12n
0404
(K12n
0404
)
A knot diagram
1
Linearized knot diagam
3 5 10 9 2 11 3 12 4 7 5 8
Solving Sequence
5,9
4
7,10
11 12 3 2 1 6 8
c
4
c
9
c
10
c
11
c
3
c
2
c
1
c
6
c
8
c
5
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
7
+ 2u
6
− 5u
5
+ 6u
4
− 6u
3
+ 4u
2
+ b − u − 1,
u
10
− 4u
9
+ 13u
8
− 26u
7
+ 42u
6
− 50u
5
+ 45u
4
− 28u
3
+ 9u
2
+ 2a + 4u − 3,
u
11
− 4u
10
+ 13u
9
− 28u
8
+ 48u
7
− 64u
6
+ 67u
5
− 52u
4
+ 29u
3
− 6u
2
− 3u + 2i
I
u
2
= hu
7
+ 3u
5
+ 2u
3
+ b − u − 1, u
9
+ 4u
7
+ u
6
+ 5u
5
+ 3u
4
+ u
3
+ 2u
2
+ a, u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 21 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
7
+2u
6
+· · · +b−1, u
10
−4u
9
+· · · +2a−3, u
11
−4u
10
+· · · −3u+2i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
7
=
−
1
2
u
10
+ 2u
9
+ ··· − 2u +
3
2
u
7
− 2u
6
+ 5u
5
− 6u
4
+ 6u
3
− 4u
2
+ u + 1
a
10
=
u
u
3
+ u
a
11
=
−
1
2
u
10
+ 2u
9
+ ··· −
5
2
u
2
+
1
2
u
10
− 3u
9
+ 9u
8
− 16u
7
+ 24u
6
− 27u
5
+ 22u
4
− 13u
3
+ 3u
2
+ 2u − 1
a
12
=
1
2
u
10
− u
9
+ ··· + 2u −
1
2
u
10
− 3u
9
+ 9u
8
− 16u
7
+ 24u
6
− 27u
5
+ 22u
4
− 13u
3
+ 3u
2
+ 2u − 1
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
−u
4
− u
2
+ 1
u
4
+ 2u
2
a
1
=
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
9u
10
− 24u
9
+ ··· + 10u − 8
a
6
=
u
8
+ 3u
6
+ u
4
− 2u
2
+ 1
−u
8
− 4u
6
− 4u
4
a
8
=
1
2
u
10
− 2u
9
+ ··· + u −
3
2
−u
8
− u
7
− 2u
6
− 5u
5
+ 2u
4
− 6u
3
+ 4u
2
− 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
10
+ 8u
9
− 26u
8
+ 54u
7
− 88u
6
+ 106u
5
− 94u
4
+ 52u
3
− 10u
2
− 18u + 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
11
+ 54u
10
+ ··· + 10929u − 576
c
2
, c
5
u
11
+ 2u
10
+ ··· + 9u − 24
c
3
, c
4
, c
9
u
11
+ 4u
10
+ ··· − 3u − 2
c
6
, c
8
, c
10
c
12
u
11
− u
10
+ ··· + u − 1
c
7
, c
11
u
11
− u
10
+ ··· − 13u − 19
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
11
− 546y
10
+ ··· + 91631457y −331776
c
2
, c
5
y
11
+ 54y
10
+ ··· + 10929y −576
c
3
, c
4
, c
9
y
11
+ 10y
10
+ ··· + 33y −4
c
6
, c
8
, c
10
c
12
y
11
+ 33y
10
+ ··· − y −1
c
7
, c
11
y
11
+ 93y
10
+ ··· + 4083y −361
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.025290 + 0.573466I
a = 0.522820 + 0.914944I
b = −2.15532 − 0.07526I
14.2401 + 3.3069I 6.51466 − 1.87736I
u = 1.025290 − 0.573466I
a = 0.522820 − 0.914944I
b = −2.15532 + 0.07526I
14.2401 − 3.3069I 6.51466 + 1.87736I
u = −0.041636 + 1.304450I
a = 0.517767 − 0.458534I
b = −0.109577 + 0.529508I
−3.44187 − 1.25408I 7.18214 + 5.23967I
u = −0.041636 − 1.304450I
a = 0.517767 + 0.458534I
b = −0.109577 − 0.529508I
−3.44187 + 1.25408I 7.18214 − 5.23967I
u = 0.564252 + 0.373580I
a = 0.013693 − 0.730377I
b = 0.832660 − 0.220165I
−1.87019 + 1.75538I 8.10394 − 4.89065I
u = 0.564252 − 0.373580I
a = 0.013693 + 0.730377I
b = 0.832660 + 0.220165I
−1.87019 − 1.75538I 8.10394 + 4.89065I
u = 0.21728 + 1.43552I
a = −1.255990 − 0.421534I
b = 1.114610 − 0.376316I
−7.66219 + 4.64924I 5.42003 − 4.56433I
u = 0.21728 − 1.43552I
a = −1.255990 + 0.421534I
b = 1.114610 + 0.376316I
−7.66219 − 4.64924I 5.42003 + 4.56433I
u = 0.40590 + 1.55278I
a = 1.63733 + 1.45401I
b = −2.10223 − 0.22168I
7.51792 + 8.56204I 4.30767 − 3.05307I
u = 0.40590 − 1.55278I
a = 1.63733 − 1.45401I
b = −2.10223 + 0.22168I
7.51792 − 8.56204I 4.30767 + 3.05307I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.342167
a = 0.628767
b = −0.160296
0.526767 18.9430
6
II. I
u
2
= hu
7
+ 3u
5
+ 2u
3
+ b − u − 1, u
9
+ 4u
7
+ u
6
+ 5u
5
+ 3u
4
+ u
3
+ 2u
2
+
a, u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1i
(i) Arc colorings
a
5
=
1
0
a
9
=
0
u
a
4
=
1
u
2
a
7
=
−u
9
− 4u
7
− u
6
− 5u
5
− 3u
4
− u
3
− 2u
2
−u
7
− 3u
5
− 2u
3
+ u + 1
a
10
=
u
u
3
+ u
a
11
=
u
9
− u
8
+ 4u
7
− 4u
6
+ 5u
5
− 4u
4
+ u
3
− 1
−u
9
− 4u
7
− 5u
5
− u
4
− u
3
− 2u
2
a
12
=
−u
8
− 4u
6
− 5u
4
− 2u
2
− 1
−u
9
− 4u
7
− 5u
5
− u
4
− u
3
− 2u
2
a
3
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
−u
4
− u
2
+ 1
u
4
+ 2u
2
a
1
=
0
−u
8
− 3u
6
− u
4
+ 2u
2
− 1
a
6
=
u
8
+ 3u
6
+ u
4
− 2u
2
+ 1
−u
8
− 4u
6
− 4u
4
a
8
=
−u
9
− 5u
7
− 8u
5
− 3u
3
+ u
u
8
− u
7
+ 4u
6
− 3u
5
+ 4u
4
− 2u
3
+ 2u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
6
− 12u
4
− 8u
2
+ 4
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
c
2
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
3
, c
4
, c
9
u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1
c
5
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
6
, c
8
, c
10
c
12
(u
2
+ 1)
5
c
7
u
10
− 2u
9
+ 5u
8
+ 7u
6
+ 10u
5
+ 24u
4
+ 30u
3
+ 37u
2
+ 40u + 29
c
11
u
10
+ 2u
9
+ 5u
8
+ 7u
6
− 10u
5
+ 24u
4
− 30u
3
+ 37u
2
− 40u + 29
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
c
2
, c
5
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
3
, c
4
, c
9
(y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
c
6
, c
8
, c
10
c
12
(y + 1)
10
c
7
, c
11
y
10
+ 6y
9
+ ··· + 546y + 841
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.217740I
a = −0.37029 − 1.58802I
b = 1.000000 + 0.766826I
−5.69095 2.51890
u = − 1.217740I
a = −0.37029 + 1.58802I
b = 1.000000 − 0.766826I
−5.69095 2.51890
u = 0.549911 + 0.309916I
a = 0.42897 − 1.54636I
b = 1.82238 − 0.33911I
−3.61897 + 1.53058I 3.48489 − 4.43065I
u = 0.549911 − 0.309916I
a = 0.42897 + 1.54636I
b = 1.82238 + 0.33911I
−3.61897 − 1.53058I 3.48489 + 4.43065I
u = −0.549911 + 0.309916I
a = −0.686530 + 0.668968I
b = 0.177625 − 0.339110I
−3.61897 − 1.53058I 3.48489 + 4.43065I
u = −0.549911 − 0.309916I
a = −0.686530 − 0.668968I
b = 0.177625 + 0.339110I
−3.61897 + 1.53058I 3.48489 − 4.43065I
u = −0.21917 + 1.41878I
a = −0.092267 + 0.641941I
b = −0.200152 − 0.455697I
−9.16243 − 4.40083I −0.74431 + 3.49859I
u = −0.21917 − 1.41878I
a = −0.092267 − 0.641941I
b = −0.200152 + 0.455697I
−9.16243 + 4.40083I −0.74431 − 3.49859I
u = 0.21917 + 1.41878I
a = −2.27989 − 1.10735I
b = 2.20015 − 0.45570I
−9.16243 + 4.40083I −0.74431 − 3.49859I
u = 0.21917 − 1.41878I
a = −2.27989 + 1.10735I
b = 2.20015 + 0.45570I
−9.16243 − 4.40083I −0.74431 + 3.49859I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)
2
)(u
11
+ 54u
10
+ ··· + 10929u − 576)
c
2
((u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
)(u
11
+ 2u
10
+ ··· + 9u − 24)
c
3
, c
4
, c
9
(u
10
+ 5u
8
+ 8u
6
+ 3u
4
− u
2
+ 1)(u
11
+ 4u
10
+ ··· − 3u − 2)
c
5
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
11
+ 2u
10
+ ··· + 9u − 24)
c
6
, c
8
, c
10
c
12
((u
2
+ 1)
5
)(u
11
− u
10
+ ··· + u − 1)
c
7
(u
10
− 2u
9
+ 5u
8
+ 7u
6
+ 10u
5
+ 24u
4
+ 30u
3
+ 37u
2
+ 40u + 29)
· (u
11
− u
10
+ ··· − 13u − 19)
c
11
(u
10
+ 2u
9
+ 5u
8
+ 7u
6
− 10u
5
+ 24u
4
− 30u
3
+ 37u
2
− 40u + 29)
· (u
11
− u
10
+ ··· − 13u − 19)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
· (y
11
− 546y
10
+ ··· + 91631457y −331776)
c
2
, c
5
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
11
+ 54y
10
+ ··· + 10929y −576)
c
3
, c
4
, c
9
((y
5
+ 5y
4
+ 8y
3
+ 3y
2
− y + 1)
2
)(y
11
+ 10y
10
+ ··· + 33y −4)
c
6
, c
8
, c
10
c
12
((y + 1)
10
)(y
11
+ 33y
10
+ ··· − y −1)
c
7
, c
11
(y
10
+ 6y
9
+ ··· + 546y + 841)(y
11
+ 93y
10
+ ··· + 4083y −361)
12
