12n
0242
(K12n
0242
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 10 11 12 3 5 6 7 8
Solving Sequence
6,10
11 7 12
3,5
2 1 4 9 8
c
10
c
6
c
11
c
5
c
2
c
1
c
4
c
9
c
8
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, u
2
+ a − 2u, u
3
− 3u
2
+ 2u + 1i
I
u
2
= hb + u, −u
2
+ a + 2, u
3
+ u
2
− 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 6 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
= hb − u, u
2
+ a − 2u, u
3
− 3u
2
+ 2u + 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
7
=
−u
−3u
2
+ 3u + 1
a
12
=
−u
2
+ 1
−5u
2
+ 7u + 3
a
3
=
−u
2
+ 2u
u
a
5
=
u
u
a
2
=
3u
2
− 3u − 2
4u
2
− 4u − 2
a
1
=
−4u
2
+ 7u + 2
7u + 2
a
4
=
−5u
2
+ 12u + 4
−8u
2
+ 17u + 6
a
9
=
−u
2
+ 1
−u
2
a
8
=
3u
2
− 4u − 1
5u
2
− 10u − 4
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
2
− u − 19
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
− 13u
2
+ 51u + 1
c
2
, c
4
u
3
− u
2
+ 7u + 1
c
3
, c
8
u
3
− 4u
2
+ 20u + 8
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
u
3
− 3u
2
+ 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
3
− 67y
2
+ 2627y −1
c
2
, c
4
y
3
+ 13y
2
+ 51y −1
c
3
, c
8
y
3
+ 24y
2
+ 464y −64
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 10y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.324718
a = −0.754878
b = −0.324718
−0.531480 −18.5700
u = 1.66236 + 0.56228I
a = 0.877439 − 0.744862I
b = 1.66236 + 0.56228I
−4.66906 − 2.82812I −18.2151 + 1.3071I
u = 1.66236 − 0.56228I
a = 0.877439 + 0.744862I
b = 1.66236 − 0.56228I
−4.66906 + 2.82812I −18.2151 − 1.3071I
5
II. I
u
2
= hb + u, −u
2
+ a + 2, u
3
+ u
2
− 2u − 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
7
=
−u
u
2
− u − 1
a
12
=
−u
2
+ 1
−u
2
+ u + 1
a
3
=
u
2
− 2
−u
a
5
=
u
u
a
2
=
u
2
− u − 2
−2u
a
1
=
−u
−u
a
4
=
u
2
− 2
−u
a
9
=
−u
2
+ 1
−u
2
a
8
=
−u
2
+ 1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
− u − 17
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
7
u
3
− u
2
− 2u + 1
c
9
, c
10
, 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
2
, c
4
(y −1)
3
c
3
, c
8
y
3
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.24698
a = −0.445042
b = −1.24698
−7.98968 −19.8020
u = −0.445042
a = −1.80194
b = 0.445042
−2.34991 −16.7530
u = −1.80194
a = 1.24698
b = 1.80194
−19.2692 −18.4450
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
3
(u
3
− 13u
2
+ 51u + 1)
c
2
(u − 1)
3
(u
3
− u
2
+ 7u + 1)
c
3
, c
8
u
3
(u
3
− 4u
2
+ 20u + 8)
c
4
(u + 1)
3
(u
3
− u
2
+ 7u + 1)
c
5
, c
6
, c
7
(u
3
− 3u
2
+ 2u + 1)(u
3
− u
2
− 2u + 1)
c
9
, c
10
, c
11
c
12
(u
3
− 3u
2
+ 2u + 1)(u
3
+ u
2
− 2u − 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
3
(y
3
− 67y
2
+ 2627y −1)
c
2
, c
4
(y −1)
3
(y
3
+ 13y
2
+ 51y −1)
c
3
, c
8
y
3
(y
3
+ 24y
2
+ 464y −64)
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
(y
3
− 5y
2
+ 6y −1)(y
3
− 5y
2
+ 10y −1)
11
