11n
30
(K11n
30
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 8 10 4 1 11 6 9
Solving Sequence
1,4
2
5,8
6 9 7 3 11 10
c
1
c
4
c
5
c
8
c
7
c
3
c
11
c
10
c
2
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
19
− 3u
18
+ ··· + 4b + 8, −3u
19
− 18u
18
+ ··· + 4a + 9, u
20
+ 5u
19
+ ··· − 6u − 1i
I
u
2
= hb
4
+ b
3
+ 3b
2
+ 2b + 1, a, u − 1i
* 2 irreducible components of dim
C
= 0, with total 24 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
19
−3u
18
+· · ·+4b+8, −3u
19
−18u
18
+· · ·+4a+9, u
20
+5u
19
+· · ·−6u−1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
8
=
3
4
u
19
+
9
2
u
18
+ ··· −
71
4
u −
9
4
−
1
4
u
19
+
3
4
u
18
+ ··· −
25
4
u − 2
a
6
=
−1
−
1
8
u
19
−
1
2
u
18
+ ··· +
21
8
u +
1
8
a
9
=
u
19
+
15
4
u
18
+ ··· −
23
2
u −
1
4
−
1
4
u
19
+
3
4
u
18
+ ··· −
25
4
u − 2
a
7
=
3
4
u
19
+
9
2
u
18
+ ··· −
71
4
u −
9
4
2u
19
+
29
4
u
18
+ ··· −
23
2
u −
11
4
a
3
=
−u
2
+ 1
u
2
a
11
=
−
1
8
u
19
−
1
2
u
18
+ ··· −
3
8
u +
17
8
1
8
u
19
+
1
2
u
18
+ ··· −
13
8
u −
1
8
a
10
=
−
1
4
u
19
−
5
4
u
18
+ ··· −
7
4
u +
5
2
−
5
2
u
19
− 10u
18
+ ··· + 13u + 3
a
10
=
−
1
4
u
19
−
5
4
u
18
+ ··· −
7
4
u +
5
2
−
5
2
u
19
− 10u
18
+ ··· + 13u + 3
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
7
2
u
19
−
65
4
u
18
+
33
4
u
17
+
241
2
u
16
+
57
4
u
15
−
877
2
u
14
− u
13
+
4221
4
u
12
− 344u
11
−
6731
4
u
10
+
2343
2
u
9
+
6133
4
u
8
−
3593
2
u
7
−491u
6
+ 1313u
5
−
457
2
u
4
−
783
2
u
3
+
281
4
u
2
+ 50u +
21
4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
20
− 5u
19
+ ··· + 6u − 1
c
2
u
20
+ 27u
19
+ ··· − 12u + 1
c
3
, c
7
u
20
+ u
19
+ ··· − 40u − 16
c
5
u
20
− 2u
19
+ ··· + 2u + 1
c
6
, c
10
u
20
− 2u
19
+ ··· + 2u + 1
c
8
, c
9
, c
11
u
20
+ 6u
19
+ ··· − 6u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
20
− 27y
19
+ ··· + 12y + 1
c
2
y
20
− 63y
19
+ ··· + 564y + 1
c
3
, c
7
y
20
− 27y
19
+ ··· + 960y + 256
c
5
y
20
− 42y
19
+ ··· − 6y + 1
c
6
, c
10
y
20
+ 6y
19
+ ··· − 6y + 1
c
8
, c
9
, c
11
y
20
+ 18y
19
+ ··· − 142y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.593945 + 0.749573I
a = 0.901534 − 0.870244I
b = −0.062242 + 1.190030I
1.80617 + 0.15475I −5.78761 + 0.24947I
u = 0.593945 − 0.749573I
a = 0.901534 + 0.870244I
b = −0.062242 − 1.190030I
1.80617 − 0.15475I −5.78761 − 0.24947I
u = 0.742600 + 0.805837I
a = −0.823991 + 0.896190I
b = −0.361535 − 1.366300I
1.34574 − 5.46019I −7.11600 + 5.63427I
u = 0.742600 − 0.805837I
a = −0.823991 − 0.896190I
b = −0.361535 + 1.366300I
1.34574 + 5.46019I −7.11600 − 5.63427I
u = 1.049030 + 0.433248I
a = −0.533800 + 0.720846I
b = −0.786836 − 0.158628I
−3.47404 − 1.25358I −14.5901 + 1.6218I
u = 1.049030 − 0.433248I
a = −0.533800 − 0.720846I
b = −0.786836 + 0.158628I
−3.47404 + 1.25358I −14.5901 − 1.6218I
u = 0.723331
a = 0.811887
b = 0.0840139
−1.09578 −8.64200
u = 1.41765 + 0.08558I
a = −0.075401 + 0.848821I
b = −0.300271 + 1.184320I
−0.40356 + 2.62035I −6.94831 − 3.53102I
u = 1.41765 − 0.08558I
a = −0.075401 − 0.848821I
b = −0.300271 − 1.184320I
−0.40356 − 2.62035I −6.94831 + 3.53102I
u = −0.494818 + 0.034941I
a = 0.071730 − 1.233390I
b = −0.12366 − 1.50920I
6.00682 − 3.10793I 1.19914 + 2.44206I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.494818 − 0.034941I
a = 0.071730 + 1.233390I
b = −0.12366 + 1.50920I
6.00682 + 3.10793I 1.19914 − 2.44206I
u = −1.67897 + 0.22535I
a = 1.055750 − 0.143916I
b = 0.293817 − 1.327320I
−6.07483 + 3.49044I −7.50331 − 0.69756I
u = −1.67897 − 0.22535I
a = 1.055750 + 0.143916I
b = 0.293817 + 1.327320I
−6.07483 − 3.49044I −7.50331 + 0.69756I
u = −1.73062
a = 1.07368
b = 0.672482
−10.3113 −7.59680
u = −1.71507 + 0.27164I
a = −1.083580 + 0.166912I
b = −0.46653 + 1.62043I
−7.04125 + 9.73657I −8.64627 − 5.28115I
u = −1.71507 − 0.27164I
a = −1.083580 − 0.166912I
b = −0.46653 − 1.62043I
−7.04125 − 9.73657I −8.64627 + 5.28115I
u = −0.098700 + 0.173726I
a = 1.16971 − 1.93438I
b = −0.407505 − 0.376718I
−0.333685 − 1.164940I −4.31355 + 5.64475I
u = −0.098700 − 0.173726I
a = 1.16971 + 1.93438I
b = −0.407505 + 0.376718I
−0.333685 + 1.164940I −4.31355 − 5.64475I
u = −1.81202 + 0.09860I
a = −1.124740 + 0.057142I
b = −1.163490 + 0.628217I
−14.0917 + 3.7151I −12.67457 − 3.10159I
u = −1.81202 − 0.09860I
a = −1.124740 − 0.057142I
b = −1.163490 − 0.628217I
−14.0917 − 3.7151I −12.67457 + 3.10159I
6
II. I
u
2
= hb
4
+ b
3
+ 3b
2
+ 2b + 1, a, u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
−1
0
a
8
=
0
b
a
6
=
−1
−b
2
a
9
=
−b
b
a
7
=
0
b
a
3
=
0
1
a
11
=
b
2
+ 1
−b
2
a
10
=
−b
3
− 2b
b
3
+ b
a
10
=
−b
3
− 2b
b
3
+ b
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2b
3
− 2b
2
− 7b − 13
7
(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
7
u
4
c
5
, c
8
, c
9
u
4
− u
3
+ 3u
2
− 2u + 1
c
6
u
4
− u
3
+ u
2
+ 1
c
10
u
4
+ u
3
+ u
2
+ 1
c
11
u
4
+ u
3
+ 3u
2
+ 2u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
4
c
3
, c
7
y
4
c
5
, c
8
, c
9
c
11
y
4
+ 5y
3
+ 7y
2
+ 2y + 1
c
6
, c
10
y
4
+ y
3
+ 3y
2
+ 2y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = −0.395123 + 0.506844I
−1.85594 + 1.41510I −10.51825 − 2.96122I
u = 1.00000
a = 0
b = −0.395123 − 0.506844I
−1.85594 − 1.41510I −10.51825 + 2.96122I
u = 1.00000
a = 0
b = −0.10488 + 1.55249I
5.14581 + 3.16396I −8.98175 − 2.83489I
u = 1.00000
a = 0
b = −0.10488 − 1.55249I
5.14581 − 3.16396I −8.98175 + 2.83489I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
4
)(u
20
− 5u
19
+ ··· + 6u − 1)
c
2
((u + 1)
4
)(u
20
+ 27u
19
+ ··· − 12u + 1)
c
3
, c
7
u
4
(u
20
+ u
19
+ ··· − 40u − 16)
c
4
((u + 1)
4
)(u
20
− 5u
19
+ ··· + 6u − 1)
c
5
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
20
− 2u
19
+ ··· + 2u + 1)
c
6
(u
4
− u
3
+ u
2
+ 1)(u
20
− 2u
19
+ ··· + 2u + 1)
c
8
, c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)(u
20
+ 6u
19
+ ··· − 6u + 1)
c
10
(u
4
+ u
3
+ u
2
+ 1)(u
20
− 2u
19
+ ··· + 2u + 1)
c
11
(u
4
+ u
3
+ 3u
2
+ 2u + 1)(u
20
+ 6u
19
+ ··· − 6u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
4
)(y
20
− 27y
19
+ ··· + 12y + 1)
c
2
((y −1)
4
)(y
20
− 63y
19
+ ··· + 564y + 1)
c
3
, c
7
y
4
(y
20
− 27y
19
+ ··· + 960y + 256)
c
5
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
20
− 42y
19
+ ··· − 6y + 1)
c
6
, c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)(y
20
+ 6y
19
+ ··· − 6y + 1)
c
8
, c
9
, c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)(y
20
+ 18y
19
+ ··· − 142y + 1)
12
