11n
57
(K11n
57
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 9 10 11 4 6 1 7
Solving Sequence
1,7
11
4,8
3 2 5 10 6 9
c
11
c
7
c
3
c
2
c
4
c
10
c
6
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
5
− u
4
+ 2u
3
− u
2
+ b + u, u
5
− u
4
+ 3u
3
− 2u
2
+ a + 2u − 1, u
8
− 2u
7
+ 5u
6
− 6u
5
+ 7u
4
− 6u
3
+ 2u
2
− u − 1i
I
u
2
= hb − 1, u
3
+ u
2
+ a + u, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 13 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
5
−u
4
+2u
3
−u
2
+b+u, u
5
−u
4
+3u
3
−2u
2
+a+2u−1, u
8
−2u
7
+· · ·−u−1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
11
=
1
u
2
a
4
=
−u
5
+ u
4
− 3u
3
+ 2u
2
− 2u + 1
−u
5
+ u
4
− 2u
3
+ u
2
− u
a
8
=
u
u
3
+ u
a
3
=
−u
6
+ u
5
− 2u
4
+ 2u
3
+ 2
u
6
+ 3u
4
− u
3
+ 2u
2
a
2
=
u
5
+ u
4
+ u
3
+ 2u
2
+ 2
u
6
+ 3u
4
− u
3
+ 2u
2
a
5
=
3u
5
+ u
4
+ 6u
3
+ u
2
+ 3u + 2
−u
7
+ 4u
6
− 3u
5
+ 9u
4
− 2u
3
+ 4u
2
+ u
a
10
=
u
2
+ 1
u
2
a
6
=
u
5
+ 2u
3
+ u
u
5
+ u
3
+ u
a
9
=
−2u
7
+ 2u
6
− 6u
5
+ 4u
4
− 6u
3
+ 2u
2
− u
−2u
7
+ 3u
6
− 6u
5
+ 5u
4
− 6u
3
+ 2u
2
− u − 1
a
9
=
−2u
7
+ 2u
6
− 6u
5
+ 4u
4
− 6u
3
+ 2u
2
− u
−2u
7
+ 3u
6
− 6u
5
+ 5u
4
− 6u
3
+ 2u
2
− u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
7
− 9u
6
+ 18u
5
− 21u
4
+ 18u
3
− 14u
2
− 7
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
8
− 6u
7
+ 7u
6
+ 9u
5
+ 6u
4
− 40u
3
− 13u
2
+ 5u − 1
c
2
u
8
+ 22u
7
+ 169u
6
+ 503u
5
+ 632u
4
+ 1860u
3
+ 557u
2
− u + 1
c
3
, c
8
u
8
− 7u
7
− 4u
6
+ 119u
5
− 212u
4
− 16u
3
+ 120u
2
+ 64u + 32
c
5
, c
6
, c
9
u
8
+ 2u
7
− 7u
6
− 12u
5
+ 7u
4
+ 2u
3
− 2u
2
− 3u − 1
c
7
, c
11
u
8
− 2u
7
+ 5u
6
− 6u
5
+ 7u
4
− 6u
3
+ 2u
2
− u − 1
c
10
u
8
+ 6u
7
+ 15u
6
+ 14u
5
− 9u
4
− 30u
3
− 22u
2
− 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
8
− 22y
7
+ 169y
6
− 503y
5
+ 632y
4
− 1860y
3
+ 557y
2
+ y + 1
c
2
y
8
− 146y
7
+ ··· + 1113y + 1
c
3
, c
8
y
8
− 57y
7
+ ··· + 3584y + 1024
c
5
, c
6
, c
9
y
8
− 18y
7
+ 111y
6
− 254y
5
+ 135y
4
− 90y
3
+ 2y
2
− 5y + 1
c
7
, c
11
y
8
+ 6y
7
+ 15y
6
+ 14y
5
− 9y
4
− 30y
3
− 22y
2
− 5y + 1
c
10
y
8
− 6y
7
+ 39y
6
− 150y
5
+ 323y
4
− 334y
3
+ 166y
2
− 69y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.09831
a = −2.90176
b = −2.68486
16.8590 −9.91680
u = 0.271970 + 0.836396I
a = 0.514998 − 0.132997I
b = −0.138100 − 0.151060I
−0.50482 + 1.32248I −5.16164 − 4.61817I
u = 0.271970 − 0.836396I
a = 0.514998 + 0.132997I
b = −0.138100 + 0.151060I
−0.50482 − 1.32248I −5.16164 + 4.61817I
u = −0.198501 + 1.220550I
a = −0.186478 + 1.015590I
b = 0.944682 + 1.046670I
−4.38598 − 2.12062I −13.41968 + 2.09452I
u = −0.198501 − 1.220550I
a = −0.186478 − 1.015590I
b = 0.944682 − 1.046670I
−4.38598 + 2.12062I −13.41968 − 2.09452I
u = 0.55241 + 1.37610I
a = −0.92476 + 1.87326I
b = −2.75351 + 0.38295I
12.57270 + 5.86054I −12.50168 − 2.57970I
u = 0.55241 − 1.37610I
a = −0.92476 − 1.87326I
b = −2.75351 − 0.38295I
12.57270 − 5.86054I −12.50168 + 2.57970I
u = −0.350076
a = 2.09424
b = 0.578712
−0.969109 −9.91720
5
II. I
u
2
= hb − 1, u
3
+ u
2
+ a + u, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
11
=
1
u
2
a
4
=
−u
3
− u
2
− u
1
a
8
=
u
u
3
+ u
a
3
=
−u
3
− u
2
− u
1
a
2
=
−u
3
− u
2
− u + 1
1
a
5
=
−1
0
a
10
=
u
2
+ 1
u
2
a
6
=
−u
4
− u
2
− 1
−u
4
− u
3
− u
2
− 1
a
9
=
u
u
3
+ u
a
9
=
u
u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
4
− u
3
− 2u − 12
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
c
2
, c
4
(u + 1)
5
c
3
, c
8
u
5
c
5
, c
6
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
7
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
9
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
10
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
11
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
8
y
5
c
5
, c
6
, c
9
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
7
, c
11
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
10
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.339110 + 0.822375I
a = 0.871221 − 1.107660I
b = 1.00000
−1.97403 + 1.53058I −12.02124 − 2.62456I
u = 0.339110 − 0.822375I
a = 0.871221 + 1.107660I
b = 1.00000
−1.97403 − 1.53058I −12.02124 + 2.62456I
u = −0.766826
a = 0.629714
b = 1.00000
−4.04602 −9.32390
u = −0.455697 + 1.200150I
a = −0.186078 + 0.874646I
b = 1.00000
−7.51750 − 4.40083I −12.31681 + 3.97407I
u = −0.455697 − 1.200150I
a = −0.186078 − 0.874646I
b = 1.00000
−7.51750 + 4.40083I −12.31681 − 3.97407I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
(u
8
− 6u
7
+ 7u
6
+ 9u
5
+ 6u
4
− 40u
3
− 13u
2
+ 5u − 1)
c
2
(u + 1)
5
· (u
8
+ 22u
7
+ 169u
6
+ 503u
5
+ 632u
4
+ 1860u
3
+ 557u
2
− u + 1)
c
3
, c
8
u
5
(u
8
− 7u
7
− 4u
6
+ 119u
5
− 212u
4
− 16u
3
+ 120u
2
+ 64u + 32)
c
4
(u + 1)
5
(u
8
− 6u
7
+ 7u
6
+ 9u
5
+ 6u
4
− 40u
3
− 13u
2
+ 5u − 1)
c
5
, c
6
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
· (u
8
+ 2u
7
− 7u
6
− 12u
5
+ 7u
4
+ 2u
3
− 2u
2
− 3u − 1)
c
7
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
· (u
8
− 2u
7
+ 5u
6
− 6u
5
+ 7u
4
− 6u
3
+ 2u
2
− u − 1)
c
9
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
· (u
8
+ 2u
7
− 7u
6
− 12u
5
+ 7u
4
+ 2u
3
− 2u
2
− 3u − 1)
c
10
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
· (u
8
+ 6u
7
+ 15u
6
+ 14u
5
− 9u
4
− 30u
3
− 22u
2
− 5u + 1)
c
11
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
· (u
8
− 2u
7
+ 5u
6
− 6u
5
+ 7u
4
− 6u
3
+ 2u
2
− u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y −1)
5
· (y
8
− 22y
7
+ 169y
6
− 503y
5
+ 632y
4
− 1860y
3
+ 557y
2
+ y + 1)
c
2
((y −1)
5
)(y
8
− 146y
7
+ ··· + 1113y + 1)
c
3
, c
8
y
5
(y
8
− 57y
7
+ ··· + 3584y + 1024)
c
5
, c
6
, c
9
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
· (y
8
− 18y
7
+ 111y
6
− 254y
5
+ 135y
4
− 90y
3
+ 2y
2
− 5y + 1)
c
7
, c
11
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
· (y
8
+ 6y
7
+ 15y
6
+ 14y
5
− 9y
4
− 30y
3
− 22y
2
− 5y + 1)
c
10
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
· (y
8
− 6y
7
+ 39y
6
− 150y
5
+ 323y
4
− 334y
3
+ 166y
2
− 69y + 1)
11
