12a
0803
(K12a
0803
)
A knot diagram
1
Linearized knot diagam
3 8 1 12 11 10 9 2 7 6 5 4
Solving Sequence
5,11
6 12 4 1 3 10 7 9 8 2
c
5
c
11
c
4
c
12
c
3
c
10
c
6
c
9
c
7
c
2
c
1
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= hu
10
+ u
9
+ 9u
8
+ 8u
7
+ 28u
6
+ 21u
5
+ 35u
4
+ 20u
3
+ 15u
2
+ 5u + 1i
* 1 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
= hu
10
+ u
9
+ 9u
8
+ 8u
7
+ 28u
6
+ 21u
5
+ 35u
4
+ 20u
3
+ 15u
2
+ 5u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
6
=
1
u
2
a
12
=
−u
u
a
4
=
u
2
+ 1
−u
2
a
1
=
−u
3
− 2u
u
3
+ u
a
3
=
u
4
+ 3u
2
+ 1
−u
4
− 2u
2
a
10
=
u
u
3
+ u
a
7
=
u
2
+ 1
u
4
+ 2u
2
a
9
=
u
3
+ 2u
u
5
+ 3u
3
+ u
a
8
=
u
4
+ 3u
2
+ 1
u
6
+ 4u
4
+ 3u
2
a
2
=
−u
5
− 4u
3
− 3u
u
5
+ 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
9
− 4u
8
− 36u
7
− 32u
6
− 112u
5
− 84u
4
− 140u
3
− 80u
2
− 60u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
u
10
+ u
9
+ 9u
8
+ 8u
7
+ 28u
6
+ 21u
5
+ 35u
4
+ 20u
3
+ 15u
2
+ 5u + 1
c
2
, c
8
u
10
− u
9
+ u
8
+ 4u
6
− 3u
5
+ 3u
4
+ 3u
2
− u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
y
10
+ 17y
9
+ ··· + 5y + 1
c
2
, c
8
y
10
+ y
9
+ 9y
8
+ 8y
7
+ 28y
6
+ 21y
5
+ 35y
4
+ 20y
3
+ 15y
2
+ 5y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.143160 + 0.750904I
2.80855 + 2.01562I 1.02004 − 5.14009I
u = −0.143160 − 0.750904I
2.80855 − 2.01562I 1.02004 + 5.14009I
u = −0.077356 + 1.254400I
9.46500 + 2.79918I 1.80410 − 3.17670I
u = −0.077356 − 1.254400I
9.46500 − 2.79918I 1.80410 + 3.17670I
u = −0.03400 + 1.65519I
−19.6410 + 3.2955I 1.95039 − 2.41562I
u = −0.03400 − 1.65519I
−19.6410 − 3.2955I 1.95039 + 2.41562I
u = −0.237002 + 0.228003I
−0.265356 + 0.793433I −6.76524 − 8.43244I
u = −0.237002 − 0.228003I
−0.265356 − 0.793433I −6.76524 + 8.43244I
u = −0.00849 + 1.91177I
−5.52666 + 3.57388I 1.99071 − 2.09226I
u = −0.00849 − 1.91177I
−5.52666 − 3.57388I 1.99071 + 2.09226I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
u
10
+ u
9
+ 9u
8
+ 8u
7
+ 28u
6
+ 21u
5
+ 35u
4
+ 20u
3
+ 15u
2
+ 5u + 1
c
2
, c
8
u
10
− u
9
+ u
8
+ 4u
6
− 3u
5
+ 3u
4
+ 3u
2
− u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
y
10
+ 17y
9
+ ··· + 5y + 1
c
2
, c
8
y
10
+ y
9
+ 9y
8
+ 8y
7
+ 28y
6
+ 21y
5
+ 35y
4
+ 20y
3
+ 15y
2
+ 5y + 1
7
