11a
55
(K11a
55
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 10 11 9 3 5 6 7
Solving Sequence
6,10
11 7
1,2
3 5 4 9 8
c
10
c
6
c
11
c
2
c
5
c
4
c
9
c
8
c
1
, c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
36
− 23u
34
+ ··· + b − 1, −u
36
+ u
35
+ ··· + a − 2, u
37
− 2u
36
+ ··· + u + 1i
I
u
2
= hb, a − u − 1, u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 39 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
= hu
36
−23u
34
+· · ·+b−1, −u
36
+u
35
+· · ·+a−2, u
37
−2u
36
+· · ·+u+1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
7
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
u
36
− u
35
+ ··· − 7u + 2
−u
36
+ 23u
34
+ ··· − 7u
2
+ 1
a
3
=
2u
36
− u
35
+ ··· − 7u + 1
−2u
36
+ 46u
34
+ ··· + u + 2
a
5
=
−u
u
a
4
=
u
35
− u
34
+ ··· + 6u − 2
−u
26
+ 16u
24
+ ··· − 5u
2
+ 2u
a
9
=
−u
2
+ 1
u
2
a
8
=
−u
7
+ 4u
5
− 4u
3
+ 2u
u
7
− 3u
5
+ u
a
8
=
−u
7
+ 4u
5
− 4u
3
+ 2u
u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −8u
36
+ 11u
35
+ ··· + 32u + 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
37
− 3u
36
+ ··· − 2u + 1
c
2
u
37
+ 19u
36
+ ··· + 4u + 1
c
3
, c
8
u
37
− u
36
+ ··· + 3u
2
+ 4
c
5
, c
6
, c
9
c
10
, c
11
u
37
− 2u
36
+ ··· + u + 1
c
7
u
37
− 15u
36
+ ··· − 24u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
37
− 19y
36
+ ··· + 4y −1
c
2
y
37
+ y
36
+ ··· − 44y −1
c
3
, c
8
y
37
+ 15y
36
+ ··· − 24y −16
c
5
, c
6
, c
9
c
10
, c
11
y
37
− 48y
36
+ ··· + 25y −1
c
7
y
37
+ 11y
36
+ ··· + 7712y −256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.957621 + 0.312318I
a = 0.073185 − 0.193326I
b = −0.598010 + 0.868889I
3.73036 − 4.62550I 7.76738 + 4.90690I
u = −0.957621 − 0.312318I
a = 0.073185 + 0.193326I
b = −0.598010 − 0.868889I
3.73036 + 4.62550I 7.76738 − 4.90690I
u = −0.949350 + 0.385280I
a = −0.633743 + 1.188910I
b = −0.19879 − 2.12627I
1.18638 − 9.75247I 4.12651 + 8.53256I
u = −0.949350 − 0.385280I
a = −0.633743 − 1.188910I
b = −0.19879 + 2.12627I
1.18638 + 9.75247I 4.12651 − 8.53256I
u = 0.883228 + 0.295441I
a = −0.51817 − 1.46272I
b = −0.41139 + 2.28971I
−0.53133 + 3.88210I 2.57643 − 5.18911I
u = 0.883228 − 0.295441I
a = −0.51817 + 1.46272I
b = −0.41139 − 2.28971I
−0.53133 − 3.88210I 2.57643 + 5.18911I
u = −1.092520 + 0.081336I
a = −0.138007 + 0.822702I
b = −0.51460 − 1.46532I
6.20655 − 2.48097I 9.67939 + 3.72325I
u = −1.092520 − 0.081336I
a = −0.138007 − 0.822702I
b = −0.51460 + 1.46532I
6.20655 + 2.48097I 9.67939 − 3.72325I
u = −0.821917 + 0.258796I
a = 1.54271 − 0.10934I
b = −0.128202 + 0.262204I
−1.03449 − 1.41041I 2.89217 + 4.96755I
u = −0.821917 − 0.258796I
a = 1.54271 + 0.10934I
b = −0.128202 − 0.262204I
−1.03449 + 1.41041I 2.89217 − 4.96755I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.819155 + 0.099014I
a = 0.507337 + 0.293800I
b = −0.977537 − 0.679950I
1.50853 + 0.14938I 6.45155 + 0.46456I
u = 0.819155 − 0.099014I
a = 0.507337 − 0.293800I
b = −0.977537 + 0.679950I
1.50853 − 0.14938I 6.45155 − 0.46456I
u = 0.669935 + 0.434127I
a = 1.373890 + 0.079928I
b = 0.053060 − 0.300111I
−0.46634 − 2.82395I 2.81248 + 2.07751I
u = 0.669935 − 0.434127I
a = 1.373890 − 0.079928I
b = 0.053060 + 0.300111I
−0.46634 + 2.82395I 2.81248 − 2.07751I
u = 0.126557 + 0.616394I
a = −0.649911 − 0.790815I
b = 0.194995 − 1.112060I
−2.10926 + 6.36685I −0.76306 − 6.73734I
u = 0.126557 − 0.616394I
a = −0.649911 + 0.790815I
b = 0.194995 + 1.112060I
−2.10926 − 6.36685I −0.76306 + 6.73734I
u = 0.446224 + 0.376427I
a = 0.324362 − 0.529872I
b = −0.123832 − 0.626016I
1.20413 + 1.03970I 6.27276 − 4.95197I
u = 0.446224 − 0.376427I
a = 0.324362 + 0.529872I
b = −0.123832 + 0.626016I
1.20413 − 1.03970I 6.27276 + 4.95197I
u = 0.164699 + 0.507419I
a = 1.129790 − 0.016387I
b = 0.171879 − 0.083354I
0.29596 + 1.82108I 2.47769 − 3.83748I
u = 0.164699 − 0.507419I
a = 1.129790 + 0.016387I
b = 0.171879 + 0.083354I
0.29596 − 1.82108I 2.47769 + 3.83748I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.041396 + 0.496138I
a = −0.92411 + 1.21544I
b = 0.418405 + 1.058430I
−3.33947 − 1.17576I −4.43128 + 1.03066I
u = −0.041396 − 0.496138I
a = −0.92411 − 1.21544I
b = 0.418405 − 1.058430I
−3.33947 + 1.17576I −4.43128 − 1.03066I
u = −1.59215 + 0.06066I
a = −0.579041 − 0.150932I
b = −0.047557 + 0.344022I
7.10974 + 1.19498I 0
u = −1.59215 − 0.06066I
a = −0.579041 + 0.150932I
b = −0.047557 − 0.344022I
7.10974 − 1.19498I 0
u = 1.67626 + 0.05941I
a = −0.503100 + 0.060196I
b = −0.271772 − 0.167857I
7.80383 + 2.56815I 0
u = 1.67626 − 0.05941I
a = −0.503100 − 0.060196I
b = −0.271772 + 0.167857I
7.80383 − 2.56815I 0
u = −1.68061 + 0.03427I
a = −1.41650 + 1.55888I
b = 1.66994 − 1.91855I
10.43060 − 0.72718I 0
u = −1.68061 − 0.03427I
a = −1.41650 − 1.55888I
b = 1.66994 + 1.91855I
10.43060 + 0.72718I 0
u = −1.68650 + 0.07280I
a = 0.09784 − 3.23140I
b = 0.52086 + 3.57531I
8.52780 − 5.28278I 0
u = −1.68650 − 0.07280I
a = 0.09784 + 3.23140I
b = 0.52086 − 3.57531I
8.52780 + 5.28278I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.70104 + 0.10270I
a = 0.45064 + 2.76832I
b = 0.21974 − 3.16301I
10.4876 + 11.6846I 0
u = 1.70104 − 0.10270I
a = 0.45064 − 2.76832I
b = 0.21974 + 3.16301I
10.4876 − 11.6846I 0
u = 1.70490 + 0.08169I
a = −0.92941 − 1.36566I
b = 1.13242 + 1.88410I
13.1327 + 6.1887I 0
u = 1.70490 − 0.08169I
a = −0.92941 + 1.36566I
b = 1.13242 − 1.88410I
13.1327 − 6.1887I 0
u = 1.73093 + 0.01518I
a = −0.67256 + 2.24479I
b = 1.13578 − 2.69126I
16.2880 + 2.8364I 0
u = 1.73093 − 0.01518I
a = −0.67256 − 2.24479I
b = 1.13578 + 2.69126I
16.2880 − 2.8364I 0
u = −0.201734
a = 3.92958
b = 0.509239
−1.30402 −9.26700
8
II. I
u
2
= hb, a − u − 1, u
2
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u − 1
a
7
=
u
−u + 1
a
1
=
u
−u
a
2
=
u + 1
0
a
3
=
1
u
a
5
=
−u
u
a
4
=
1
u
a
9
=
u
−u + 1
a
8
=
u
−u + 1
a
8
=
u
−u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
c
2
, c
4
(u + 1)
2
c
3
, c
7
, c
8
u
2
c
5
, c
6
u
2
− u − 1
c
9
, c
10
, c
11
u
2
+ u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y − 1)
2
c
3
, c
7
, c
8
y
2
c
5
, c
6
, c
9
c
10
, c
11
y
2
− 3y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.61803
b = 0
−0.657974 5.00000
u = −1.61803
a = −0.618034
b = 0
7.23771 5.00000
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
2
)(u
37
− 3u
36
+ ··· − 2u + 1)
c
2
((u + 1)
2
)(u
37
+ 19u
36
+ ··· + 4u + 1)
c
3
, c
8
u
2
(u
37
− u
36
+ ··· + 3u
2
+ 4)
c
4
((u + 1)
2
)(u
37
− 3u
36
+ ··· − 2u + 1)
c
5
, c
6
(u
2
− u − 1)(u
37
− 2u
36
+ ··· + u + 1)
c
7
u
2
(u
37
− 15u
36
+ ··· − 24u + 16)
c
9
, c
10
, c
11
(u
2
+ u − 1)(u
37
− 2u
36
+ ··· + u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y − 1)
2
)(y
37
− 19y
36
+ ··· + 4y − 1)
c
2
((y − 1)
2
)(y
37
+ y
36
+ ··· − 44y − 1)
c
3
, c
8
y
2
(y
37
+ 15y
36
+ ··· − 24y − 16)
c
5
, c
6
, c
9
c
10
, c
11
(y
2
− 3y + 1)(y
37
− 48y
36
+ ··· + 25y − 1)
c
7
y
2
(y
37
+ 11y
36
+ ··· + 7712y − 256)
14
