11n
51
(K11n
51
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 9 10 3 1 11 6 7
Solving Sequence
6,10 3,7
4 11 1 2 5 9 8
c
6
c
3
c
10
c
11
c
2
c
4
c
9
c
8
c
1
, c
5
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
18
− 3u
17
+ ··· + b − 3u, −u
18
+ 3u
17
+ ··· + a + 1, u
19
− 2u
18
+ ··· + 4u − 1i
I
u
2
= h−u
4
− u
3
− u
2
+ b, −u
2
+ a − u − 1, u
5
+ u
4
+ 2u
3
+ u
2
+ 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
18
−3u
17
+· · ·+b−3u, −u
18
+3u
17
+· · ·+a+1, u
19
−2u
18
+· · ·+4u−1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
3
=
u
18
− 3u
17
+ ··· − 3u − 1
−u
18
+ 3u
17
+ ··· − 6u
2
+ 3u
a
7
=
1
u
2
a
4
=
−u
17
+ u
16
+ ··· − u − 2
u
17
− u
16
+ ··· − 3u
2
+ 2u
a
11
=
−u
u
a
1
=
u
3
u
5
+ u
3
+ u
a
2
=
−u
14
+ u
13
+ ··· + u − 2
−u
16
+ u
15
+ ··· − u
2
+ 2u
a
5
=
u
6
+ u
4
− 1
−u
6
− 2u
4
− u
2
a
9
=
−u
3
u
3
+ u
a
8
=
−u
11
− 2u
9
− 2u
7
− u
3
−u
13
− 3u
11
− 5u
9
− 4u
7
− 2u
5
+ u
3
+ u
a
8
=
−u
11
− 2u
9
− 2u
7
− u
3
−u
13
− 3u
11
− 5u
9
− 4u
7
− 2u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
18
+ 7u
17
− 30u
16
+ 40u
15
− 92u
14
+ 103u
13
− 146u
12
+
136u
11
− 103u
10
+ 75u
9
+ 15u
8
− 26u
7
+ 68u
6
− 42u
5
+ 11u
4
+ 2u
3
− 25u
2
+ 13u − 11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
19
− 6u
18
+ ··· − 6u + 1
c
2
u
19
+ 22u
17
+ ··· + 10u + 1
c
3
, c
7
u
19
+ u
18
+ ··· + 32u + 32
c
5
, c
11
u
19
− 2u
18
+ ··· + 2u + 1
c
6
, c
10
u
19
+ 2u
18
+ ··· + 4u + 1
c
8
u
19
+ 8u
18
+ ··· + 3614u − 53
c
9
u
19
− 12u
18
+ ··· + 8u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
19
+ 22y
17
+ ··· + 10y −1
c
2
y
19
+ 44y
18
+ ··· − 82y −1
c
3
, c
7
y
19
+ 33y
18
+ ··· − 6656y −1024
c
5
, c
11
y
19
− 28y
18
+ ··· + 8y −1
c
6
, c
10
y
19
+ 12y
18
+ ··· + 8y −1
c
8
y
19
− 88y
18
+ ··· + 11357576y −2809
c
9
y
19
− 8y
18
+ ··· + 148y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.958201 + 0.037511I
a = −0.09691 + 2.45307I
b = −0.68781 + 4.36675I
13.37810 + 4.28212I −1.00628 − 2.00074I
u = 0.958201 − 0.037511I
a = −0.09691 − 2.45307I
b = −0.68781 − 4.36675I
13.37810 − 4.28212I −1.00628 + 2.00074I
u = 0.257925 + 1.029280I
a = −0.974517 − 0.715624I
b = 0.400502 + 0.133322I
1.26128 − 2.36565I 1.22099 + 4.76618I
u = 0.257925 − 1.029280I
a = −0.974517 + 0.715624I
b = 0.400502 − 0.133322I
1.26128 + 2.36565I 1.22099 − 4.76618I
u = 0.411726 + 0.802360I
a = −0.613331 + 0.154728I
b = 0.0749982 + 0.0666033I
0.05095 − 1.76235I 0.18768 + 4.49049I
u = 0.411726 − 0.802360I
a = −0.613331 − 0.154728I
b = 0.0749982 − 0.0666033I
0.05095 + 1.76235I 0.18768 − 4.49049I
u = −0.136067 + 0.851256I
a = −0.34442 + 2.57576I
b = 1.21717 − 1.16867I
−0.904771 + 0.899537I 0.47063 + 1.75855I
u = −0.136067 − 0.851256I
a = −0.34442 − 2.57576I
b = 1.21717 + 1.16867I
−0.904771 − 0.899537I 0.47063 − 1.75855I
u = −0.751661 + 0.156600I
a = −0.485692 + 0.793209I
b = −0.42510 + 1.60730I
2.51975 − 1.53406I −0.31883 + 1.85733I
u = −0.751661 − 0.156600I
a = −0.485692 − 0.793209I
b = −0.42510 − 1.60730I
2.51975 + 1.53406I −0.31883 − 1.85733I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.503922 + 1.144700I
a = 1.50073 + 1.71154I
b = 0.50812 − 2.33885I
5.36309 + 6.16703I 2.22619 − 6.06641I
u = −0.503922 − 1.144700I
a = 1.50073 − 1.71154I
b = 0.50812 + 2.33885I
5.36309 − 6.16703I 2.22619 + 6.06641I
u = −0.349482 + 1.221390I
a = 0.51320 − 2.86957I
b = −2.48582 + 2.22515I
6.61331 + 2.23643I 3.68670 − 1.85634I
u = −0.349482 − 1.221390I
a = 0.51320 + 2.86957I
b = −2.48582 − 2.22515I
6.61331 − 2.23643I 3.68670 + 1.85634I
u = 0.505857 + 1.287090I
a = −0.86298 + 4.44842I
b = −4.03195 − 5.22546I
17.2204 − 9.5042I 1.84766 + 4.86373I
u = 0.505857 − 1.287090I
a = −0.86298 − 4.44842I
b = −4.03195 + 5.22546I
17.2204 + 9.5042I 1.84766 − 4.86373I
u = 0.461672 + 1.308730I
a = 1.77953 − 4.18615I
b = 2.72777 + 6.18474I
17.5665 − 0.7457I 2.31305 + 0.82283I
u = 0.461672 − 1.308730I
a = 1.77953 + 4.18615I
b = 2.72777 − 6.18474I
17.5665 + 0.7457I 2.31305 − 0.82283I
u = 0.291501
a = −1.83122
b = 0.404221
−1.12234 −9.25560
6
II. I
u
2
= h−u
4
− u
3
− u
2
+ b, −u
2
+ a − u − 1, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
6
=
1
0
a
10
=
0
u
a
3
=
u
2
+ u + 1
u
4
+ u
3
+ u
2
a
7
=
1
u
2
a
4
=
u
2
+ u + 1
u
4
+ u
3
+ u
2
a
11
=
−u
u
a
1
=
u
3
−u
4
− u
3
− u
2
− 1
a
2
=
u
3
+ u
2
+ u + 1
−1
a
5
=
−u
3
u
4
+ u
3
+ u
2
+ 1
a
9
=
−u
3
u
3
+ u
a
8
=
1
u
2
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
4
+ 7u
3
+ 8u
2
+ 6u
7
(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
7
u
5
c
5
, c
8
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
6
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
9
u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
c
10
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
11
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
7
y
5
c
5
, c
8
, c
11
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
6
, c
10
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
9
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.339110 + 0.822375I
a = 0.77780 + 1.38013I
b = −1.206350 − 0.340852I
−1.31583 − 1.53058I −6.99101 + 6.23673I
u = 0.339110 − 0.822375I
a = 0.77780 − 1.38013I
b = −1.206350 + 0.340852I
−1.31583 + 1.53058I −6.99101 − 6.23673I
u = −0.766826
a = 0.821196
b = 0.482881
0.756147 −2.36160
u = −0.455697 + 1.200150I
a = −0.688402 + 0.106340I
b = 0.964913 + 0.621896I
4.22763 + 4.40083I 1.17182 − 3.02310I
u = −0.455697 − 1.200150I
a = −0.688402 − 0.106340I
b = 0.964913 − 0.621896I
4.22763 − 4.40083I 1.17182 + 3.02310I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
19
− 6u
18
+ ··· − 6u + 1)
c
2
((u + 1)
5
)(u
19
+ 22u
17
+ ··· + 10u + 1)
c
3
, c
7
u
5
(u
19
+ u
18
+ ··· + 32u + 32)
c
4
((u + 1)
5
)(u
19
− 6u
18
+ ··· − 6u + 1)
c
5
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
19
− 2u
18
+ ··· + 2u + 1)
c
6
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
19
+ 2u
18
+ ··· + 4u + 1)
c
8
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
19
+ 8u
18
+ ··· + 3614u − 53)
c
9
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)(u
19
− 12u
18
+ ··· + 8u + 1)
c
10
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
19
+ 2u
18
+ ··· + 4u + 1)
c
11
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
19
− 2u
18
+ ··· + 2u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y −1)
5
)(y
19
+ 22y
17
+ ··· + 10y −1)
c
2
((y −1)
5
)(y
19
+ 44y
18
+ ··· − 82y −1)
c
3
, c
7
y
5
(y
19
+ 33y
18
+ ··· − 6656y −1024)
c
5
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
19
− 28y
18
+ ··· + 8y −1)
c
6
, c
10
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
19
+ 12y
18
+ ··· + 8y −1)
c
8
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
19
− 88y
18
+ ··· + 11357576y −2809)
c
9
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
19
− 8y
18
+ ··· + 148y −1)
12
