2u
2
+ 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
11
− 24y
10
+ ··· + 182y −1
c
2
y
11
− 36y
10
+ ··· + 32578y −1
c
3
, c
7
y
11
+ 21y
10
+ ··· + 7680y −1024
c
5
y
11
+ 26y
10
+ ··· − 108562y −10201
c
6
y
11
− 30y
10
+ ··· − 15893686y −1841449
c
8
, c
10
y
11
− 27y
10
+ ··· − 171y −1
c
9
y
11
+ 27y
10
+ ··· + 4096y −4096
c
11
y
11
+ 16y
9
+ 86y
7
+ 160y
5
+ 25y
3
− 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.16062
a = −0.487360
b = −0.271903
2.30902 2.53950
u = −0.570873 + 0.314013I
a = −1.11819 + 1.14047I
b = −0.204727 + 0.543309I
0.69226 − 1.35881I 4.43349 + 4.96761I
u = −0.570873 − 0.314013I
a = −1.11819 − 1.14047I
b = −0.204727 − 0.543309I
0.69226 + 1.35881I 4.43349 − 4.96761I
u = 0.0123536 + 0.1046970I
a = −7.99180 + 4.91916I
b = 0.392173 + 0.533181I
−1.88779 − 0.79699I −5.15274 − 0.95060I
u = 0.0123536 − 0.1046970I
a = −7.99180 − 4.91916I
b = 0.392173 − 0.533181I
−1.88779 + 0.79699I −5.15274 + 0.95060I
u = 1.99230 + 1.10149I
a = −0.812502 + 0.545736I
b = 2.19746 + 2.02033I
−17.0622 + 11.2191I 1.86536 − 4.34062I
u = 1.99230 − 1.10149I
a = −0.812502 − 0.545736I
b = 2.19746 − 2.02033I
−17.0622 − 11.2191I 1.86536 + 4.34062I
u = 2.11551 + 1.00650I
a = 0.775984 − 0.370969I
b = −1.98959 − 2.25555I
−17.0176 + 3.4378I 1.85943 − 0.49918I
u = 2.11551 − 1.00650I
a = 0.775984 + 0.370969I
b = −1.98959 + 2.25555I
−17.0176 − 3.4378I 1.85943 + 0.49918I
u = 2.53102 + 0.11992I
a = −0.109810 − 0.279561I
b = 0.24063 + 3.13515I
2.86702 + 4.05320I 1.72472 − 1.91622I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 2.53102 − 0.11992I
a = −0.109810 + 0.279561I
b = 0.24063 − 3.13515I
2.86702 − 4.05320I 1.72472 + 1.91622I
6
II. I
u
2
=
h3a
5
−13a
4
+7a
3
+17a
2
+13b+21a−7, a
6
−6a
5
+11a
4
−4a
3
−a
2
−a+1, u+1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
−1
a
4
=
a
−
3
13
a
5
+ a
4
+ ··· −
21
13
a +
7
13
a
9
=
1
−1
a
3
=
3
13
a
5
− a
4
+ ··· +
34
13
a −
7
13
−
3
13
a
5
+ a
4
+ ··· −
21
13
a +
7
13
a
7
=
−1.53846a
5
+ 8a
4
+ ··· + 1.23077a + 1.92308
15
13
a
5
− 6a
4
+ ··· −
12
13
a −
9
13
a
10
=
1
−1
a
6
=
−1.15385a
5
+ 6a
4
+ ··· + 0.923077a + 0.692308
10
13
a
5
− 4a
4
+ ··· −
8
13
a +
7
13
a
1
=
−2.07692a
5
+ 11a
4
+ ··· + 2.46154a + 2.84615
2.07692a
5
− 11a
4
+ ··· − 2.46154a − 2.84615
a
2
=
−1.92308a
5
+ 10a
4
+ ··· + 1.53846a + 2.15385
25
13
a
5
− 10a
4
+ ··· −
7
13
a −
28
13
a
5
=
−1.53846a
5
+ 8a
4
+ ··· + 1.23077a + 1.92308
15
13
a
5
− 6a
4
+ ··· −
12
13
a −
9
13
a
5
=
−1.53846a
5
+ 8a
4
+ ··· + 1.23077a + 1.92308
15
13
a
5
− 6a
4
+ ··· −
12
13
a −
9
13
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
92
13
a
5
+ 37a
4
−
622
13
a
3
−
179
13
a
2
−
20
13
a +
180
13
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
2
, c
11
u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1
c
3
, c
4
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
5
, c
6
u
6
+ u
5
+ 2u
4
+ 4u
3
+ 5u
2
+ 3u + 1
c
8
(u + 1)
6
c
9
u
6
c
10
(u − 1)
6
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
c
2
, c
11
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
c
5
, c
6
y
6
+ 3y
5
+ 6y
4
+ 5y
2
+ y + 1
c
8
, c
10
(y −1)
6
c
9
y
6
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 0.658836 + 0.177500I
b = −1.073950 − 0.558752I
1.64493 + 5.69302I 0.29418 − 8.33058I
u = −1.00000
a = 0.658836 − 0.177500I
b = −1.073950 + 0.558752I
1.64493 − 5.69302I 0.29418 + 8.33058I
u = −1.00000
a = −0.346225 + 0.393823I
b = 1.002190 − 0.295542I
3.53554 − 0.92430I 6.31051 + 0.25702I
u = −1.00000
a = −0.346225 − 0.393823I
b = 1.002190 + 0.295542I
3.53554 + 0.92430I 6.31051 − 0.25702I
u = −1.00000
a = 2.68739 + 0.76772I
b = −0.428243 + 0.664531I
−0.245672 + 0.924305I −0.60470 + 5.55069I
u = −1.00000
a = 2.68739 − 0.76772I
b = −0.428243 − 0.664531I
−0.245672 − 0.924305I −0.60470 − 5.55069I
10
III. I
u
3
= hb, −u
4
+ 2u
3
+ u
2
+ a − 2u − 1, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
8
=
1
0
a
11
=
0
u
a
4
=
u
4
− 2u
3
− u
2
+ 2u + 1
0
a
9
=
1
−u
2
a
3
=
u
4
− 2u
3
− u
2
+ 2u + 1
0
a
7
=
1
0
a
10
=
−u
u
a
6
=
−u
2
+ 1
u
2
a
1
=
u
4
− u
2
− 1
−u
4
− u
3
+ u
2
+ 2u + 1
a
2
=
2u
4
− 2u
3
− 2u
2
+ 2u
−u
4
− u
3
+ u
2
+ 2u + 1
a
5
=
−u
4
+ u
2
+ 1
u
4
+ u
3
− u
2
− 2u − 1
a
5
=
−u
4
+ u
2
+ 1
u
4
+ u
3
− u
2
− 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
4
− u
3
+ 2u
2
+ 10u + 5
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
c
2
, c
4
(u + 1)
5
c
3
, c
7
u
5
c
5
, c
9
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
6
, c
8
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
10
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
11
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
7
y
5
c
5
, c
9
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
6
, c
8
, c
10
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
11
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.21774
a = 2.89210
b = 0
0.756147 −9.00270
u = −0.309916 + 0.549911I
a = 0.01014 + 1.59703I
b = 0
−1.31583 − 1.53058I 1.45754 + 4.40323I
u = −0.309916 − 0.549911I
a = 0.01014 − 1.59703I
b = 0
−1.31583 + 1.53058I 1.45754 − 4.40323I
u = 1.41878 + 0.21917I
a = 0.043806 − 0.365575I
b = 0
4.22763 + 4.40083I 10.04378 − 5.20937I
u = 1.41878 − 0.21917I
a = 0.043806 + 0.365575I
b = 0
4.22763 − 4.40083I 10.04378 + 5.20937I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
6
+ u
5
+ ··· + u + 1)(u
11
− 10u
10
+ ··· + 10u − 1)
c
2
(u + 1)
5
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
· (u
11
+ 24u
10
+ ··· + 182u + 1)
c
3
u
5
(u
6
− u
5
+ ··· − u + 1)(u
11
+ u
10
+ ··· + 96u − 32)
c
4
((u + 1)
5
)(u
6
− u
5
+ ··· − u + 1)(u
11
− 10u
10
+ ··· + 10u − 1)
c
5
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
6
+ u
5
+ 2u
4
+ 4u
3
+ 5u
2
+ 3u + 1)
· (u
11
+ 13u
9
+ ··· + 66u − 101)
c
6
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
6
+ u
5
+ 2u
4
+ 4u
3
+ 5u
2
+ 3u + 1)
· (u
11
− 2u
10
+ ··· + 136u − 1357)
c
7
u
5
(u
6
+ u
5
+ ··· + u + 1)(u
11
+ u
10
+ ··· + 96u − 32)
c
8
((u + 1)
6
)(u
5
− u
4
+ ··· + u + 1)(u
11
+ 11u
10
+ ··· − u − 1)
c
9
u
6
(u
5
+ u
4
+ ··· + u + 1)(u
11
− u
10
+ ··· − 192u − 64)
c
10
((u − 1)
6
)(u
5
+ u
4
+ ··· + u − 1)(u
11
+ 11u
10
+ ··· − u − 1)
c
11
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)
· (u
11
+ 2u
10
+ 2u
9
+ 6u
7
+ 12u
6
+ 12u
5
+ u
3
+ 2u
2
+ 2u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y −1)
5
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
· (y
11
− 24y
10
+ ··· + 182y −1)
c
2
((y −1)
5
)(y
6
+ y
5
+ ··· + 3y + 1)(y
11
− 36y
10
+ ··· + 32578y −1)
c
3
, c
7
y
5
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
· (y
11
+ 21y
10
+ ··· + 7680y −1024)
c
5
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
6
+ 3y
5
+ 6y
4
+ 5y
2
+ y + 1)
· (y
11
+ 26y
10
+ ··· − 108562y −10201)
c
6
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
6
+ 3y
5
+ 6y
4
+ 5y
2
+ y + 1)
· (y
11
− 30y
10
+ ··· − 15893686y −1841449)
c
8
, c
10
((y −1)
6
)(y
5
− 5y
4
+ ··· − y −1)(y
11
− 27y
10
+ ··· − 171y −1)
c
9
y
6
(y
5
+ 3y
4
+ ··· − y −1)(y
11
+ 27y
10
+ ··· + 4096y −4096)
c
11
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
· (y
11
+ 16y
9
+ 86y
7
+ 160y
5
+ 25y
3
− 1)
16
