12n
0235
(K12n
0235
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 10 11 12 4 5 6 7 9
Solving Sequence
5,10
6 11
3,7
2 1 4 9 12 8
c
5
c
10
c
6
c
2
c
1
c
4
c
9
c
12
c
7
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
8
+ u
7
− 6u
6
− 5u
5
+ 11u
4
+ 7u
3
− 6u
2
+ b − u + 1,
− 2u
8
− 2u
7
+ 12u
6
+ 10u
5
− 22u
4
− 14u
3
+ 12u
2
+ a + 2u − 3,
u
9
+ 2u
8
− 6u
7
− 12u
6
+ 11u
5
+ 22u
4
− 6u
3
− 11u
2
+ 3u + 1i
I
u
2
= hb + 1, a − 1, u
3
+ u
2
− 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 12 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
8
+ u
7
+ · · · + b + 1, −2u
8
− 2u
7
+ · · · + a − 3, u
9
+ 2u
8
+ · · · + 3u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
u
−u
3
+ u
a
3
=
2u
8
+ 2u
7
− 12u
6
− 10u
5
+ 22u
4
+ 14u
3
− 12u
2
− 2u + 3
−u
8
− u
7
+ 6u
6
+ 5u
5
− 11u
4
− 7u
3
+ 6u
2
+ u − 1
a
7
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
u
8
+ u
7
− 6u
6
− 5u
5
+ 11u
4
+ 7u
3
− 6u
2
− u + 2
−u
8
− u
7
+ 6u
6
+ 5u
5
− 11u
4
− 7u
3
+ 6u
2
+ u − 1
a
1
=
u
7
− 4u
5
+ 2u
3
+ 2u
−u
7
+ 5u
5
− 6u
3
+ u
a
4
=
u
7
− 6u
5
+ 10u
3
− 4u + 2
u
8
− 6u
6
+ u
5
+ 11u
4
− 3u
3
− 6u
2
+ 3u
a
9
=
−u
u
a
12
=
−u
3
+ 2u
u
5
− 3u
3
+ u
a
8
=
u
4
− 3u
2
+ 1
−u
6
+ 4u
4
− 3u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −5u
8
− 8u
7
+ 31u
6
+ 46u
5
− 61u
4
− 79u
3
+ 40u
2
+ 32u − 7
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
− 2u
8
+ 17u
7
− 27u
6
+ 64u
5
− 32u
4
− 18u
3
+ 48u
2
− 12u + 1
c
2
, c
4
u
9
− 4u
8
+ 9u
7
− 9u
6
+ 4u
5
+ 6u
4
− 6u
3
+ 6u
2
+ 1
c
3
, c
8
u
9
− u
8
+ 11u
7
− 4u
6
+ 36u
5
− 7u
4
+ 35u
3
− 24u
2
+ 4u + 8
c
5
, c
6
, c
7
c
9
, c
10
, c
11
u
9
+ 2u
8
− 6u
7
− 12u
6
+ 11u
5
+ 22u
4
− 6u
3
− 11u
2
+ 3u + 1
c
12
u
9
− 8u
8
+ ··· + 563u − 55
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
+ 30y
8
+ ··· + 48y −1
c
2
, c
4
y
9
+ 2y
8
+ 17y
7
+ 27y
6
+ 64y
5
+ 32y
4
− 18y
3
− 48y
2
− 12y −1
c
3
, c
8
y
9
+ 21y
8
+ ··· + 400y −64
c
5
, c
6
, c
7
c
9
, c
10
, c
11
y
9
− 16y
8
+ ··· + 31y −1
c
12
y
9
− 76y
8
+ ··· + 218299y −3025
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.179250 + 0.234667I
a = 0.27808 − 1.83091I
b = 0.360958 + 0.915457I
6.67894 − 1.77536I 10.01363 + 2.14949I
u = −1.179250 − 0.234667I
a = 0.27808 + 1.83091I
b = 0.360958 − 0.915457I
6.67894 + 1.77536I 10.01363 − 2.14949I
u = 0.551791 + 0.168482I
a = 1.190020 + 0.762701I
b = −0.095011 − 0.381350I
1.001300 + 0.199242I 9.75217 − 1.35811I
u = 0.551791 − 0.168482I
a = 1.190020 − 0.762701I
b = −0.095011 + 0.381350I
1.001300 − 0.199242I 9.75217 + 1.35811I
u = 1.64788 + 0.14930I
a = −0.74609 + 2.36291I
b = 0.87305 − 1.18145I
16.6165 + 3.5415I 9.41596 − 2.15533I
u = 1.64788 − 0.14930I
a = −0.74609 − 2.36291I
b = 0.87305 + 1.18145I
16.6165 − 3.5415I 9.41596 + 2.15533I
u = −0.206388
a = 2.82125
b = −0.910627
−1.31799 −11.3310
u = −1.91723 + 0.04388I
a = −1.63264 − 2.58420I
b = 1.31632 + 1.29210I
−8.83332 − 4.85466I 8.98397 + 1.82769I
u = −1.91723 − 0.04388I
a = −1.63264 + 2.58420I
b = 1.31632 − 1.29210I
−8.83332 + 4.85466I 8.98397 − 1.82769I
5
II. I
u
2
= hb + 1, a − 1, u
3
+ u
2
− 2u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
−u
2
a
11
=
u
u
2
− u − 1
a
3
=
1
−1
a
7
=
−u
2
+ 1
u
2
− u − 1
a
2
=
0
−1
a
1
=
−1
0
a
4
=
1
−1
a
9
=
−u
u
a
12
=
u
2
− 1
−u
2
a
8
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
− u + 11
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 = 1.00000
b = −1.00000
4.69981 8.19810
u = −0.445042
a = 1.00000
b = −1.00000
−0.939962 11.2470
u = −1.80194
a = 1.00000
b = −1.00000
15.9794 9.55500
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
3
· (u
9
− 2u
8
+ 17u
7
− 27u
6
+ 64u
5
− 32u
4
− 18u
3
+ 48u
2
− 12u + 1)
c
2
(u − 1)
3
(u
9
− 4u
8
+ 9u
7
− 9u
6
+ 4u
5
+ 6u
4
− 6u
3
+ 6u
2
+ 1)
c
3
, c
8
u
3
(u
9
− u
8
+ 11u
7
− 4u
6
+ 36u
5
− 7u
4
+ 35u
3
− 24u
2
+ 4u + 8)
c
4
(u + 1)
3
(u
9
− 4u
8
+ 9u
7
− 9u
6
+ 4u
5
+ 6u
4
− 6u
3
+ 6u
2
+ 1)
c
5
, c
6
, c
7
(u
3
+ u
2
− 2u − 1)
· (u
9
+ 2u
8
− 6u
7
− 12u
6
+ 11u
5
+ 22u
4
− 6u
3
− 11u
2
+ 3u + 1)
c
9
, c
10
, c
11
(u
3
− u
2
− 2u + 1)
· (u
9
+ 2u
8
− 6u
7
− 12u
6
+ 11u
5
+ 22u
4
− 6u
3
− 11u
2
+ 3u + 1)
c
12
(u
3
− u
2
− 2u + 1)(u
9
− 8u
8
+ ··· + 563u − 55)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
3
)(y
9
+ 30y
8
+ ··· + 48y −1)
c
2
, c
4
(y −1)
3
· (y
9
+ 2y
8
+ 17y
7
+ 27y
6
+ 64y
5
+ 32y
4
− 18y
3
− 48y
2
− 12y −1)
c
3
, c
8
y
3
(y
9
+ 21y
8
+ ··· + 400y −64)
c
5
, c
6
, c
7
c
9
, c
10
, c
11
(y
3
− 5y
2
+ 6y −1)(y
9
− 16y
8
+ ··· + 31y −1)
c
12
(y
3
− 5y
2
+ 6y −1)(y
9
− 76y
8
+ ··· + 218299y −3025)
11
