11a
333
(K11a
333
)
A knot diagram
1
Linearized knot diagam
8 7 1 11 10 3 2 4 6 5 9
Solving Sequence
5,10
6 11 4 9 1 3 8 2 7
c
5
c
10
c
4
c
9
c
11
c
3
c
8
c
1
c
7
c
2
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
8
+ 5u
6
+ 7u
4
+ u
3
+ 2u
2
+ 2u + 1i
I
u
2
= hu
24
+ u
23
+ ··· − u
3
+ 1i
* 2 irreducible components of dim
C
= 0, with total 32 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
8
+ 5u
6
+ 7u
4
+ u
3
+ 2u
2
+ 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
u
u
a
4
=
u
2
+ 1
u
2
a
9
=
−u
u
3
+ u
a
1
=
−u
5
− 2u
3
+ u
u
7
+ 3u
5
+ 2u
3
+ u
a
3
=
u
7
+ 4u
5
+ u
4
+ 4u
3
+ 3u
2
+ u + 1
u
7
− u
6
+ 3u
5
− 2u
4
+ u
3
+ u
2
a
8
=
u
7
+ 4u
5
+ 4u
3
u
7
+ 3u
5
+ 2u
3
+ u
a
2
=
−u
6
− u
5
− 3u
4
− 3u
3
− 2u
2
− u − 1
u
7
− 2u
6
+ 3u
5
− 5u
4
+ u
3
− u
2
− u − 1
a
7
=
−u
6
+ u
5
− 3u
4
+ 2u
3
− u
2
− u
−u
7
− 2u
6
− 2u
5
− 6u
4
− 3u
2
− 2u − 1
a
7
=
−u
6
+ u
5
− 3u
4
+ 2u
3
− u
2
− u
−u
7
− 2u
6
− 2u
5
− 6u
4
− 3u
2
− 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
7
− 4u
6
+ 20u
5
− 16u
4
+ 24u
3
− 12u
2
− 4u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
u
8
+ 5u
6
+ 7u
4
− u
3
+ 2u
2
− 2u + 1
c
3
, c
11
u
8
− 2u
7
+ 3u
6
+ 5u
4
− 5u
3
+ 6u
2
− 2u + 1
c
8
u
8
− 5u
7
+ 11u
6
− 16u
5
+ 22u
4
− 27u
3
+ 23u
2
− 12u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
y
8
+ 10y
7
+ 39y
6
+ 74y
5
+ 71y
4
+ 37y
3
+ 14y
2
+ 1
c
3
, c
11
y
8
+ 2y
7
+ 19y
6
+ 22y
5
+ 55y
4
+ 41y
3
+ 26y
2
+ 8y + 1
c
8
y
8
− 3y
7
+ 5y
6
+ 4y
5
+ 14y
4
− 13y
3
+ 57y
2
+ 40y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.461135 + 0.691908I
−1.71296 + 5.69915I 0.47037 − 9.01967I
u = 0.461135 − 0.691908I
−1.71296 − 5.69915I 0.47037 + 9.01967I
u = 0.08626 + 1.49661I
−11.28300 + 3.64910I −1.10964 − 3.07905I
u = 0.08626 − 1.49661I
−11.28300 − 3.64910I −1.10964 + 3.07905I
u = −0.404853 + 0.285137I
0.853870 − 0.627235I 8.80552 + 5.03557I
u = −0.404853 − 0.285137I
0.853870 + 0.627235I 8.80552 − 5.03557I
u = −0.14255 + 1.61382I
−17.4667 − 10.2751I −4.16626 + 5.30618I
u = −0.14255 − 1.61382I
−17.4667 + 10.2751I −4.16626 − 5.30618I
5
II. I
u
2
= hu
24
+ u
23
+ · · · − u
3
+ 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
u
u
a
4
=
u
2
+ 1
u
2
a
9
=
−u
u
3
+ u
a
1
=
−u
5
− 2u
3
+ u
u
7
+ 3u
5
+ 2u
3
+ u
a
3
=
u
14
+ 7u
12
+ 16u
10
+ 11u
8
− 2u
6
+ 1
−u
16
− 8u
14
− 24u
12
− 34u
10
− 26u
8
− 14u
6
− 4u
4
a
8
=
u
7
+ 4u
5
+ 4u
3
u
7
+ 3u
5
+ 2u
3
+ u
a
2
=
u
21
+ 12u
19
+ ··· − 2u
3
+ u
u
21
+ 11u
19
+ ··· + u
3
+ u
a
7
=
2u
23
+ u
22
+ ··· + u + 2
−u
22
− 12u
20
+ ··· − 2u
3
+ u
2
a
7
=
2u
23
+ u
22
+ ··· + u + 2
−u
22
− 12u
20
+ ··· − 2u
3
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 4u
23
+ 52u
21
+ 276u
19
− 4u
18
+ 760u
17
− 40u
16
+ 1136u
15
− 148u
14
+ 880u
13
−
232u
12
+ 328u
11
−96u
10
+ 84u
9
+ 92u
8
+ 4u
7
+ 64u
6
−4u
5
+ 16u
4
+ 4u
3
−4u
2
+ 8u + 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
u
24
− u
23
+ ··· + u
3
+ 1
c
3
, c
11
u
24
− 7u
23
+ ··· − 94u + 17
c
8
(u
12
+ 2u
11
+ ··· + 2u + 3)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
y
24
+ 27y
23
+ ··· − 2y
2
+ 1
c
3
, c
11
y
24
− 9y
23
+ ··· − 2376y + 289
c
8
(y
12
− 6y
11
+ ··· − 46y + 9)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.288696 + 0.833188I
−10.86470 + 1.36952I −4.42656 + 0.88523I
u = −0.288696 − 0.833188I
−10.86470 − 1.36952I −4.42656 − 0.88523I
u = −0.489583 + 0.725679I
−9.51380 − 7.90456I −1.91927 + 6.92574I
u = −0.489583 − 0.725679I
−9.51380 + 7.90456I −1.91927 − 6.92574I
u = 0.271534 + 0.725672I
−2.90121 −3.57147 + 0.I
u = 0.271534 − 0.725672I
−2.90121 −3.57147 + 0.I
u = −0.409437 + 0.638189I
−0.18849 − 2.30634I 4.56865 + 4.07548I
u = −0.409437 − 0.638189I
−0.18849 + 2.30634I 4.56865 − 4.07548I
u = 0.493302 + 0.448019I
−4.94432 + 1.72225I 2.81956 − 4.07903I
u = 0.493302 − 0.448019I
−4.94432 − 1.72225I 2.81956 + 4.07903I
u = −0.591891 + 0.137722I
−7.79349 + 4.22631I 1.67942 − 2.13120I
u = −0.591891 − 0.137722I
−7.79349 − 4.22631I 1.67942 + 2.13120I
u = 0.516875 + 0.160721I
−0.18849 − 2.30634I 4.56865 + 4.07548I
u = 0.516875 − 0.160721I
−0.18849 + 2.30634I 4.56865 − 4.07548I
u = −0.02617 + 1.49212I
−4.94432 − 1.72225I 2.81956 + 4.07903I
u = −0.02617 − 1.49212I
−4.94432 + 1.72225I 2.81956 − 4.07903I
u = −0.11519 + 1.59101I
−7.79349 − 4.22631I 1.67942 + 2.13120I
u = −0.11519 − 1.59101I
−7.79349 + 4.22631I 1.67942 − 2.13120I
u = 0.08387 + 1.60577I
−10.86470 + 1.36952I −4.42656 + 0.88523I
u = 0.08387 − 1.60577I
−10.86470 − 1.36952I −4.42656 − 0.88523I
u = 0.13255 + 1.60291I
−9.51380 + 7.90456I −1.91927 − 6.92574I
u = 0.13255 − 1.60291I
−9.51380 − 7.90456I −1.91927 + 6.92574I
u = −0.07716 + 1.63217I
−19.3156 −5.87212 + 0.I
u = −0.07716 − 1.63217I
−19.3156 −5.87212 + 0.I
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
(u
8
+ 5u
6
+ 7u
4
− u
3
+ 2u
2
− 2u + 1)(u
24
− u
23
+ ··· + u
3
+ 1)
c
3
, c
11
(u
8
− 2u
7
+ 3u
6
+ 5u
4
− 5u
3
+ 6u
2
− 2u + 1)
· (u
24
− 7u
23
+ ··· − 94u + 17)
c
8
(u
8
− 5u
7
+ 11u
6
− 16u
5
+ 22u
4
− 27u
3
+ 23u
2
− 12u + 4)
· (u
12
+ 2u
11
+ ··· + 2u + 3)
2
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
7
c
9
, c
10
(y
8
+ 10y
7
+ 39y
6
+ 74y
5
+ 71y
4
+ 37y
3
+ 14y
2
+ 1)
· (y
24
+ 27y
23
+ ··· − 2y
2
+ 1)
c
3
, c
11
(y
8
+ 2y
7
+ 19y
6
+ 22y
5
+ 55y
4
+ 41y
3
+ 26y
2
+ 8y + 1)
· (y
24
− 9y
23
+ ··· − 2376y + 289)
c
8
(y
8
− 3y
7
+ 5y
6
+ 4y
5
+ 14y
4
− 13y
3
+ 57y
2
+ 40y + 16)
· (y
12
− 6y
11
+ ··· − 46y + 9)
2
11
