12n
0243
(K12n
0243
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 10 12 11 3 5 6 7 8
Solving Sequence
6,12
7 11 8 1 10
3,5
2 4 9
c
6
c
11
c
7
c
12
c
10
c
5
c
2
c
4
c
9
c
1
, c
3
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
5
+ u
4
− 2u
3
+ u
2
+ b − u, −u
5
+ u
4
− 3u
3
+ 2u
2
+ a − 2u + 1,
u
8
− 2u
7
+ 5u
6
− 6u
5
+ 7u
4
− 7u
3
+ 4u
2
− 4u + 1i
I
u
2
= hu
5
+ u
4
+ 2u
3
+ u
2
+ b + u, u
5
+ u
4
+ 3u
3
+ 2u
2
+ a + 2u + 1, u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 2 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
= h−u
5
+ u
4
− 2u
3
+ u
2
+ b − u, −u
5
+ u
4
− 3u
3
+ 2u
2
+ a − 2u +
1, u
8
− 2u
7
+ · · · − 4u + 1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
7
=
1
u
2
a
11
=
u
u
3
+ u
a
8
=
u
2
+ 1
u
4
+ 2u
2
a
1
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
10
=
u
3
+ 2u
u
3
+ u
a
3
=
u
5
− u
4
+ 3u
3
− 2u
2
+ 2u − 1
u
5
− u
4
+ 2u
3
− u
2
+ u
a
5
=
−u
6
− 3u
4
− 2u
2
+ 1
−u
6
− 2u
4
− u
2
a
2
=
u
7
+ 8u
5
− 2u
4
+ 13u
3
− 3u
2
+ 5u − 2
u
7
+ 2u
6
+ 4u
5
+ 3u
4
+ 4u
3
+ u
2
a
4
=
2u
7
+ 6u
6
+ 11u
5
+ 15u
4
+ 17u
3
+ 8u
2
+ 8u − 3
4u
7
+ 7u
6
+ 10u
5
+ 14u
4
+ 8u
3
+ 5u
2
− 1
a
9
=
−3u
7
+ 4u
6
− 10u
5
+ 7u
4
− 10u
3
+ 4u
2
− 4u + 2
−2u
7
+ 4u
6
− 8u
5
+ 7u
4
− 10u
3
+ 4u
2
− 6u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
7
− 8u
6
+ 16u
5
− 17u
4
+ 14u
3
− 15u
2
+ 6u − 19
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ 33u
7
+ 402u
6
+ 2159u
5
+ 4922u
4
+ 6895u
3
− 302u
2
+ 33u + 1
c
2
, c
4
u
8
− 7u
7
+ 8u
6
+ 27u
5
− 20u
4
− 81u
3
+ 12u
2
− 3u − 1
c
3
, c
8
u
8
+ 7u
7
− 19u
6
− 256u
5
− 600u
4
− 536u
3
− 32u
2
− 128u − 64
c
5
, c
9
, c
10
c
12
u
8
− 2u
7
− 7u
6
+ 12u
5
+ 5u
4
+ 3u
3
− 2u
2
+ 2u + 1
c
6
, c
7
, c
11
u
8
+ 2u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 7u
3
+ 4u
2
+ 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
− 285y
7
+ ··· − 1693y + 1
c
2
, c
4
y
8
− 33y
7
+ 402y
6
− 2159y
5
+ 4922y
4
− 6895y
3
− 302y
2
− 33y + 1
c
3
, c
8
y
8
− 87y
7
+ ··· − 12288y + 4096
c
5
, c
9
, c
10
c
12
y
8
− 18y
7
+ 107y
6
− 206y
5
− 9y
4
− 91y
3
+ 2y
2
− 8y + 1
c
6
, c
7
, c
11
y
8
+ 6y
7
+ 15y
6
+ 14y
5
− 9y
4
− 31y
3
− 26y
2
− 8y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.381025 + 0.877247I
a = 1.30622 + 1.00951I
b = 1.238510 − 0.243220I
−1.28153 + 1.66195I −14.7384 − 2.2086I
u = −0.381025 − 0.877247I
a = 1.30622 − 1.00951I
b = 1.238510 + 0.243220I
−1.28153 − 1.66195I −14.7384 + 2.2086I
u = 1.11498
a = 3.07969
b = 2.82176
9.42637 −17.0560
u = 0.126694 + 1.193160I
a = −0.183567 − 0.143629I
b = −0.178784 + 0.606721I
2.78716 − 1.62541I −7.16123 + 3.74390I
u = 0.126694 − 1.193160I
a = −0.183567 + 0.143629I
b = −0.178784 − 0.606721I
2.78716 + 1.62541I −7.16123 − 3.74390I
u = 0.54402 + 1.39007I
a = 1.08549 − 1.80102I
b = 2.89776 − 0.22684I
13.7911 − 5.9041I −14.4329 + 2.5359I
u = 0.54402 − 1.39007I
a = 1.08549 + 1.80102I
b = 2.89776 + 0.22684I
13.7911 + 5.9041I −14.4329 − 2.5359I
u = 0.305633
a = −0.495968
b = 0.263262
−0.541319 −18.2790
5
II. I
u
2
= hu
5
+ u
4
+ 2u
3
+ u
2
+ b + u, u
5
+ u
4
+ 3u
3
+ 2u
2
+ a + 2u + 1, u
6
+
u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
7
=
1
u
2
a
11
=
u
u
3
+ u
a
8
=
u
2
+ 1
u
4
+ 2u
2
a
1
=
−u
5
− 2u
3
− u
−u
5
− u
4
− 2u
3
− u
2
− u + 1
a
10
=
u
3
+ 2u
u
3
+ u
a
3
=
−u
5
− u
4
− 3u
3
− 2u
2
− 2u − 1
−u
5
− u
4
− 2u
3
− u
2
− u
a
5
=
u
5
+ 2u
3
+ u
u
5
+ u
4
+ 2u
3
+ u
2
+ u − 1
a
2
=
−2u
5
− u
4
− 5u
3
− 2u
2
− 3u − 1
−2u
5
− 2u
4
− 4u
3
− 2u
2
− 2u + 1
a
4
=
−u
5
− u
4
− 3u
3
− 2u
2
− 2u − 1
−u
5
− u
4
− 2u
3
− u
2
− u
a
9
=
u
2
+ 1
u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
4
− 2u
3
− 5u
2
− 2u − 15
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
8
u
6
c
4
(u + 1)
6
c
5
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
6
, c
7
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
9
, c
10
, c
12
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
11
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
8
y
6
c
5
, c
9
, c
10
c
12
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
6
, c
7
, c
11
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.873214
a = 1.14519
b = 1.36865
−9.30502 −17.4790
u = 0.138835 + 1.234450I
a = −0.089969 + 0.799962I
b = −1.087730 + 0.567441I
1.31531 − 1.97241I −12.92955 + 2.53106I
u = 0.138835 − 1.234450I
a = −0.089969 − 0.799962I
b = −1.087730 − 0.567441I
1.31531 + 1.97241I −12.92955 − 2.53106I
u = −0.408802 + 1.276380I
a = 0.227586 + 0.710576I
b = 1.286430 − 0.496092I
−5.34051 + 4.59213I −13.8770 − 3.6103I
u = −0.408802 − 1.276380I
a = 0.227586 − 0.710576I
b = 1.286430 + 0.496092I
−5.34051 − 4.59213I −13.8770 + 3.6103I
u = 0.413150
a = −2.42043
b = −0.766061
−2.38379 −16.9080
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
6
· (u
8
+ 33u
7
+ 402u
6
+ 2159u
5
+ 4922u
4
+ 6895u
3
− 302u
2
+ 33u + 1)
c
2
(u − 1)
6
(u
8
− 7u
7
+ 8u
6
+ 27u
5
− 20u
4
− 81u
3
+ 12u
2
− 3u − 1)
c
3
, c
8
u
6
(u
8
+ 7u
7
− 19u
6
− 256u
5
− 600u
4
− 536u
3
− 32u
2
− 128u − 64)
c
4
(u + 1)
6
(u
8
− 7u
7
+ 8u
6
+ 27u
5
− 20u
4
− 81u
3
+ 12u
2
− 3u − 1)
c
5
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
8
− 2u
7
− 7u
6
+ 12u
5
+ 5u
4
+ 3u
3
− 2u
2
+ 2u + 1)
c
6
, c
7
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
8
+ 2u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 7u
3
+ 4u
2
+ 4u + 1)
c
9
, c
10
, c
12
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
8
− 2u
7
− 7u
6
+ 12u
5
+ 5u
4
+ 3u
3
− 2u
2
+ 2u + 1)
c
11
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
8
+ 2u
7
+ 5u
6
+ 6u
5
+ 7u
4
+ 7u
3
+ 4u
2
+ 4u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
8
− 285y
7
+ ··· − 1693y + 1)
c
2
, c
4
(y −1)
6
· (y
8
− 33y
7
+ 402y
6
− 2159y
5
+ 4922y
4
− 6895y
3
− 302y
2
− 33y + 1)
c
3
, c
8
y
6
(y
8
− 87y
7
+ ··· − 12288y + 4096)
c
5
, c
9
, c
10
c
12
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
8
− 18y
7
+ 107y
6
− 206y
5
− 9y
4
− 91y
3
+ 2y
2
− 8y + 1)
c
6
, c
7
, c
11
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
8
+ 6y
7
+ 15y
6
+ 14y
5
− 9y
4
− 31y
3
− 26y
2
− 8y + 1)
11
