11n
122
(K11n
122
)
A knot diagram
1
Linearized knot diagam
7 1 9 7 10 2 11 4 1 5 8
Solving Sequence
7,11
8 1 2
3,5
4 6 10 9
c
7
c
11
c
1
c
2
c
4
c
6
c
10
c
9
c
3
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
15
− u
14
− 5u
13
+ 5u
12
+ 7u
11
− 9u
10
+ 7u
9
− u
8
− 25u
7
+ 17u
6
+ 9u
5
− 7u
4
+ 17u
3
− 13u
2
+ 4b − 2u − 4,
− u
15
+ 6u
13
+ ··· + 4a − 4, u
16
− 2u
15
+ ··· + u + 2i
I
u
2
= h−u
4
− u
3
+ u
2
+ b + u, −u
4
+ 2u
2
+ a − 1, u
6
− 3u
4
+ 2u
2
+ 1i
I
u
3
= ha
2
+ b, a
3
+ a − 1, u + 1i
* 3 irreducible components of dim
C
= 0, with total 25 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
15
−u
14
+· · ·+4b−4, −u
15
+6u
13
+· · ·+4a−4, u
16
−2u
15
+· · ·+u+2i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
1
=
u
−u
3
+ u
a
2
=
−u
3
+ 2u
−u
3
+ u
a
3
=
u
7
− 2u
5
+ 2u
−u
9
+ 3u
7
− 3u
5
+ u
a
5
=
1
4
u
15
−
3
2
u
13
+ ··· −
9
4
u + 1
−
1
4
u
15
+
1
4
u
14
+ ··· +
1
2
u + 1
a
4
=
1
2
u
15
−
1
4
u
14
+ ··· +
1
4
u
2
−
11
4
u
−
1
4
u
15
+
1
4
u
14
+ ··· +
1
2
u + 1
a
6
=
u
6
− 3u
4
+ 2u
2
+ 1
u
6
− 2u
4
+ u
2
a
10
=
1
4
u
13
−
5
4
u
11
+ ··· +
3
4
u + 1
1
2
u
10
− 2u
8
+ ··· −
5
2
u
2
+
1
2
u
a
9
=
−
1
2
u
15
+
1
4
u
14
+ ··· +
7
4
u + 2
1
2
u
15
−
1
2
u
14
+ ··· −
1
4
u −
1
2
a
9
=
−
1
2
u
15
+
1
4
u
14
+ ··· +
7
4
u + 2
1
2
u
15
−
1
2
u
14
+ ··· −
1
4
u −
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
15
+ 12u
13
− 2u
12
− 28u
11
+ 10u
10
+ 20u
9
− 18u
8
+ 28u
7
+
10u
6
− 52u
5
+ 6u
4
+ 10u
3
− 6u
2
+ 16u + 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
16
+ 3u
15
+ ··· − 163u + 62
c
2
u
16
+ 29u
15
+ ··· + 16707u + 3844
c
3
, c
8
u
16
− u
15
+ ··· + 14u + 5
c
4
u
16
+ 5u
15
+ ··· − 6u + 67
c
5
, c
10
u
16
− u
15
+ ··· + 8u + 5
c
7
, c
11
u
16
+ 2u
15
+ ··· − u + 2
c
9
u
16
− u
15
+ ··· − 2824u + 1117
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
16
+ 29y
15
+ ··· + 16707y + 3844
c
2
y
16
− 75y
15
+ ··· + 939185823y + 14776336
c
3
, c
8
y
16
+ 27y
15
+ ··· − 96y + 25
c
4
y
16
+ 19y
15
+ ··· + 15374y + 4489
c
5
, c
10
y
16
− y
15
+ ··· − 64y + 25
c
7
, c
11
y
16
− 12y
15
+ ··· + 19y + 4
c
9
y
16
+ 51y
15
+ ··· − 7186374y + 1247689
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.077517 + 1.005540I
a = −1.07927 − 1.29543I
b = −0.62616 − 1.56703I
−15.2325 + 4.4644I 1.08918 − 2.21387I
u = 0.077517 − 1.005540I
a = −1.07927 + 1.29543I
b = −0.62616 + 1.56703I
−15.2325 − 4.4644I 1.08918 + 2.21387I
u = 0.170392 + 0.771288I
a = −0.408921 + 1.021250I
b = −0.482279 + 1.104540I
−4.05827 − 0.49300I −0.617664 + 0.214534I
u = 0.170392 − 0.771288I
a = −0.408921 − 1.021250I
b = −0.482279 − 1.104540I
−4.05827 + 0.49300I −0.617664 − 0.214534I
u = 1.160690 + 0.407151I
a = 0.600692 − 0.658208I
b = −1.126220 − 0.798721I
−1.06445 + 4.80370I 3.93778 − 5.08204I
u = 1.160690 − 0.407151I
a = 0.600692 + 0.658208I
b = −1.126220 + 0.798721I
−1.06445 − 4.80370I 3.93778 + 5.08204I
u = 1.293170 + 0.155822I
a = −0.056229 + 0.786374I
b = 0.86906 + 1.53568I
5.01976 + 2.82849I 13.14002 − 4.04275I
u = 1.293170 − 0.155822I
a = −0.056229 − 0.786374I
b = 0.86906 − 1.53568I
5.01976 − 2.82849I 13.14002 + 4.04275I
u = 1.269320 + 0.545322I
a = −1.162000 − 0.309874I
b = −0.241796 + 0.780806I
−11.56800 + 1.02407I 3.53875 − 0.89724I
u = 1.269320 − 0.545322I
a = −1.162000 + 0.309874I
b = −0.241796 − 0.780806I
−11.56800 − 1.02407I 3.53875 + 0.89724I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.374820 + 0.254049I
a = −0.397419 − 0.220831I
b = 0.02694 − 1.60521I
0.90149 − 3.13168I 3.23299 + 2.68195I
u = −1.374820 − 0.254049I
a = −0.397419 + 0.220831I
b = 0.02694 + 1.60521I
0.90149 + 3.13168I 3.23299 − 2.68195I
u = −1.37116 + 0.47203I
a = 0.424287 + 1.067010I
b = −1.31943 + 2.03484I
−10.6907 − 9.7305I 4.40505 + 4.74516I
u = −1.37116 − 0.47203I
a = 0.424287 − 1.067010I
b = −1.31943 − 2.03484I
−10.6907 + 9.7305I 4.40505 − 4.74516I
u = −0.225111 + 0.325313I
a = 1.32887 − 1.40325I
b = 0.399881 − 0.330960I
0.504151 − 0.997325I 7.27390 + 6.88407I
u = −0.225111 − 0.325313I
a = 1.32887 + 1.40325I
b = 0.399881 + 0.330960I
0.504151 + 0.997325I 7.27390 − 6.88407I
6
II. I
u
2
= h−u
4
− u
3
+ u
2
+ b + u, −u
4
+ 2u
2
+ a − 1, u
6
− 3u
4
+ 2u
2
+ 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
8
=
1
−u
2
a
1
=
u
−u
3
+ u
a
2
=
−u
3
+ 2u
−u
3
+ u
a
3
=
u
5
− 2u
3
+ u
−u
5
+ u
3
+ u
a
5
=
u
4
− 2u
2
+ 1
u
4
+ u
3
− u
2
− u
a
4
=
−u
3
− u
2
+ u + 1
u
4
+ u
3
− u
2
− u
a
6
=
0
u
4
− u
2
− 1
a
10
=
u
5
− 3u
3
+ 2u
u
5
+ u
4
− 2u
3
− 2u
2
+ u
a
9
=
−u
4
− u
3
+ 2u
2
+ 2u + 1
u
5
+ u
4
− 2u
3
− 3u
2
a
9
=
−u
4
− u
3
+ 2u
2
+ 2u + 1
u
5
+ u
4
− 2u
3
− 3u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
4
+ 8u
2
+ 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
6
+ u
4
+ 2u
2
+ 1
c
2
(u
3
+ u
2
+ 2u + 1)
2
c
3
, c
5
, c
8
c
10
(u
2
+ 1)
3
c
4
u
6
+ 4u
5
+ 11u
4
+ 10u
3
+ 8u
2
+ 2u + 1
c
7
, c
11
u
6
− 3u
4
+ 2u
2
+ 1
c
9
u
6
− 2u
5
− u
4
+ 8u
3
+ 12u
2
+ 6u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
3
+ y
2
+ 2y + 1)
2
c
2
(y
3
+ 3y
2
+ 2y −1)
2
c
3
, c
5
, c
8
c
10
(y + 1)
6
c
4
y
6
+ 6y
5
+ 57y
4
+ 62y
3
+ 46y
2
+ 12y + 1
c
7
, c
11
(y
3
− 3y
2
+ 2y + 1)
2
c
9
y
6
− 6y
5
+ 57y
4
− 62y
3
+ 46y
2
− 12y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.307140 + 0.215080I
a = 0.122561 + 0.744862I
b = 1.52978 + 2.18458I
3.02413 + 2.82812I 7.50976 − 2.97945I
u = 1.307140 − 0.215080I
a = 0.122561 − 0.744862I
b = 1.52978 − 2.18458I
3.02413 − 2.82812I 7.50976 + 2.97945I
u = −1.307140 + 0.215080I
a = 0.122561 − 0.744862I
b = 0.040058 − 0.429702I
3.02413 − 2.82812I 7.50976 + 2.97945I
u = −1.307140 − 0.215080I
a = 0.122561 + 0.744862I
b = 0.040058 + 0.429702I
3.02413 + 2.82812I 7.50976 − 2.97945I
u = 0.569840I
a = 1.75488
b = 0.430160 − 0.754878I
−1.11345 0.980490
u = − 0.569840I
a = 1.75488
b = 0.430160 + 0.754878I
−1.11345 0.980490
10
III. I
u
3
= ha
2
+ b, a
3
+ a − 1, u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
−1
a
8
=
1
−1
a
1
=
−1
0
a
2
=
−1
0
a
3
=
−1
0
a
5
=
a
−a
2
a
4
=
a
2
+ a
−a
2
a
6
=
1
0
a
10
=
−a
2
−a
a
9
=
−a
2
+ a
−a
a
9
=
−a
2
+ a
−a
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
3
c
3
, c
5
, c
8
c
9
, c
10
u
3
+ u − 1
c
4
u
3
− 2u
2
+ u + 1
c
7
, c
11
(u − 1)
3
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y
3
c
3
, c
5
, c
8
c
9
, c
10
y
3
+ 2y
2
+ y −1
c
4
y
3
− 2y
2
+ 5y −1
c
7
, c
11
(y −1)
3
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −0.341164 + 1.161540I
b = 1.23279 + 0.79255I
1.64493 6.00000
u = −1.00000
a = −0.341164 − 1.161540I
b = 1.23279 − 0.79255I
1.64493 6.00000
u = −1.00000
a = 0.682328
b = −0.465571
1.64493 6.00000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u
3
(u
6
+ u
4
+ 2u
2
+ 1)(u
16
+ 3u
15
+ ··· − 163u + 62)
c
2
u
3
(u
3
+ u
2
+ 2u + 1)
2
(u
16
+ 29u
15
+ ··· + 16707u + 3844)
c
3
, c
8
((u
2
+ 1)
3
)(u
3
+ u − 1)(u
16
− u
15
+ ··· + 14u + 5)
c
4
(u
3
− 2u
2
+ u + 1)(u
6
+ 4u
5
+ 11u
4
+ 10u
3
+ 8u
2
+ 2u + 1)
· (u
16
+ 5u
15
+ ··· − 6u + 67)
c
5
, c
10
((u
2
+ 1)
3
)(u
3
+ u − 1)(u
16
− u
15
+ ··· + 8u + 5)
c
7
, c
11
((u − 1)
3
)(u
6
− 3u
4
+ 2u
2
+ 1)(u
16
+ 2u
15
+ ··· − u + 2)
c
9
(u
3
+ u − 1)(u
6
− 2u
5
− u
4
+ 8u
3
+ 12u
2
+ 6u + 1)
· (u
16
− u
15
+ ··· − 2824u + 1117)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
3
(y
3
+ y
2
+ 2y + 1)
2
(y
16
+ 29y
15
+ ··· + 16707y + 3844)
c
2
y
3
(y
3
+ 3y
2
+ 2y −1)
2
· (y
16
− 75y
15
+ ··· + 939185823y + 14776336)
c
3
, c
8
((y + 1)
6
)(y
3
+ 2y
2
+ y −1)(y
16
+ 27y
15
+ ··· − 96y + 25)
c
4
(y
3
− 2y
2
+ 5y −1)(y
6
+ 6y
5
+ 57y
4
+ 62y
3
+ 46y
2
+ 12y + 1)
· (y
16
+ 19y
15
+ ··· + 15374y + 4489)
c
5
, c
10
((y + 1)
6
)(y
3
+ 2y
2
+ y −1)(y
16
− y
15
+ ··· − 64y + 25)
c
7
, c
11
((y −1)
3
)(y
3
− 3y
2
+ 2y + 1)
2
(y
16
− 12y
15
+ ··· + 19y + 4)
c
9
(y
3
+ 2y
2
+ y −1)(y
6
− 6y
5
+ 57y
4
− 62y
3
+ 46y
2
− 12y + 1)
· (y
16
+ 51y
15
+ ··· − 7186374y + 1247689)
16
