12n
0121
(K12n
0121
)
A knot diagram
1
Linearized knot diagam
3 5 6 2 10 3 12 11 5 9 8 7
Solving Sequence
5,9
10 6
3,11
2 1 4 8 12 7
c
9
c
5
c
10
c
2
c
1
c
4
c
8
c
11
c
7
c
3
, c
6
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
4
− 3u
3
− 2u
2
+ b + 1, −u
4
− 3u
3
− 2u
2
+ a + u + 1, u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1i
I
u
2
= h−u
4
+ u
3
+ b − 1, −u
4
+ u
3
+ a − u − 1, u
5
− u
4
+ u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 10 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
4
−3u
3
−2u
2
+b+1, −u
4
−3u
3
−2u
2
+a+u+1, u
5
+3u
4
+4u
3
+u
2
−u−1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
6
=
u
−u
3
+ u
a
3
=
u
4
+ 3u
3
+ 2u
2
− u − 1
u
4
+ 3u
3
+ 2u
2
− 1
a
11
=
−u
2
+ 1
−u
2
a
2
=
u
4
+ 3u
3
+ 2u
2
− u − 1
3u
4
+ 5u
3
+ 2u
2
− u − 1
a
1
=
−20u
4
− 44u
3
− 16u
2
+ 8u + 12
−26u
4
− 46u
3
− 16u
2
+ 11u + 12
a
4
=
−3u
4
− 9u
3
− 4u
2
+ u + 3
−13u
4
− 15u
3
− 4u
2
+ 6u + 3
a
8
=
u
4
− u
2
+ 1
u
4
a
12
=
−4u
4
− 11u
3
− 6u
2
+ 2u + 4
−5u
4
− 11u
3
− 5u
2
+ 2u + 3
a
7
=
−6u
4
− 2u
3
+ 3u
−2u
4
+ 9u
3
+ 6u
2
+ u − 4
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
4
− 3u
3
+ u
2
+ 11u + 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
+ 37u
4
+ 682u
3
− 278u
2
+ 101u + 1
c
2
, c
4
u
5
− 9u
4
+ 22u
3
+ 10u
2
+ 9u − 1
c
3
, c
6
u
5
+ 12u
4
+ 120u
3
− 120u
2
+ 128u − 32
c
5
, c
9
u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
c
7
, c
8
, c
10
c
11
, c
12
u
5
− u
4
+ 8u
3
− 3u
2
+ 3u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
− 5y
4
+ 485898y
3
+ 60406y
2
+ 10757y −1
c
2
, c
4
y
5
− 37y
4
+ 682y
3
+ 278y
2
+ 101y −1
c
3
, c
6
y
5
+ 96y
4
+ 17536y
3
+ 17088y
2
+ 8704y −1024
c
5
, c
9
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
c
7
, c
8
, c
10
c
11
, c
12
y
5
+ 15y
4
+ 64y
3
+ 37y
2
+ 3y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.561306 + 0.557752I
a = 0.218218 − 0.753989I
b = −0.343087 − 0.196237I
−1.31583 − 1.53058I 1.57269 + 4.45807I
u = −0.561306 − 0.557752I
a = 0.218218 + 0.753989I
b = −0.343087 + 0.196237I
−1.31583 + 1.53058I 1.57269 − 4.45807I
u = 0.588022
a = −0.166966
b = 0.421056
0.756147 13.9650
u = −1.23271 + 1.09381I
a = 1.36526 + 2.80304I
b = 0.13256 + 3.89685I
4.22763 − 4.40083I 1.44484 + 1.78781I
u = −1.23271 − 1.09381I
a = 1.36526 − 2.80304I
b = 0.13256 − 3.89685I
4.22763 + 4.40083I 1.44484 − 1.78781I
5
II. I
u
2
= h−u
4
+ u
3
+ b − 1, −u
4
+ u
3
+ a − u − 1, u
5
− u
4
+ u
2
+ u − 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
6
=
u
−u
3
+ u
a
3
=
u
4
− u
3
+ u + 1
u
4
− u
3
+ 1
a
11
=
−u
2
+ 1
−u
2
a
2
=
u
4
− u
3
+ u + 1
u
4
− u
3
− u + 1
a
1
=
0
−u
a
4
=
u
4
− u
3
+ u + 1
u
4
− u
3
+ 1
a
8
=
u
4
− u
2
+ 1
u
4
a
12
=
u
3
−u
4
+ u
3
+ u
2
− 1
a
7
=
u
−u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
3
− 3u
2
+ u + 2
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
6
u
5
c
4
(u + 1)
5
c
5
u
5
+ u
4
− u
2
+ u + 1
c
7
, c
8
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
9
u
5
− u
4
+ u
2
+ u − 1
c
10
, c
11
, c
12
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 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
6
y
5
c
5
, c
9
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
c
7
, c
8
, c
10
c
11
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.758138 + 0.584034I
a = −0.827780 − 0.637683I
b = −0.069642 − 1.221720I
−3.46474 − 2.21397I 0.88087 + 4.04855I
u = −0.758138 − 0.584034I
a = −0.827780 + 0.637683I
b = −0.069642 + 1.221720I
−3.46474 + 2.21397I 0.88087 − 4.04855I
u = 0.935538 + 0.903908I
a = 0.552827 − 0.534136I
b = −0.38271 − 1.43804I
−12.60320 + 3.33174I 1.28666 − 2.53508I
u = 0.935538 − 0.903908I
a = 0.552827 + 0.534136I
b = −0.38271 + 1.43804I
−12.60320 − 3.33174I 1.28666 + 2.53508I
u = 0.645200
a = 1.54991
b = 0.904706
−0.762751 1.66490
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
(u
5
+ 37u
4
+ 682u
3
− 278u
2
+ 101u + 1)
c
2
(u − 1)
5
(u
5
− 9u
4
+ 22u
3
+ 10u
2
+ 9u − 1)
c
3
, c
6
u
5
(u
5
+ 12u
4
+ 120u
3
− 120u
2
+ 128u − 32)
c
4
(u + 1)
5
(u
5
− 9u
4
+ 22u
3
+ 10u
2
+ 9u − 1)
c
5
(u
5
+ u
4
− u
2
+ u + 1)(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
c
7
, c
8
(u
5
− u
4
+ 8u
3
− 3u
2
+ 3u − 1)(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
c
9
(u
5
− u
4
+ u
2
+ u − 1)(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
c
10
, c
11
, c
12
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
5
− u
4
+ 8u
3
− 3u
2
+ 3u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
5
(y
5
− 5y
4
+ 485898y
3
+ 60406y
2
+ 10757y −1)
c
2
, c
4
(y −1)
5
(y
5
− 37y
4
+ 682y
3
+ 278y
2
+ 101y −1)
c
3
, c
6
y
5
(y
5
+ 96y
4
+ 17536y
3
+ 17088y
2
+ 8704y −1024)
c
5
, c
9
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
c
7
, c
8
, c
10
c
11
, c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
5
+ 15y
4
+ 64y
3
+ 37y
2
+ 3y −1)
11
