11n
29
(K11n
29
)
A knot diagram
1
Linearized knot diagam
4 1 6 2 10 4 5 11 6 9 8
Solving Sequence
5,10 2,6
4 7 1 3 9 11 8
c
5
c
4
c
6
c
1
c
3
c
9
c
10
c
8
c
2
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
27
− u
26
+ ··· + b + 2u, u
25
− u
24
+ ··· + a − 2, u
29
− 2u
28
+ ··· + 3u − 1i
I
u
2
= hb + 1, −u
2
+ a − u, u
4
+ u
3
+ u
2
+ 1i
* 2 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
= hu
27
− u
26
+ · · · + b + 2u, u
25
− u
24
+ · · · + a − 2, u
29
− 2u
28
+ · · · + 3u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
−u
25
+ u
24
+ ··· − 3u + 2
−u
27
+ u
26
+ ··· + 2u
2
− 2u
a
6
=
1
u
2
a
4
=
−u
27
+ u
26
+ ··· − 4u + 3
−u
27
+ u
26
+ ··· + u
2
− u
a
7
=
−u
7
− 2u
3
u
7
+ u
5
+ 2u
3
+ u
a
1
=
−u
7
− 2u
3
−u
9
− u
7
− 3u
5
− 2u
3
− u
a
3
=
u
28
− 3u
27
+ ··· − 8u + 4
−u
28
− u
27
+ ··· − 6u
3
− u
2
a
9
=
u
u
3
+ u
a
11
=
u
3
u
5
+ u
3
+ u
a
8
=
u
5
+ u
u
7
+ u
5
+ 2u
3
+ u
a
8
=
u
5
+ u
u
7
+ u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
28
+ 4u
27
−19u
26
+ 14u
25
−67u
24
+ 43u
23
−162u
22
+ 82u
21
−
317u
20
+ 116u
19
− 481u
18
+ 100u
17
− 607u
16
+ 2u
15
− 600u
14
− 158u
13
− 488u
12
−
304u
11
−310u
10
−342u
9
−172u
8
−265u
7
−92u
6
−122u
5
−65u
4
−24u
3
−29u
2
+ 4u −7
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
29
− 5u
28
+ ··· − 5u + 1
c
2
u
29
+ 9u
28
+ ··· + 13u + 1
c
3
, c
6
u
29
− u
28
+ ··· + 8u + 16
c
5
, c
9
u
29
+ 2u
28
+ ··· + 3u + 1
c
7
u
29
+ 2u
28
+ ··· + 3u + 1
c
8
, c
10
, c
11
u
29
− 8u
28
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
29
− 9y
28
+ ··· + 13y −1
c
2
y
29
+ 27y
28
+ ··· − 111y −1
c
3
, c
6
y
29
+ 27y
28
+ ··· − 2752y −256
c
5
, c
9
y
29
+ 8y
28
+ ··· + 3y −1
c
7
y
29
− 32y
28
+ ··· + 3y −1
c
8
, c
10
, c
11
y
29
+ 28y
28
+ ··· + 123y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.231265 + 1.046420I
a = −1.57334 − 1.28254I
b = 0.858634 + 0.959440I
7.61033 − 0.10509I 1.75140 − 0.82130I
u = −0.231265 − 1.046420I
a = −1.57334 + 1.28254I
b = 0.858634 − 0.959440I
7.61033 + 0.10509I 1.75140 + 0.82130I
u = −0.312663 + 1.045390I
a = −0.10596 + 2.52791I
b = 1.014760 − 0.890779I
7.11886 + 6.67995I 0.53493 − 6.07824I
u = −0.312663 − 1.045390I
a = −0.10596 − 2.52791I
b = 1.014760 + 0.890779I
7.11886 − 6.67995I 0.53493 + 6.07824I
u = 0.822501 + 0.730493I
a = 0.512567 − 0.480050I
b = 0.633017 + 0.915825I
0.636180 − 0.689036I −3.76307 + 1.94423I
u = 0.822501 − 0.730493I
a = 0.512567 + 0.480050I
b = 0.633017 − 0.915825I
0.636180 + 0.689036I −3.76307 − 1.94423I
u = 0.194951 + 0.848946I
a = 0.90551 + 1.74398I
b = −0.405971 − 0.466803I
1.06975 − 1.85093I 1.30743 + 5.79968I
u = 0.194951 − 0.848946I
a = 0.90551 − 1.74398I
b = −0.405971 + 0.466803I
1.06975 + 1.85093I 1.30743 − 5.79968I
u = 0.450225 + 0.741417I
a = 0.877674 − 0.501097I
b = 0.282831 + 0.220896I
−0.03811 − 1.72919I −0.45461 + 4.60784I
u = 0.450225 − 0.741417I
a = 0.877674 + 0.501097I
b = 0.282831 − 0.220896I
−0.03811 + 1.72919I −0.45461 − 4.60784I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.788076 + 0.837379I
a = 0.263911 + 0.556243I
b = −0.840182 − 0.602881I
−4.79824 + 0.50805I −6.89911 − 0.01309I
u = −0.788076 − 0.837379I
a = 0.263911 − 0.556243I
b = −0.840182 + 0.602881I
−4.79824 − 0.50805I −6.89911 + 0.01309I
u = 0.882629 + 0.776302I
a = 0.523074 + 0.300180I
b = 1.103310 − 0.772041I
−0.78075 + 5.57785I −5.50657 − 2.77090I
u = 0.882629 − 0.776302I
a = 0.523074 − 0.300180I
b = 1.103310 + 0.772041I
−0.78075 − 5.57785I −5.50657 + 2.77090I
u = 0.784724 + 0.886082I
a = −1.19979 + 1.01167I
b = −1.337190 − 0.029086I
−6.30586 − 2.95151I −5.76823 + 2.64939I
u = 0.784724 − 0.886082I
a = −1.19979 − 1.01167I
b = −1.337190 + 0.029086I
−6.30586 + 2.95151I −5.76823 − 2.64939I
u = −0.767855 + 0.926785I
a = −0.44228 − 1.68632I
b = −0.771267 + 0.660663I
−4.52351 + 5.35315I −5.93805 − 5.66710I
u = −0.767855 − 0.926785I
a = −0.44228 + 1.68632I
b = −0.771267 − 0.660663I
−4.52351 − 5.35315I −5.93805 + 5.66710I
u = 0.750471 + 0.994480I
a = −0.996828 + 0.026809I
b = 0.648434 − 1.004480I
1.43095 − 5.20261I −2.56531 + 3.25116I
u = 0.750471 − 0.994480I
a = −0.996828 − 0.026809I
b = 0.648434 + 1.004480I
1.43095 + 5.20261I −2.56531 − 3.25116I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.866458 + 0.916553I
a = 0.906367 + 0.416512I
b = 0.675975 − 0.021605I
−7.73749 + 3.20954I −0.41591 − 2.86957I
u = −0.866458 − 0.916553I
a = 0.906367 − 0.416512I
b = 0.675975 + 0.021605I
−7.73749 − 3.20954I −0.41591 + 2.86957I
u = −0.719374 + 0.070912I
a = 0.523981 − 0.369051I
b = 0.914734 + 0.838366I
3.96205 − 3.12839I −4.70122 + 2.58517I
u = −0.719374 − 0.070912I
a = 0.523981 + 0.369051I
b = 0.914734 − 0.838366I
3.96205 + 3.12839I −4.70122 − 2.58517I
u = 0.793942 + 1.004110I
a = 1.14222 − 1.86384I
b = 1.130840 + 0.799307I
−0.06814 − 11.79740I −4.44971 + 7.37898I
u = 0.793942 − 1.004110I
a = 1.14222 + 1.86384I
b = 1.130840 − 0.799307I
−0.06814 + 11.79740I −4.44971 − 7.37898I
u = −0.146225 + 0.649247I
a = 0.11069 − 2.17375I
b = −1.073810 + 0.142900I
−1.181700 + 0.773921I −1.52981 + 2.72477I
u = −0.146225 − 0.649247I
a = 0.11069 + 2.17375I
b = −1.073810 − 0.142900I
−1.181700 − 0.773921I −1.52981 − 2.72477I
u = 0.304949
a = 1.10441
b = −0.668226
−1.01334 −10.2040
7
II. I
u
2
= hb + 1, −u
2
+ a − u, u
4
+ u
3
+ u
2
+ 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
2
=
u
2
+ u
−1
a
6
=
1
u
2
a
4
=
u
2
+ u + 1
−1
a
7
=
1
u
2
a
1
=
−1
0
a
3
=
u
2
+ u + 1
−1
a
9
=
u
u
3
+ u
a
11
=
u
3
u
3
+ u
2
+ 1
a
8
=
u
2
+ 1
u
2
a
8
=
u
2
+ 1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
2
+ 6u − 7
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
4
c
2
, c
4
(u + 1)
4
c
3
, c
6
u
4
c
5
u
4
+ u
3
+ u
2
+ 1
c
7
, c
10
, c
11
u
4
− u
3
+ 3u
2
− 2u + 1
c
8
u
4
+ u
3
+ 3u
2
+ 2u + 1
c
9
u
4
− u
3
+ u
2
+ 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
6
y
4
c
5
, c
9
y
4
+ y
3
+ 3y
2
+ 2y + 1
c
7
, c
8
, c
10
c
11
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351808 + 0.720342I
a = −0.043315 + 1.227190I
b = −1.00000
−1.43393 − 1.41510I −6.86477 + 6.85627I
u = 0.351808 − 0.720342I
a = −0.043315 − 1.227190I
b = −1.00000
−1.43393 + 1.41510I −6.86477 − 6.85627I
u = −0.851808 + 0.911292I
a = −0.956685 − 0.641200I
b = −1.00000
−8.43568 + 3.16396I −12.63523 − 2.29471I
u = −0.851808 − 0.911292I
a = −0.956685 + 0.641200I
b = −1.00000
−8.43568 − 3.16396I −12.63523 + 2.29471I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
29
− 5u
28
+ ··· − 5u + 1)
c
2
((u + 1)
4
)(u
29
+ 9u
28
+ ··· + 13u + 1)
c
3
, c
6
u
4
(u
29
− u
28
+ ··· + 8u + 16)
c
4
((u + 1)
4
)(u
29
− 5u
28
+ ··· − 5u + 1)
c
5
(u
4
+ u
3
+ u
2
+ 1)(u
29
+ 2u
28
+ ··· + 3u + 1)
c
7
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
29
+ 2u
28
+ ··· + 3u + 1)
c
8
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
29
− 8u
28
+ ··· + 3u + 1)
c
9
(u
4
− u
3
+ u
2
+ 1)(u
29
+ 2u
28
+ ··· + 3u + 1)
c
10
, c
11
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
29
− 8u
28
+ ··· + 3u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
29
− 9y
28
+ ··· + 13y −1)
c
2
((y −1)
4
)(y
29
+ 27y
28
+ ··· − 111y −1)
c
3
, c
6
y
4
(y
29
+ 27y
28
+ ··· − 2752y −256)
c
5
, c
9
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
29
+ 8y
28
+ ··· + 3y −1)
c
7
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
29
− 32y
28
+ ··· + 3y −1)
c
8
, c
10
, c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
29
+ 28y
28
+ ··· + 123y −1)
13
