12n
0475
(K12n
0475
)
A knot diagram
1
Linearized knot diagam
3 6 11 10 2 11 12 1 3 4 7 8
Solving Sequence
3,11 4,6
7 12 2 1 5 10 9 8
c
3
c
6
c
11
c
2
c
1
c
5
c
10
c
9
c
8
c
4
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
7
− u
6
+ 6u
5
− 3u
4
+ 8u
3
− 6u
2
+ 2b + 2u − 2, −u
7
+ 2u
6
− 7u
5
+ 7u
4
− 9u
3
+ 4u
2
+ 4a − 6u − 2,
u
9
− u
8
+ 10u
7
− 3u
6
+ 30u
5
+ 4u
4
+ 36u
3
+ 8u − 4i
I
u
2
= hb + 1, 2a
2
− au + 1, u
2
+ 2i
I
v
1
= ha, b − 1, v
2
+ v −1i
* 3 irreducible components of dim
C
= 0, with total 15 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
7
− u
6
+ 6u
5
− 3u
4
+ 8u
3
− 6u
2
+ 2b + 2u − 2, −u
7
+ 2u
6
+ · · · +
4a − 2, u
9
− u
8
+ · · · + 8u − 4i
(i) Arc colorings
a
3
=
1
0
a
11
=
0
u
a
4
=
1
u
2
a
6
=
1
4
u
7
−
1
2
u
6
+ ··· +
3
2
u +
1
2
−
1
2
u
7
+
1
2
u
6
+ ··· − u + 1
a
7
=
1
4
u
7
−
1
2
u
6
+ ··· +
3
2
u +
1
2
1
4
u
8
+
1
4
u
7
+ ··· +
5
2
u
2
+ u
a
12
=
1
4
u
7
+
7
4
u
5
+ ··· +
3
2
u −
1
2
1
4
u
7
−
1
4
u
6
+ ··· + 2u
3
+
3
2
u
a
2
=
−
1
4
u
8
− 2u
6
+ ··· +
1
2
u +
1
2
u
7
+ 5u
5
+ 3u
4
+ 6u
3
− 1
a
1
=
−
1
4
u
8
+ u
7
+ ··· +
1
2
u −
1
2
u
7
+ 5u
5
+ 3u
4
+ 6u
3
− 1
a
5
=
u
2
+ 1
u
4
+ 2u
2
a
10
=
u
u
3
+ u
a
9
=
u
3
+ 2u
u
3
+ u
a
8
=
−
1
4
u
8
−
3
2
u
6
+ ··· +
3
2
u +
1
2
−
1
4
u
8
−
5
4
u
6
+ ··· +
1
2
u
2
+
3
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
8
+ u
7
− 10u
6
+ 4u
5
− 31u
4
− 37u
2
− 2u − 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
− 5u
8
+ ··· + 20u + 121
c
2
, c
5
u
9
+ 3u
8
+ 7u
7
+ 8u
6
+ 9u
5
− 14u
4
+ 29u
3
+ 8u
2
− 14u + 11
c
3
, c
4
, c
10
u
9
+ u
8
+ 10u
7
+ 3u
6
+ 30u
5
− 4u
4
+ 36u
3
+ 8u + 4
c
6
, c
7
, c
8
c
11
, c
12
u
9
+ 2u
8
− 7u
7
− 14u
6
+ 16u
5
+ 33u
4
− 5u
3
− 13u
2
+ 7u + 3
c
9
u
9
− 19u
8
+ ··· + 1064u + 212
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
+ 13y
8
+ ··· − 137056y −14641
c
2
, c
5
y
9
+ 5y
8
+ ··· + 20y −121
c
3
, c
4
, c
10
y
9
+ 19y
8
+ ··· + 64y −16
c
6
, c
7
, c
8
c
11
, c
12
y
9
− 18y
8
+ ··· + 127y −9
c
9
y
9
+ 7y
8
+ ··· + 50048y −44944
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.356661 + 1.085780I
a = −0.402756 − 1.012080I
b = −0.703349 + 0.767146I
6.51508 + 2.26295I 5.26586 − 3.19513I
u = −0.356661 − 1.085780I
a = −0.402756 + 1.012080I
b = −0.703349 − 0.767146I
6.51508 − 2.26295I 5.26586 + 3.19513I
u = −0.218744 + 0.648601I
a = 0.771686 + 0.526493I
b = −0.432002 − 0.509023I
1.07917 − 0.96089I 3.53169 + 3.58492I
u = −0.218744 − 0.648601I
a = 0.771686 − 0.526493I
b = −0.432002 + 0.509023I
1.07917 + 0.96089I 3.53169 − 3.58492I
u = 0.325488
a = 0.946130
b = 0.860684
−1.12741 −12.9160
u = 0.47728 + 2.04979I
a = −0.410722 + 0.825671I
b = 0.27104 − 2.15157I
17.8250 + 0.9503I 4.58159 − 0.40162I
u = 0.47728 − 2.04979I
a = −0.410722 − 0.825671I
b = 0.27104 + 2.15157I
17.8250 − 0.9503I 4.58159 + 0.40162I
u = 0.43538 + 2.08426I
a = 0.068727 − 0.946296I
b = 1.93397 + 1.37424I
−7.58379 − 6.45137I 4.07876 + 2.00413I
u = 0.43538 − 2.08426I
a = 0.068727 + 0.946296I
b = 1.93397 − 1.37424I
−7.58379 + 6.45137I 4.07876 − 2.00413I
5
II. I
u
2
= hb + 1, 2a
2
− au + 1, u
2
+ 2i
(i) Arc colorings
a
3
=
1
0
a
11
=
0
u
a
4
=
1
−2
a
6
=
a
−1
a
7
=
a
2a − 1
a
12
=
−a −
1
2
u
−au − 2a
a
2
=
a + 1
−1
a
1
=
a
−1
a
5
=
−1
0
a
10
=
u
−u
a
9
=
0
−u
a
8
=
−a −
1
2
u
−au − u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
9
c
10
(u
2
+ 2)
2
c
6
, c
7
, c
8
(u
2
+ u − 1)
2
c
11
, c
12
(u
2
− u − 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
9
c
10
(y + 2)
4
c
6
, c
7
, c
8
c
11
, c
12
(y
2
− 3y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.414210I
a = 1.144120I
b = −1.00000
12.1725 4.00000
u = 1.414210I
a = − 0.437016I
b = −1.00000
4.27683 4.00000
u = − 1.414210I
a = − 1.144120I
b = −1.00000
12.1725 4.00000
u = − 1.414210I
a = 0.437016I
b = −1.00000
4.27683 4.00000
9
III. I
v
1
= ha, b − 1, v
2
+ v − 1i
(i) Arc colorings
a
3
=
1
0
a
11
=
v
0
a
4
=
1
0
a
6
=
0
1
a
7
=
−v + 1
1
a
12
=
−v + 1
−v
a
2
=
1
−1
a
1
=
0
−1
a
5
=
1
0
a
10
=
v
0
a
9
=
v
0
a
8
=
v
v
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
9
c
10
u
2
c
5
(u + 1)
2
c
6
, c
7
, c
8
u
2
− u − 1
c
11
, c
12
u
2
+ u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y − 1)
2
c
3
, c
4
, c
9
c
10
y
2
c
6
, c
7
, c
8
c
11
, c
12
y
2
− 3y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 0.618034
a = 0
b = 1.00000
−0.657974 6.00000
v = −1.61803
a = 0
b = 1.00000
7.23771 6.00000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
9
− 5u
8
+ ··· + 20u + 121)
c
2
(u − 1)
2
(u + 1)
4
· (u
9
+ 3u
8
+ 7u
7
+ 8u
6
+ 9u
5
− 14u
4
+ 29u
3
+ 8u
2
− 14u + 11)
c
3
, c
4
, c
10
u
2
(u
2
+ 2)
2
(u
9
+ u
8
+ 10u
7
+ 3u
6
+ 30u
5
− 4u
4
+ 36u
3
+ 8u + 4)
c
5
(u − 1)
4
(u + 1)
2
· (u
9
+ 3u
8
+ 7u
7
+ 8u
6
+ 9u
5
− 14u
4
+ 29u
3
+ 8u
2
− 14u + 11)
c
6
, c
7
, c
8
(u
2
− u − 1)(u
2
+ u − 1)
2
· (u
9
+ 2u
8
− 7u
7
− 14u
6
+ 16u
5
+ 33u
4
− 5u
3
− 13u
2
+ 7u + 3)
c
9
u
2
(u
2
+ 2)
2
(u
9
− 19u
8
+ ··· + 1064u + 212)
c
11
, c
12
(u
2
− u − 1)
2
(u
2
+ u − 1)
· (u
9
+ 2u
8
− 7u
7
− 14u
6
+ 16u
5
+ 33u
4
− 5u
3
− 13u
2
+ 7u + 3)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
6
)(y
9
+ 13y
8
+ ··· − 137056y − 14641)
c
2
, c
5
((y − 1)
6
)(y
9
+ 5y
8
+ ··· + 20y − 121)
c
3
, c
4
, c
10
y
2
(y + 2)
4
(y
9
+ 19y
8
+ ··· + 64y − 16)
c
6
, c
7
, c
8
c
11
, c
12
((y
2
− 3y + 1)
3
)(y
9
− 18y
8
+ ··· + 127y − 9)
c
9
y
2
(y + 2)
4
(y
9
+ 7y
8
+ ··· + 50048y − 44944)
15
