12a
0722
(K12a
0722
)
A knot diagram
1
Linearized knot diagam
3 8 9 10 11 12 1 2 4 5 6 7
Solving Sequence
2,9
8 3 4 10 5 11 1 7 12 6
c
8
c
2
c
3
c
9
c
4
c
10
c
1
c
7
c
12
c
6
c
5
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
14
− u
13
+ 5u
12
− 4u
11
+ 10u
10
− 7u
9
+ 7u
8
− 4u
7
− 4u
6
+ 2u
5
− 8u
4
+ 4u
3
− 2u
2
+ u + 1i
* 1 irreducible components of dim
C
= 0, with total 14 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
14
− u
13
+ 5u
12
− 4u
11
+ 10u
10
− 7u
9
+ 7u
8
− 4u
7
− 4u
6
+ 2u
5
−
8u
4
+ 4u
3
− 2u
2
+ u + 1i
(i) Arc colorings
a
2
=
0
u
a
9
=
1
0
a
8
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
10
=
−u
6
− u
4
+ 1
u
6
+ 2u
4
+ u
2
a
5
=
u
9
+ 2u
7
+ u
5
− 2u
3
− u
−u
9
− 3u
7
− 3u
5
+ u
a
11
=
u
12
+ 3u
10
+ 3u
8
− 2u
6
− 4u
4
− u
2
+ 1
−u
12
− 4u
10
− 6u
8
− 2u
6
+ 3u
4
+ 2u
2
a
1
=
u
3
u
5
+ u
3
+ u
a
7
=
−u
6
− u
4
+ 1
−u
8
− 2u
6
− 2u
4
a
12
=
−u
9
− 2u
7
− u
5
+ 2u
3
+ u
−u
11
− 3u
9
− 4u
7
− u
5
+ u
3
+ u
a
6
=
u
12
+ 3u
10
+ 3u
8
− 2u
6
− 4u
4
− u
2
+ 1
u
13
− u
12
+ ··· − u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
13
+ 4u
12
−16u
11
+ 12u
10
−24u
9
+ 16u
8
−4u
7
+ 20u
5
−8u
4
+ 12u
3
−8u
2
−4u −14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
+ 9u
13
+ ··· − 5u + 1
c
2
, c
8
u
14
− u
13
+ ··· + u + 1
c
3
, c
4
, c
5
c
6
, c
7
, c
9
c
10
, c
11
, c
12
u
14
+ u
13
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 7y
13
+ ··· − 65y + 1
c
2
, c
8
y
14
+ 9y
13
+ ··· − 5y + 1
c
3
, c
4
, c
5
c
6
, c
7
, c
9
c
10
, c
11
, c
12
y
14
− 23y
13
+ ··· − 5y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.972298
11.0809 −15.9440
u = −0.306114 + 1.029060I
−3.14696 − 2.76430I −17.6509 + 6.3298I
u = −0.306114 − 1.029060I
−3.14696 + 2.76430I −17.6509 − 6.3298I
u = −0.884219
−14.9042 −15.7550
u = 0.159123 + 0.837990I
−0.675258 + 0.985154I −10.63652 − 6.07794I
u = 0.159123 − 0.837990I
−0.675258 − 0.985154I −10.63652 + 6.07794I
u = 0.396353 + 1.167340I
−9.01747 + 3.99409I −18.9152 − 4.1194I
u = 0.396353 − 1.167340I
−9.01747 − 3.99409I −18.9152 + 4.1194I
u = 0.713918
−5.61914 −15.3500
u = −0.455547 + 1.256230I
−18.7410 − 4.7668I −19.0242 + 3.1632I
u = −0.455547 − 1.256230I
−18.7410 + 4.7668I −19.0242 − 3.1632I
u = 0.489108 + 1.306520I
7.03257 + 5.19559I −19.0102 − 2.7600I
u = 0.489108 − 1.306520I
7.03257 − 5.19559I −19.0102 + 2.7600I
u = −0.367845
−0.678832 −14.4770
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
14
+ 9u
13
+ ··· − 5u + 1
c
2
, c
8
u
14
− u
13
+ ··· + u + 1
c
3
, c
4
, c
5
c
6
, c
7
, c
9
c
10
, c
11
, c
12
u
14
+ u
13
+ ··· + 3u + 1
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
14
− 7y
13
+ ··· − 65y + 1
c
2
, c
8
y
14
+ 9y
13
+ ··· − 5y + 1
c
3
, c
4
, c
5
c
6
, c
7
, c
9
c
10
, c
11
, c
12
y
14
− 23y
13
+ ··· − 5y + 1
7
