12n
0293
(K12n
0293
)
A knot diagram
1
Linearized knot diagam
3 6 7 8 2 5 11 5 12 7 10 9
Solving Sequence
7,11 5,8
4 3 6 2 1 10 12 9
c
7
c
4
c
3
c
6
c
2
c
1
c
10
c
11
c
9
c
5
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
2
+ b, −u
5
+ u
4
− u
2
+ a − 2u + 1, u
6
− 2u
5
+ u
4
+ 2u
3
− 2u + 1i
I
u
2
= hu
2
+ b, a + 1, u
3
+ u
2
− 1i
I
u
3
= h−u
2
a + b, a
2
+ au + 2u
2
+ 3u + 2, u
3
+ u
2
− 1i
I
u
4
= hu
4
− 2u
3
+ u
2
+ 2b − u + 1, −u
5
+ 3u
4
− 5u
3
+ 4u
2
+ 2a − 6u + 3, u
6
− 2u
5
+ 3u
4
− 2u
3
+ 4u
2
− 2u − 1i
* 4 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
= hu
2
+ b, −u
5
+ u
4
− u
2
+ a − 2u + 1, u
6
− 2u
5
+ u
4
+ 2u
3
− 2u + 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
5
=
u
5
− u
4
+ u
2
+ 2u − 1
−u
2
a
8
=
1
u
2
a
4
=
2u
5
− 3u
4
+ 3u
2
+ 3u − 2
−u
5
+ u
3
− u
a
3
=
u
5
− 3u
4
+ u
3
+ 3u
2
+ 2u − 2
−u
5
+ u
3
− u
a
6
=
u
5
− 2u
4
+ u
2
+ u
−u
4
a
2
=
u
5
− 2u
4
+ 2u
3
+ 2u
2
− 1
u
3
− u
a
1
=
3u
5
− 4u
4
− 2u
3
+ 2u
2
+ 3u − 2
2u
5
− 4u
4
− 2u
3
+ 2u
2
+ 2u − 2
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6u
5
− 8u
4
− 2u
3
+ 18u
2
+ 6u − 16
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
9
c
11
, c
12
u
6
+ 2u
5
+ 9u
4
+ 10u
3
+ 10u
2
+ 4u + 1
c
2
, c
5
, c
7
c
10
u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u + 1
c
3
u
6
− 12u
5
+ 79u
4
+ 50u
3
− 2u
2
− 2u + 1
c
4
, c
8
u
6
+ 10u
5
+ 38u
4
+ 56u
3
+ 44u
2
+ 24u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
9
c
11
, c
12
y
6
+ 14y
5
+ 61y
4
+ 66y
3
+ 38y
2
+ 4y + 1
c
2
, c
5
, c
7
c
10
y
6
− 2y
5
+ 9y
4
− 10y
3
+ 10y
2
− 4y + 1
c
3
y
6
+ 14y
5
+ 7437y
4
− 2862y
3
+ 362y
2
− 8y + 1
c
4
, c
8
y
6
− 24y
5
+ 412y
4
− 256y
3
− 144y
2
+ 128y + 64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.801169 + 0.530454I
a = −0.849606 + 1.000430I
b = −0.360490 + 0.849967I
1.85230 + 4.21966I −5.10387 − 7.89854I
u = −0.801169 − 0.530454I
a = −0.849606 − 1.000430I
b = −0.360490 − 0.849967I
1.85230 − 4.21966I −5.10387 + 7.89854I
u = 0.586664 + 0.275361I
a = 0.407311 + 0.793222I
b = −0.268351 − 0.323088I
−0.92569 − 1.07524I −7.92809 + 6.11055I
u = 0.586664 − 0.275361I
a = 0.407311 − 0.793222I
b = −0.268351 + 0.323088I
−0.92569 + 1.07524I −7.92809 − 6.11055I
u = 1.21451 + 1.05065I
a = −1.55771 − 1.63833I
b = −0.37116 − 2.55204I
−13.2636 − 8.5731I −4.96804 + 3.72288I
u = 1.21451 − 1.05065I
a = −1.55771 + 1.63833I
b = −0.37116 + 2.55204I
−13.2636 + 8.5731I −4.96804 − 3.72288I
5
II. I
u
2
= hu
2
+ b, a + 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
5
=
−1
−u
2
a
8
=
1
u
2
a
4
=
−1
−u
2
a
3
=
−u
2
− 1
−u
2
a
6
=
−u
2
+ 1
−u
2
− u + 1
a
2
=
−2u
2
− u
−u
2
− u + 1
a
1
=
−1
0
a
10
=
u
u
a
12
=
u
2
− 1
u
2
+ u − 1
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u − 12
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
9
u
3
− u
2
+ 2u − 1
c
2
, c
7
u
3
+ u
2
− 1
c
4
, c
8
u
3
c
5
, c
10
u
3
− u
2
+ 1
c
6
, c
11
, c
12
u
3
+ u
2
+ 2u + 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
9
, c
11
, c
12
y
3
+ 3y
2
+ 2y −1
c
2
, c
5
, c
7
c
10
y
3
− y
2
+ 2y −1
c
4
, c
8
y
3
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −1.00000
b = −0.215080 + 1.307140I
6.04826 + 5.65624I −4.98049 − 5.95889I
u = −0.877439 − 0.744862I
a = −1.00000
b = −0.215080 − 1.307140I
6.04826 − 5.65624I −4.98049 + 5.95889I
u = 0.754878
a = −1.00000
b = −0.569840
−2.22691 −18.0390
9
III. I
u
3
= h−u
2
a + b, a
2
+ au + 2u
2
+ 3u + 2, u
3
+ u
2
− 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
5
=
a
u
2
a
a
8
=
1
u
2
a
4
=
a
u
2
a
a
3
=
u
2
a + a
u
2
a
a
6
=
−u
2
a + u
2
+ a + 2u + 2
−au + a + u + 1
a
2
=
−u
2
a − au + u
2
+ 2a + 3u + 1
−au + a + u + 1
a
1
=
−1
0
a
10
=
u
u
a
12
=
u
2
− 1
u
2
+ u − 1
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
a + u
2
− 5
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
9
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
7
(u
3
+ u
2
− 1)
2
c
4
, c
8
u
6
c
5
, c
10
(u
3
− u
2
+ 1)
2
c
6
, c
11
, c
12
(u
3
+ u
2
+ 2u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
c
9
, c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
, c
7
c
10
(y
3
− y
2
+ 2y −1)
2
c
4
, c
8
y
6
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.947279 − 0.320410I
b = −0.215080 − 1.307140I
6.04826 −4.56984 + 0.I
u = −0.877439 + 0.744862I
a = −0.069840 − 0.424452I
b = −0.569840
1.91067 + 2.82812I −4.21508 − 1.30714I
u = −0.877439 − 0.744862I
a = 0.947279 + 0.320410I
b = −0.215080 + 1.307140I
6.04826 −4.56984 + 0.I
u = −0.877439 − 0.744862I
a = −0.069840 + 0.424452I
b = −0.569840
1.91067 − 2.82812I −4.21508 + 1.30714I
u = 0.754878
a = −0.37744 + 2.29387I
b = −0.215080 + 1.307140I
1.91067 + 2.82812I −4.21508 − 1.30714I
u = 0.754878
a = −0.37744 − 2.29387I
b = −0.215080 − 1.307140I
1.91067 − 2.82812I −4.21508 + 1.30714I
13
IV. I
u
4
= hu
4
− 2u
3
+ u
2
+ 2b − u + 1, −u
5
+ 3u
4
− 5u
3
+ 4u
2
+ 2a − 6u +
3, u
6
− 2u
5
+ 3u
4
− 2u
3
+ 4u
2
− 2u − 1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
5
=
1
2
u
5
−
3
2
u
4
+ ··· + 3u −
3
2
−
1
2
u
4
+ u
3
−
1
2
u
2
+
1
2
u −
1
2
a
8
=
1
u
2
a
4
=
1
2
u
5
−
1
2
u
4
+
3
2
u
3
+ 2u −
3
2
u
5
−
3
2
u
4
+ ··· +
5
2
u +
1
2
a
3
=
3
2
u
5
− 2u
4
+ ··· +
9
2
u − 1
u
5
−
3
2
u
4
+ ··· +
5
2
u +
1
2
a
6
=
1
2
u
5
− u
4
+
1
2
u
3
−
3
2
u
2
+
5
2
u
−
1
2
u
4
−
1
2
u
3
− u
2
+
3
2
u
a
2
=
1
2
u
5
−
1
2
u
4
+ 2u
3
−
3
2
u
2
+ 3u − 1
1
2
u
4
+
1
2
u
3
− u
2
+
1
2
u
a
1
=
u
5
− 4u
4
+ 2u
3
− 6u
2
+ 5u + 2
−4u
4
+ 2u
3
− 6u
2
+ 4u + 2
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
1
2
u
4
+ u
3
−
3
2
u
2
+
1
2
u −
11
2
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
9
c
11
, c
12
u
6
− 2u
5
+ 9u
4
− 10u
3
+ 2u
2
+ 12u + 1
c
2
, c
5
, c
7
c
10
u
6
+ 2u
5
+ 3u
4
+ 2u
3
+ 4u
2
+ 2u − 1
c
3
u
6
− 12u
5
+ 217u
4
− 1458u
3
+ 3038u
2
+ 1786u − 673
c
4
, c
8
(u
3
− 8u
2
+ 12u + 8)
2
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
9
c
11
, c
12
y
6
+ 14y
5
+ 45y
4
− 14y
3
+ 262y
2
− 140y + 1
c
2
, c
5
, c
7
c
10
y
6
+ 2y
5
+ 9y
4
+ 10y
3
+ 2y
2
− 12y + 1
c
3
y
6
+ 290y
5
+ ··· − 7278944y + 452929
c
4
, c
8
(y
3
− 40y
2
+ 272y −64)
2
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.846666
a = 0.570369
b = −0.0850937
−1.40994 −5.80190
u = −0.400969 + 1.133260I
a = 1.346010 − 0.279891I
b = 1.12349 − 0.90880I
4.22983 −2.75302 + 0.I
u = −0.400969 − 1.133260I
a = 1.346010 + 0.279891I
b = 1.12349 + 0.90880I
4.22983 −2.75302 + 0.I
u = 1.12349 + 1.24085I
a = 1.17845 + 1.79308I
b = 0.27748 + 2.78816I
−12.6895 −4.44504 + 0.I
u = 1.12349 − 1.24085I
a = 1.17845 − 1.79308I
b = 0.27748 − 2.78816I
−12.6895 −4.44504 + 0.I
u = −0.291708
a = −2.61929
b = −0.716844
−1.40994 −5.80190
17
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
9
(u
3
− u
2
+ 2u − 1)
3
(u
6
− 2u
5
+ 9u
4
− 10u
3
+ 2u
2
+ 12u + 1)
· (u
6
+ 2u
5
+ 9u
4
+ 10u
3
+ 10u
2
+ 4u + 1)
c
2
, c
7
(u
3
+ u
2
− 1)
3
(u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u + 1)
· (u
6
+ 2u
5
+ 3u
4
+ 2u
3
+ 4u
2
+ 2u − 1)
c
3
(u
3
− u
2
+ 2u − 1)
3
(u
6
− 12u
5
+ 79u
4
+ 50u
3
− 2u
2
− 2u + 1)
· (u
6
− 12u
5
+ 217u
4
− 1458u
3
+ 3038u
2
+ 1786u − 673)
c
4
, c
8
u
9
(u
3
− 8u
2
+ 12u + 8)
2
(u
6
+ 10u
5
+ ··· + 24u + 8)
c
5
, c
10
(u
3
− u
2
+ 1)
3
(u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u + 1)
· (u
6
+ 2u
5
+ 3u
4
+ 2u
3
+ 4u
2
+ 2u − 1)
c
6
, c
11
, c
12
(u
3
+ u
2
+ 2u + 1)
3
(u
6
− 2u
5
+ 9u
4
− 10u
3
+ 2u
2
+ 12u + 1)
· (u
6
+ 2u
5
+ 9u
4
+ 10u
3
+ 10u
2
+ 4u + 1)
18
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
9
c
11
, c
12
(y
3
+ 3y
2
+ 2y −1)
3
(y
6
+ 14y
5
+ 45y
4
− 14y
3
+ 262y
2
− 140y + 1)
· (y
6
+ 14y
5
+ 61y
4
+ 66y
3
+ 38y
2
+ 4y + 1)
c
2
, c
5
, c
7
c
10
(y
3
− y
2
+ 2y −1)
3
(y
6
− 2y
5
+ 9y
4
− 10y
3
+ 10y
2
− 4y + 1)
· (y
6
+ 2y
5
+ 9y
4
+ 10y
3
+ 2y
2
− 12y + 1)
c
3
((y
3
+ 3y
2
+ 2y −1)
3
)(y
6
+ 14y
5
+ ··· − 8y + 1)
· (y
6
+ 290y
5
+ ··· − 7278944y + 452929)
c
4
, c
8
y
9
(y
3
− 40y
2
+ 272y −64)
2
· (y
6
− 24y
5
+ 412y
4
− 256y
3
− 144y
2
+ 128y + 64)
19
