12n
0332
(K12n
0332
)
A knot diagram
1
Linearized knot diagam
3 5 7 10 2 4 3 12 11 5 9 8
Solving Sequence
5,11
10
4,7
3 2 1 6 9 12 8
c
10
c
4
c
3
c
2
c
1
c
6
c
9
c
11
c
8
c
5
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
8
− u
7
− u
6
+ 3u
5
− u
4
− u
3
+ u
2
+ b + u − 1, −u
9
+ u
8
+ u
7
− 3u
6
+ u
5
+ u
4
− 2u
3
+ 2a + u − 1,
u
10
− 3u
9
+ 3u
8
+ 3u
7
− 9u
6
+ 7u
5
+ 2u
4
− 4u
3
− u
2
+ 5u − 2i
I
u
2
= hu
7
− u
5
+ 3u
3
+ u
2
+ b − u, −u
7
+ u
6
+ u
5
− u
4
− 3u
3
+ 2u
2
+ a + 2u − 1, u
8
− u
6
+ 3u
4
− 2u
2
+ 1i
I
u
3
= hb
2
+ b + 2, a − 1, u + 1i
* 3 irreducible components of dim
C
= 0, with total 20 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
− u
6
+ 3u
5
− u
4
− u
3
+ u
2
+ b + u − 1, −u
9
+ u
8
+ · · · +
2a − 1, u
10
− 3u
9
+ · · · + 5u − 2i
(i) Arc colorings
a
5
=
0
u
a
11
=
1
0
a
10
=
1
u
2
a
4
=
−u
−u
3
+ u
a
7
=
1
2
u
9
−
1
2
u
8
+ ··· −
1
2
u +
1
2
−u
8
+ u
7
+ u
6
− 3u
5
+ u
4
+ u
3
− u
2
− u + 1
a
3
=
1
2
u
9
−
3
2
u
8
+ ··· −
5
2
u +
1
2
−u
9
+ 2u
8
− 4u
6
+ 4u
5
+ u
4
− 3u
3
+ 3u − 1
a
2
=
1
2
u
9
−
3
2
u
8
+ ··· −
5
2
u +
1
2
−2u
9
+ 3u
8
+ u
7
− 8u
6
+ 5u
5
+ 3u
4
− 5u
3
− 2u
2
+ 4u − 1
a
1
=
−u
8
+ u
6
− 3u
4
+ 2u
2
− 1
u
8
+ 2u
4
a
6
=
−
1
2
u
9
+
1
2
u
8
+ ··· +
1
2
u +
1
2
−u
9
+ 3u
8
− 2u
7
− 4u
6
+ 10u
5
− 3u
4
− 5u
3
+ 4u
2
+ 6u − 3
a
9
=
−u
2
+ 1
u
2
a
12
=
u
4
− u
2
+ 1
−u
4
a
8
=
−u
6
+ u
4
− 2u
2
+ 1
u
6
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
9
−10u
8
+ 6u
7
+ 18u
6
−30u
5
+ 10u
4
+ 20u
3
−12u
2
−10u + 22
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
− 10u
9
+ ··· + 2u + 1
c
2
, c
3
, c
5
c
6
, c
7
u
10
+ 2u
9
− 3u
8
− 6u
7
+ 8u
6
+ 26u
5
− 12u
4
+ 22u
3
− 9u
2
+ 4u − 1
c
4
, c
10
u
10
− 3u
9
+ 3u
8
+ 3u
7
− 9u
6
+ 7u
5
+ 2u
4
− 4u
3
− u
2
+ 5u − 2
c
8
, c
9
, c
11
c
12
u
10
− 3u
9
+ ··· − 21u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 2y
9
+ ··· − 146y + 1
c
2
, c
3
, c
5
c
6
, c
7
y
10
− 10y
9
+ ··· + 2y + 1
c
4
, c
10
y
10
− 3y
9
+ ··· − 21y + 4
c
8
, c
9
, c
11
c
12
y
10
+ 9y
9
+ ··· − 177y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.578093 + 0.999236I
a = 0.42866 − 1.39259I
b = 0.90785 + 1.25726I
1.59788 − 2.80907I 5.86935 + 0.88784I
u = 0.578093 − 0.999236I
a = 0.42866 + 1.39259I
b = 0.90785 − 1.25726I
1.59788 + 2.80907I 5.86935 − 0.88784I
u = −0.702617 + 0.466190I
a = 0.287658 + 0.593730I
b = −0.219312 − 0.226300I
−1.04423 − 1.80881I 6.53522 + 6.24906I
u = −0.702617 − 0.466190I
a = 0.287658 − 0.593730I
b = −0.219312 + 0.226300I
−1.04423 + 1.80881I 6.53522 − 6.24906I
u = 0.916845 + 0.866673I
a = −0.785318 + 0.843397I
b = −0.26510 − 1.64874I
−8.72520 + 3.21048I 7.41352 − 2.75592I
u = 0.916845 − 0.866673I
a = −0.785318 − 0.843397I
b = −0.26510 + 1.64874I
−8.72520 − 3.21048I 7.41352 + 2.75592I
u = 1.144580 + 0.768721I
a = 1.275550 − 0.486093I
b = −0.96622 + 2.20358I
3.33426 + 9.25636I 6.94023 − 4.73549I
u = 1.144580 − 0.768721I
a = 1.275550 + 0.486093I
b = −0.96622 − 2.20358I
3.33426 − 9.25636I 6.94023 + 4.73549I
u = −1.37948
a = −1.26013
b = 0.730548
9.04363 10.0430
u = 0.505678
a = 0.347038
b = 0.355011
0.630953 16.4400
5
II. I
u
2
= hu
7
− u
5
+ 3u
3
+ u
2
+ b − u, −u
7
+ u
6
+ u
5
− u
4
− 3u
3
+ 2u
2
+ a +
2u − 1, u
8
− u
6
+ 3u
4
− 2u
2
+ 1i
(i) Arc colorings
a
5
=
0
u
a
11
=
1
0
a
10
=
1
u
2
a
4
=
−u
−u
3
+ u
a
7
=
u
7
− u
6
− u
5
+ u
4
+ 3u
3
− 2u
2
− 2u + 1
−u
7
+ u
5
− 3u
3
− u
2
+ u
a
3
=
−u
6
− 2u
2
−u
7
+ u
5
+ u
4
− 3u
3
+ 2u + 1
a
2
=
−u
6
− 2u
2
−u
7
− u
6
+ u
5
+ 2u
4
− 3u
3
− 2u
2
+ 2u + 2
a
1
=
0
−u
6
+ u
4
− 2u
2
+ 1
a
6
=
u
7
− u
5
+ 3u
3
− 2u
−u
7
− u
6
+ u
5
− 3u
3
− 2u
2
+ u
a
9
=
−u
2
+ 1
u
2
a
12
=
u
4
− u
2
+ 1
−u
4
a
8
=
−u
6
+ u
4
− 2u
2
+ 1
u
6
+ u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
6
+ 4u
4
− 12u
2
+ 8
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
8
c
2
, c
3
, c
5
c
6
, c
7
(u
2
+ 1)
4
c
4
, c
10
u
8
− u
6
+ 3u
4
− 2u
2
+ 1
c
8
, c
9
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
c
11
, c
12
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y − 1)
8
c
2
, c
3
, c
5
c
6
, c
7
(y + 1)
8
c
4
, c
10
(y
4
− y
3
+ 3y
2
− 2y + 1)
2
c
8
, c
9
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.720342 + 0.351808I
a = −0.769066 − 0.172918I
b = 0.005408 − 1.406080I
−3.07886 + 1.41510I 3.82674 − 4.90874I
u = 0.720342 − 0.351808I
a = −0.769066 + 0.172918I
b = 0.005408 + 1.406080I
−3.07886 − 1.41510I 3.82674 + 4.90874I
u = −0.720342 + 0.351808I
a = 1.47268 + 1.26777I
b = −0.795655 − 0.392388I
−3.07886 − 1.41510I 3.82674 + 4.90874I
u = −0.720342 − 0.351808I
a = 1.47268 − 1.26777I
b = −0.795655 + 0.392388I
−3.07886 + 1.41510I 3.82674 − 4.90874I
u = 0.911292 + 0.851808I
a = −1.43746 + 1.45872I
b = −0.43052 − 2.95172I
−10.08060 + 3.16396I 0.17326 − 2.56480I
u = 0.911292 − 0.851808I
a = −1.43746 − 1.45872I
b = −0.43052 + 2.95172I
−10.08060 − 3.16396I 0.17326 + 2.56480I
u = −0.911292 + 0.851808I
a = −0.266156 − 0.363868I
b = 0.220764 + 0.153260I
−10.08060 − 3.16396I 0.17326 + 2.56480I
u = −0.911292 − 0.851808I
a = −0.266156 + 0.363868I
b = 0.220764 − 0.153260I
−10.08060 + 3.16396I 0.17326 − 2.56480I
9
III. I
u
3
= hb
2
+ b + 2, a − 1, u + 1i
(i) Arc colorings
a
5
=
0
−1
a
11
=
1
0
a
10
=
1
1
a
4
=
1
0
a
7
=
1
b
a
3
=
b + 1
−b − 2
a
2
=
b + 1
−1
a
1
=
−2
3
a
6
=
−b + 1
b
a
9
=
0
1
a
12
=
1
−1
a
8
=
−1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 10
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
2
+ 3u + 4
c
2
, c
3
, c
5
c
6
, c
7
u
2
− u + 2
c
4
, c
10
(u + 1)
2
c
8
, c
9
, c
11
c
12
(u − 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
2
− y + 16
c
2
, c
3
, c
5
c
6
, c
7
y
2
+ 3y + 4
c
4
, c
8
, c
9
c
10
, c
11
, c
12
(y − 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = −0.50000 + 1.32288I
−1.64493 10.0000
u = −1.00000
a = 1.00000
b = −0.50000 − 1.32288I
−1.64493 10.0000
13
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
2
+ 3u + 4)(u
10
− 10u
9
+ ··· + 2u + 1)
c
2
, c
3
, c
5
c
6
, c
7
(u
2
+ 1)
4
(u
2
− u + 2)
· (u
10
+ 2u
9
− 3u
8
− 6u
7
+ 8u
6
+ 26u
5
− 12u
4
+ 22u
3
− 9u
2
+ 4u − 1)
c
4
, c
10
(u + 1)
2
(u
8
− u
6
+ 3u
4
− 2u
2
+ 1)
· (u
10
− 3u
9
+ 3u
8
+ 3u
7
− 9u
6
+ 7u
5
+ 2u
4
− 4u
3
− u
2
+ 5u − 2)
c
8
, c
9
((u − 1)
2
)(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
(u
10
− 3u
9
+ ··· − 21u + 4)
c
11
, c
12
((u − 1)
2
)(u
4
− u
3
+ 3u
2
− 2u + 1)
2
(u
10
− 3u
9
+ ··· − 21u + 4)
14
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
8
)(y
2
− y + 16)(y
10
− 2y
9
+ ··· − 146y + 1)
c
2
, c
3
, c
5
c
6
, c
7
((y + 1)
8
)(y
2
+ 3y + 4)(y
10
− 10y
9
+ ··· + 2y + 1)
c
4
, c
10
((y − 1)
2
)(y
4
− y
3
+ 3y
2
− 2y + 1)
2
(y
10
− 3y
9
+ ··· − 21y + 4)
c
8
, c
9
, c
11
c
12
((y − 1)
2
)(y
4
+ 5y
3
+ ··· + 2y + 1)
2
(y
10
+ 9y
9
+ ··· − 177y + 16)
15
