11n
42
(K11n
42
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 10 11 3 6 1 8 9
Solving Sequence
1,4
2
5,10
6 9 8 3 11 7
c
1
c
4
c
5
c
9
c
8
c
3
c
11
c
6
c
2
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
10
− 9u
9
− 29u
8
− 28u
7
+ 45u
6
+ 98u
5
− 16u
4
− 100u
3
+ 18u
2
+ 16b + 23u − 17,
− 17u
10
− 169u
9
+ ··· + 16a − 209,
u
11
+ 10u
10
+ 38u
9
+ 57u
8
− 17u
7
− 143u
6
− 82u
5
+ 116u
4
+ 82u
3
− 41u
2
+ 10u + 1i
I
u
2
= hb − 1, −u
5
− u
4
+ u
3
+ 2u
2
+ a, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
I
u
3
= hb − a + 1, a
5
− 4a
4
+ 4a
3
+ a
2
− 2a − 1, 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−u
10
− 9u
9
+ · · · + 16b − 17, −17u
10
− 169u
9
+ · · · + 16a −
209, u
11
+ 10u
10
+ · · · + 10u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
10
=
1.06250u
10
+ 10.5625u
9
+ ··· − 42.4375u + 13.0625
0.0625000u
10
+ 0.562500u
9
+ ··· − 1.43750u + 1.06250
a
6
=
−1.18750u
10
− 11.8125u
9
+ ··· + 54.3125u − 17.3125
−
1
8
u
9
− u
8
+ ··· + 6u −
9
8
a
9
=
u
10
+ 10u
9
+ ··· − 41u + 12
0.0625000u
10
+ 0.562500u
9
+ ··· − 1.43750u + 1.06250
a
8
=
1
4
u
10
+
5
2
u
9
+ ··· −
29
4
u + 3
−
1
4
u
8
−
5
4
u
7
+ ··· −
3
4
u +
1
4
a
3
=
−u
2
+ 1
u
2
a
11
=
1.18750u
10
+ 11.6875u
9
+ ··· − 45.3125u + 15.1875
−
1
8
u
10
− u
9
+ ··· −
33
8
u + 1
a
7
=
−
5
4
u
10
−
19
2
u
9
+ ··· +
17
4
u − 3
u
10
+
27
4
u
9
+ ··· −
31
4
u
2
+
13
4
u
a
7
=
−
5
4
u
10
−
19
2
u
9
+ ··· +
17
4
u − 3
u
10
+
27
4
u
9
+ ··· −
31
4
u
2
+
13
4
u
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
1
4
u
10
+
21
8
u
9
+ 11u
8
+
165
8
u
7
+
55
8
u
6
−
137
4
u
5
−
83
2
u
4
+
15
2
u
3
+
55
2
u
2
+
13
2
u +
23
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
− 2u
10
+ ··· + 136u − 1357
c
6
u
11
+ 13u
9
+ ··· + 66u − 101
c
8
u
11
+ 2u
10
+ 2u
9
+ 6u
7
+ 12u
6
+ 12u
5
+ u
3
+ 2u
2
+ 2u + 1
c
9
, c
11
u
11
+ 11u
10
+ ··· − u − 1
c
10
u
11
− u
10
+ ··· − 192u − 64
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
− 30y
10
+ ··· − 15893686y −1841449
c
6
y
11
+ 26y
10
+ ··· − 108562y −10201
c
8
y
11
+ 16y
9
+ 86y
7
+ 160y
5
+ 25y
3
− 1
c
9
, c
11
y
11
− 27y
10
+ ··· − 171y −1
c
10
y
11
+ 27y
10
+ ··· + 4096y −4096
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002510 + 0.212279I
a = 1.032960 + 0.097456I
b = −0.0123536 − 0.1046970I
−1.88779 − 0.79699I −5.15274 − 0.95060I
u = 1.002510 − 0.212279I
a = 1.032960 − 0.097456I
b = −0.0123536 + 0.1046970I
−1.88779 + 0.79699I −5.15274 + 0.95060I
u = 0.224257 + 0.244726I
a = 0.53554 + 1.90709I
b = 0.570873 − 0.314013I
0.69226 − 1.35881I 4.43349 + 4.96761I
u = 0.224257 − 0.244726I
a = 0.53554 − 1.90709I
b = 0.570873 + 0.314013I
0.69226 + 1.35881I 4.43349 − 4.96761I
u = −0.0743419
a = 16.6120
b = 1.16062
2.30902 2.53950
u = −1.90293 + 1.00229I
a = 0.419077 − 0.884818I
b = −1.99230 − 1.10149I
−17.0622 + 11.2191I 1.86536 − 4.34062I
u = −1.90293 − 1.00229I
a = 0.419077 + 0.884818I
b = −1.99230 + 1.10149I
−17.0622 − 11.2191I 1.86536 + 4.34062I
u = −1.98831 + 0.89173I
a = 0.303205 − 0.818713I
b = −2.11551 − 1.00650I
−17.0176 + 3.4378I 1.85943 − 0.49918I
u = −1.98831 − 0.89173I
a = 0.303205 + 0.818713I
b = −2.11551 + 1.00650I
−17.0176 − 3.4378I 1.85943 + 0.49918I
u = −2.29836 + 0.10169I
a = −0.0967724 − 0.1007040I
b = −2.53102 − 0.11992I
2.86702 + 4.05320I 1.72472 − 1.91622I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −2.29836 − 0.10169I
a = −0.0967724 + 0.1007040I
b = −2.53102 + 0.11992I
2.86702 − 4.05320I 1.72472 + 1.91622I
6
II. I
u
2
= hb − 1, −u
5
− u
4
+ u
3
+ 2u
2
+ a, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
10
=
u
5
+ u
4
− u
3
− 2u
2
1
a
6
=
−u
5
− u
4
− 2u
u
2
+ u − 1
a
9
=
u
5
+ u
4
− u
3
− 2u
2
− 1
1
a
8
=
−u
4
+ u
2
− 1
u
5
+ u
4
− 2u
3
− u
2
+ u + 1
a
3
=
−u
2
+ 1
u
2
a
11
=
u
5
+ u
4
− u
3
− 2u
2
1
a
7
=
−u
−u
3
+ u
a
7
=
−u
−u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
5
+ u
4
− u
3
− 2u
2
− 3u + 5
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
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
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
c
9
(u + 1)
6
c
10
u
6
c
11
(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
8
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
9
, c
11
(y −1)
6
c
10
y
6
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002190 + 0.295542I
a = −1.91798 + 0.27071I
b = 1.00000
−0.245672 − 0.924305I −0.60470 − 5.55069I
u = 1.002190 − 0.295542I
a = −1.91798 − 0.27071I
b = 1.00000
−0.245672 + 0.924305I −0.60470 + 5.55069I
u = −0.428243 + 0.664531I
a = −0.314804 + 1.063260I
b = 1.00000
3.53554 − 0.92430I 6.31051 + 0.25702I
u = −0.428243 − 0.664531I
a = −0.314804 − 1.063260I
b = 1.00000
3.53554 + 0.92430I 6.31051 − 0.25702I
u = −1.073950 + 0.558752I
a = −0.267214 + 0.381252I
b = 1.00000
1.64493 + 5.69302I 0.29418 − 8.33058I
u = −1.073950 − 0.558752I
a = −0.267214 − 0.381252I
b = 1.00000
1.64493 − 5.69302I 0.29418 + 8.33058I
10
III. I
u
3
= hb − a + 1, a
5
− 4a
4
+ 4a
3
+ a
2
− 2a − 1, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
10
=
a
a − 1
a
6
=
a
2
− a − 1
a
2
− 2a + 1
a
9
=
1
a − 1
a
8
=
0
a
4
− 5a
3
+ 8a
2
− 3a − 2
a
3
=
0
1
a
11
=
a
a
2
− 2a + 1
a
7
=
0
a
4
− 5a
3
+ 8a
2
− 3a − 2
a
7
=
0
a
4
− 5a
3
+ 8a
2
− 3a − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3a
4
+ 13a
3
− 19a
2
+ a + 13
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
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
8
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
10
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
11
u
5
+ u
4
− 2u
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
, c
11
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
6
, c
10
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
8
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 = 1.00000
a = 1.30992 + 0.54991I
b = 0.309916 + 0.549911I
−1.31583 + 1.53058I 1.45754 − 4.40323I
u = 1.00000
a = 1.30992 − 0.54991I
b = 0.309916 − 0.549911I
−1.31583 − 1.53058I 1.45754 + 4.40323I
u = 1.00000
a = −0.418784 + 0.219165I
b = −1.41878 + 0.21917I
4.22763 − 4.40083I 10.04378 + 5.20937I
u = 1.00000
a = −0.418784 − 0.219165I
b = −1.41878 − 0.21917I
4.22763 + 4.40083I 10.04378 − 5.20937I
u = 1.00000
a = 2.21774
b = 1.21774
0.756147 −9.00270
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
− 2u
10
+ ··· + 136u − 1357)
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
+ 13u
9
+ ··· + 66u − 101)
c
7
u
5
(u
6
+ u
5
+ ··· + u + 1)(u
11
+ u
10
+ ··· + 96u − 32)
c
8
(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)
c
9
((u + 1)
6
)(u
5
− u
4
+ ··· + u + 1)(u
11
+ 11u
10
+ ··· − u − 1)
c
10
u
6
(u
5
− u
4
+ ··· + u − 1)(u
11
− u
10
+ ··· − 192u − 64)
c
11
((u − 1)
6
)(u
5
+ u
4
+ ··· + u − 1)(u
11
+ 11u
10
+ ··· − u − 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
− 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
6
(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
8
(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)
c
9
, c
11
((y −1)
6
)(y
5
− 5y
4
+ ··· − y −1)(y
11
− 27y
10
+ ··· − 171y −1)
c
10
y
6
(y
5
+ 3y
4
+ ··· − y −1)(y
11
+ 27y
10
+ ··· + 4096y −4096)
16
