11n
138
(K11n
138
)
A knot diagram
1
Linearized knot diagam
8 1 9 8 10 11 2 4 1 6 5
Solving Sequence
1,8 2,4
5 9 10 3 7 11 6
c
1
c
4
c
8
c
9
c
3
c
7
c
11
c
6
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
8
− u
7
− 8u
6
− 5u
5
− 12u
4
− 3u
3
+ 20u
2
+ 8b + u + 1, a − 1,
u
9
+ 9u
7
− 3u
6
+ 23u
5
− 15u
4
+ 7u
3
− 5u
2
− 1i
I
u
2
= hb
3
+ b
2
u − 3b
2
− 2bu + 3b + 2u − 1, a + 1, u
2
+ 1i
I
u
3
= hb − 1, u
3
+ 6u
2
+ 15a + 4u + 20, u
4
+ u
3
+ 4u
2
+ 5i
* 3 irreducible components of dim
C
= 0, with total 19 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
8
−u
7
+· · ·+8b+1, a−1, u
9
+9u
7
−3u
6
+23u
5
−15u
4
+7u
3
−5u
2
−1i
(i) Arc colorings
a
1
=
1
0
a
8
=
0
u
a
2
=
1
−u
2
a
4
=
1
1
8
u
8
+
1
8
u
7
+ ··· −
1
8
u −
1
8
a
5
=
1
1
8
u
8
+
1
8
u
7
+ ··· −
1
8
u −
1
8
a
9
=
u
1
8
u
8
−
1
8
u
7
+ ··· +
7
8
u +
1
8
a
10
=
1
8
u
8
−
1
8
u
7
+ ··· +
15
8
u +
1
8
1
8
u
8
−
1
8
u
7
+ ··· +
7
8
u +
1
8
a
3
=
u
2
+ 1
−u
2
a
7
=
−u
u
3
+ u
a
11
=
−
1
8
u
8
−
1
8
u
7
+ ··· +
1
8
u +
9
8
−
3
8
u
8
−
3
8
u
7
+ ··· +
5
8
u +
5
8
a
6
=
1
8
u
8
+
3
8
u
7
+ ··· −
7
8
u −
1
8
1
8
u
8
+
3
8
u
7
+ ··· −
5
8
u −
7
8
a
6
=
1
8
u
8
+
3
8
u
7
+ ··· −
7
8
u −
1
8
1
8
u
8
+
3
8
u
7
+ ··· −
5
8
u −
7
8
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
3
2
u
8
+ 2u
7
−
27
2
u
6
+ 22u
5
− 40u
4
+ 64u
3
−
73
2
u
2
+ 10u −
9
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
u
9
+ 9u
7
+ 3u
6
+ 23u
5
+ 15u
4
+ 7u
3
+ 5u
2
+ 1
c
2
u
9
+ 18u
8
+ ··· − 10u − 1
c
5
, c
6
, c
10
u
9
− 3u
8
+ 5u
6
+ u
5
− 2u
4
− 9u
3
+ 5u
2
+ u + 2
c
9
u
9
+ u
8
+ 22u
7
+ 19u
6
+ 127u
5
+ 84u
4
+ 67u
3
− 41u
2
+ 23u + 8
c
11
u
9
+ 9u
8
+ 38u
7
+ 85u
6
+ 87u
5
− 18u
4
− 147u
3
− 167u
2
− 85u − 26
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
y
9
+ 18y
8
+ ··· − 10y −1
c
2
y
9
− 70y
8
+ ··· − 10y −1
c
5
, c
6
, c
10
y
9
− 9y
8
+ 32y
7
− 55y
6
+ 53y
5
− 60y
4
+ 83y
3
− 35y
2
− 19y −4
c
9
y
9
+ 43y
8
+ ··· + 1185y −64
c
11
y
9
− 5y
8
+ ··· − 1459y −676
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.721273
a = 1.00000
b = −0.684414
−3.38429 −0.599760
u = 0.159982 + 0.567821I
a = 1.00000
b = 0.666256 − 0.688894I
−4.68408 + 3.45373I −2.44779 − 5.78928I
u = 0.159982 − 0.567821I
a = 1.00000
b = 0.666256 + 0.688894I
−4.68408 − 3.45373I −2.44779 + 5.78928I
u = −0.198901 + 0.378443I
a = 1.00000
b = 0.170075 + 0.364475I
0.131099 − 0.964036I 2.44921 + 7.22651I
u = −0.198901 − 0.378443I
a = 1.00000
b = 0.170075 − 0.364475I
0.131099 + 0.964036I 2.44921 − 7.22651I
u = 0.14689 + 2.12129I
a = 1.00000
b = −2.55711 − 0.09982I
−17.5620 + 3.0332I −3.21143 − 2.16261I
u = 0.14689 − 2.12129I
a = 1.00000
b = −2.55711 + 0.09982I
−17.5620 − 3.0332I −3.21143 + 2.16261I
u = −0.46861 + 2.14498I
a = 1.00000
b = −2.43701 + 0.25982I
14.7600 − 7.7767I −5.49011 + 2.86525I
u = −0.46861 − 2.14498I
a = 1.00000
b = −2.43701 − 0.25982I
14.7600 + 7.7767I −5.49011 − 2.86525I
5
II. I
u
2
= hb
3
+ b
2
u − 3b
2
− 2bu + 3b + 2u − 1, a + 1, u
2
+ 1i
(i) Arc colorings
a
1
=
1
0
a
8
=
0
u
a
2
=
1
1
a
4
=
−1
b
a
5
=
−1
b − 1
a
9
=
u
−bu + u
a
10
=
−bu + 2u
−bu + u
a
3
=
0
1
a
7
=
−u
0
a
11
=
b
−b
2
+ 2b − 1
a
6
=
−b
2
u − b
2
+ 2bu + 2b − 2u − 2
−b
2
u + 2bu + b − 2u − 1
a
6
=
−b
2
u − b
2
+ 2bu + 2b − 2u − 2
−b
2
u + 2bu + b − 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4bu + 4u − 4
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
3
c
2
(u + 1)
6
c
5
, c
6
, c
10
u
6
− 3u
4
+ 2u
2
+ 1
c
9
(u
3
− u
2
+ 1)
2
c
11
u
6
+ u
4
+ 2u
2
+ 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(y + 1)
6
c
2
(y −1)
6
c
5
, c
6
, c
10
(y
3
− 3y
2
+ 2y + 1)
2
c
9
(y
3
− y
2
+ 2y −1)
2
c
11
(y
3
+ y
2
+ 2y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.000000I
a = −1.00000
b = 0.255138 − 0.877439I
−6.31400 − 2.82812I −7.50976 + 2.97945I
u = 1.000000I
a = −1.00000
b = 1.000000 + 0.754878I
−2.17641 −6 − 0.980489 + 0.10I
u = 1.000000I
a = −1.00000
b = 1.74486 − 0.87744I
−6.31400 + 2.82812I −7.50976 − 2.97945I
u = − 1.000000I
a = −1.00000
b = 0.255138 + 0.877439I
−6.31400 + 2.82812I −7.50976 − 2.97945I
u = − 1.000000I
a = −1.00000
b = 1.000000 − 0.754878I
−2.17641 −6 − 0.980489 + 0.10I
u = − 1.000000I
a = −1.00000
b = 1.74486 + 0.87744I
−6.31400 − 2.82812I −7.50976 + 2.97945I
9
III. I
u
3
= hb − 1, u
3
+ 6u
2
+ 15a + 4u + 20, u
4
+ u
3
+ 4u
2
+ 5i
(i) Arc colorings
a
1
=
1
0
a
8
=
0
u
a
2
=
1
−u
2
a
4
=
−
1
15
u
3
−
2
5
u
2
−
4
15
u −
4
3
1
a
5
=
−
1
15
u
3
−
2
5
u
2
−
4
15
u −
4
3
−
1
3
u
3
−
1
3
u −
2
3
a
9
=
8
15
u
3
+
1
5
u
2
+
17
15
u −
1
3
−
1
3
u
3
−
1
3
u +
1
3
a
10
=
1
5
u
3
+
1
5
u
2
+
4
5
u
−
1
3
u
3
−
1
3
u +
1
3
a
3
=
u
2
+ 1
−u
2
a
7
=
−u
u
3
+ u
a
11
=
−
4
15
u
3
−
3
5
u
2
−
1
15
u −
1
3
−u
3
− u − 1
a
6
=
−
2
15
u
3
−
4
5
u
2
−
8
15
u −
5
3
−
2
3
u
3
−
2
3
u −
4
3
a
6
=
−
2
15
u
3
−
4
5
u
2
−
8
15
u −
5
3
−
2
3
u
3
−
2
3
u −
4
3
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
u
4
− u
3
+ 4u
2
+ 5
c
2
u
4
+ 7u
3
+ 26u
2
+ 40u + 25
c
5
, c
6
, c
10
(u
2
+ u − 1)
2
c
9
(u
2
− u − 1)
2
c
11
(u
2
− 3u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
y
4
+ 7y
3
+ 26y
2
+ 40y + 25
c
2
y
4
+ 3y
3
+ 166y
2
− 300y + 625
c
5
, c
6
, c
9
c
10
(y
2
− 3y + 1)
2
c
11
(y
2
− 7y + 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.309017 + 1.134230I
a = −0.861803 − 0.507242I
b = 1.00000
−4.27683 −6.00000
u = 0.309017 − 1.134230I
a = −0.861803 + 0.507242I
b = 1.00000
−4.27683 −6.00000
u = −0.80902 + 1.72149I
a = −0.638197 + 0.769873I
b = 1.00000
−12.1725 −6.00000
u = −0.80902 − 1.72149I
a = −0.638197 − 0.769873I
b = 1.00000
−12.1725 −6.00000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
(u
2
+ 1)
3
(u
4
− u
3
+ 4u
2
+ 5)
· (u
9
+ 9u
7
+ 3u
6
+ 23u
5
+ 15u
4
+ 7u
3
+ 5u
2
+ 1)
c
2
((u + 1)
6
)(u
4
+ 7u
3
+ ··· + 40u + 25)(u
9
+ 18u
8
+ ··· − 10u − 1)
c
5
, c
6
, c
10
(u
2
+ u − 1)
2
(u
6
− 3u
4
+ 2u
2
+ 1)
· (u
9
− 3u
8
+ 5u
6
+ u
5
− 2u
4
− 9u
3
+ 5u
2
+ u + 2)
c
9
(u
2
− u − 1)
2
(u
3
− u
2
+ 1)
2
· (u
9
+ u
8
+ 22u
7
+ 19u
6
+ 127u
5
+ 84u
4
+ 67u
3
− 41u
2
+ 23u + 8)
c
11
(u
2
− 3u + 1)
2
(u
6
+ u
4
+ 2u
2
+ 1)
· (u
9
+ 9u
8
+ 38u
7
+ 85u
6
+ 87u
5
− 18u
4
− 147u
3
− 167u
2
− 85u − 26)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
7
, c
8
((y + 1)
6
)(y
4
+ 7y
3
+ ··· + 40y + 25)(y
9
+ 18y
8
+ ··· − 10y −1)
c
2
((y −1)
6
)(y
4
+ 3y
3
+ ··· − 300y + 625)(y
9
− 70y
8
+ ··· − 10y −1)
c
5
, c
6
, c
10
(y
2
− 3y + 1)
2
(y
3
− 3y
2
+ 2y + 1)
2
· (y
9
− 9y
8
+ 32y
7
− 55y
6
+ 53y
5
− 60y
4
+ 83y
3
− 35y
2
− 19y −4)
c
9
((y
2
− 3y + 1)
2
)(y
3
− y
2
+ 2y −1)
2
(y
9
+ 43y
8
+ ··· + 1185y −64)
c
11
((y
2
− 7y + 1)
2
)(y
3
+ y
2
+ 2y + 1)
2
(y
9
− 5y
8
+ ··· − 1459y −676)
15
