12a
0511
(K12a
0511
)
A knot diagram
1
Linearized knot diagam
3 7 8 9 11 2 1 4 5 12 6 10
Solving Sequence
5,11
6 12 10 1 9 4 8 3 7 2
c
5
c
11
c
10
c
12
c
9
c
4
c
8
c
3
c
7
c
2
c
1
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
52
− u
51
+ ··· − u
2
+ 1i
I
u
2
= hu
10
− 2u
8
+ 3u
6
+ u
5
− 2u
4
− u
3
+ u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 62 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
52
− u
51
+ · · · − u
2
+ 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
6
=
1
−u
2
a
12
=
u
−u
3
+ u
a
10
=
−u
3
u
5
− u
3
+ u
a
1
=
u
5
+ u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
−u
5
− u
u
5
− u
3
+ u
a
4
=
−u
10
+ u
8
− 2u
6
+ u
4
− u
2
+ 1
u
10
− 2u
8
+ 3u
6
− 2u
4
+ u
2
a
8
=
u
15
− 2u
13
+ 4u
11
− 4u
9
+ 4u
7
− 4u
5
+ 2u
3
− 2u
−u
15
+ 3u
13
− 6u
11
+ 7u
9
− 6u
7
+ 4u
5
− 2u
3
+ u
a
3
=
u
20
− 3u
18
+ ··· − 3u
2
+ 1
−u
20
+ 4u
18
+ ··· − 5u
4
+ 2u
2
a
7
=
−u
27
+ 4u
25
+ ··· − u
3
− 2u
u
29
− 5u
27
+ ··· − 5u
3
+ u
a
2
=
u
47
− 8u
45
+ ··· − 42u
5
+ 10u
3
−u
47
+ 9u
45
+ ··· − 4u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
51
+ 40u
49
+ ··· + 16u + 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
52
+ 23u
51
+ ··· + 2u + 1
c
2
, c
6
u
52
− u
51
+ ··· − 2u + 1
c
3
, c
4
, c
8
c
9
u
52
− 4u
51
+ ··· + 36u + 4
c
5
, c
11
u
52
− u
51
+ ··· − u
2
+ 1
c
7
u
52
− 3u
51
+ ··· − 8u + 5
c
10
, c
12
u
52
− 19u
51
+ ··· − 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
52
+ 13y
51
+ ··· − 6y + 1
c
2
, c
6
y
52
− 23y
51
+ ··· − 2y + 1
c
3
, c
4
, c
8
c
9
y
52
− 60y
51
+ ··· − 696y + 16
c
5
, c
11
y
52
− 19y
51
+ ··· − 2y + 1
c
7
y
52
− 3y
51
+ ··· − 534y + 25
c
10
, c
12
y
52
+ 29y
51
+ ··· − 6y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.998941 + 0.045978I
5.28620 − 0.93871I 15.9673 + 0.9512I
u = −0.998941 − 0.045978I
5.28620 + 0.93871I 15.9673 − 0.9512I
u = −0.728323 + 0.685940I
−3.57521 − 0.97987I 0.260565 + 0.640305I
u = −0.728323 − 0.685940I
−3.57521 + 0.97987I 0.260565 − 0.640305I
u = 1.002230 + 0.089379I
3.67626 + 5.72177I 12.7013 − 6.5868I
u = 1.002230 − 0.089379I
3.67626 − 5.72177I 12.7013 + 6.5868I
u = −0.665641 + 0.720471I
−1.78289 + 5.70739I 3.97022 − 5.89201I
u = −0.665641 − 0.720471I
−1.78289 − 5.70739I 3.97022 + 5.89201I
u = 0.530837 + 0.816404I
6.92563 − 8.79045I 7.25155 + 4.85962I
u = 0.530837 − 0.816404I
6.92563 + 8.79045I 7.25155 − 4.85962I
u = −0.522819 + 0.813333I
8.75278 + 3.37930I 9.83163 − 0.35758I
u = −0.522819 − 0.813333I
8.75278 − 3.37930I 9.83163 + 0.35758I
u = 0.521713 + 0.791233I
3.07937 − 1.57482I 4.02266 + 0.18175I
u = 0.521713 − 0.791233I
3.07937 + 1.57482I 4.02266 − 0.18175I
u = −0.833280 + 0.443412I
0.09032 − 4.10436I 9.88532 + 7.11286I
u = −0.833280 − 0.443412I
0.09032 + 4.10436I 9.88532 − 7.11286I
u = 0.650819 + 0.682385I
0.204162 − 1.159700I 7.88295 + 1.51609I
u = 0.650819 − 0.682385I
0.204162 + 1.159700I 7.88295 − 1.51609I
u = 0.492028 + 0.801217I
7.16166 + 5.41348I 7.59584 − 4.65169I
u = 0.492028 − 0.801217I
7.16166 − 5.41348I 7.59584 + 4.65169I
u = −0.850769 + 0.649712I
−2.04278 − 2.52764I 4.28217 + 3.73621I
u = −0.850769 − 0.649712I
−2.04278 + 2.52764I 4.28217 − 3.73621I
u = 0.823932 + 0.690742I
−4.74803 − 0.75860I −0.678268 + 1.202668I
u = 0.823932 − 0.690742I
−4.74803 + 0.75860I −0.678268 − 1.202668I
u = 0.874578 + 0.686205I
−4.59465 + 6.05228I 0. − 7.84880I
u = 0.874578 − 0.686205I
−4.59465 − 6.05228I 0. + 7.84880I
u = 0.982865 + 0.602859I
2.10133 + 4.69499I 11.88053 − 6.10182I
u = 0.982865 − 0.602859I
2.10133 − 4.69499I 11.88053 + 6.10182I
u = −0.946937 + 0.662515I
−2.91580 − 4.23415I 0. + 5.26094I
u = −0.946937 − 0.662515I
−2.91580 + 4.23415I 0. − 5.26094I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.158760 + 0.014478I
12.9367 − 7.2052I 13.29023 + 4.77271I
u = −1.158760 − 0.014478I
12.9367 + 7.2052I 13.29023 − 4.77271I
u = 1.159210 + 0.007998I
14.7139 + 1.7466I 15.7254 + 0.I
u = 1.159210 − 0.007998I
14.7139 − 1.7466I 15.7254 + 0.I
u = 0.984648 + 0.652855I
1.18503 + 6.34235I 6.00000 − 6.57485I
u = 0.984648 − 0.652855I
1.18503 − 6.34235I 6.00000 + 6.57485I
u = −0.986168 + 0.671138I
−0.83243 − 11.04580I 0. + 10.84993I
u = −0.986168 − 0.671138I
−0.83243 + 11.04580I 0. − 10.84993I
u = 1.064000 + 0.654185I
4.67776 + 7.01331I 0
u = 1.064000 − 0.654185I
4.67776 − 7.01331I 0
u = −1.074140 + 0.648969I
10.57820 − 5.44672I 0
u = −1.074140 − 0.648969I
10.57820 + 5.44672I 0
u = −1.072330 + 0.660031I
10.38760 − 8.89731I 0
u = −1.072330 − 0.660031I
10.38760 + 8.89731I 0
u = 1.071430 + 0.664123I
8.5369 + 14.3344I 0
u = 1.071430 − 0.664123I
8.5369 − 14.3344I 0
u = 0.649830 + 0.190765I
0.943472 + 0.087273I 11.64609 − 1.04296I
u = 0.649830 − 0.190765I
0.943472 − 0.087273I 11.64609 + 1.04296I
u = −0.355914 + 0.492873I
−0.32921 − 4.26537I 5.60545 + 7.03160I
u = −0.355914 − 0.492873I
−0.32921 + 4.26537I 5.60545 − 7.03160I
u = −0.114095 + 0.393922I
−1.45941 + 1.41253I 0.827325 − 0.785575I
u = −0.114095 − 0.393922I
−1.45941 − 1.41253I 0.827325 + 0.785575I
6
II. I
u
2
= hu
10
− 2u
8
+ 3u
6
+ u
5
− 2u
4
− u
3
+ u
2
+ u − 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
6
=
1
−u
2
a
12
=
u
−u
3
+ u
a
10
=
−u
3
u
5
− u
3
+ u
a
1
=
u
5
+ u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
−u
5
− u
u
5
− u
3
+ u
a
4
=
−u
8
+ u
6
+ u
5
− u
4
− u
3
+ u
−u
5
+ u
3
− u + 1
a
8
=
−u
8
+ u
6
− u
4
− 1
−u
5
+ u
3
− u + 1
a
3
=
−u
3
u
5
− u
3
+ u
a
7
=
−u
2
− 1
u
4
a
2
=
u
−u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
+ 4u
9
+ 10u
8
+ 16u
7
+ 19u
6
+ 19u
5
+ 16u
4
+ 13u
3
+ 7u
2
+ 3u + 1
c
2
, c
5
, c
6
c
11
u
10
− 2u
8
+ 3u
6
+ u
5
− 2u
4
− u
3
+ u
2
+ u − 1
c
3
, c
4
, c
8
c
9
(u
2
+ u − 1)
5
c
7
u
10
− 2u
8
+ 2u
7
+ 9u
6
− 5u
5
− 12u
4
+ u
3
+ 13u
2
− 7u + 1
c
10
, c
12
u
10
− 4u
9
+ 10u
8
− 16u
7
+ 19u
6
− 19u
5
+ 16u
4
− 13u
3
+ 7u
2
− 3u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
, c
12
y
10
+ 4y
9
+ 10y
8
+ 4y
7
− 17y
6
− 51y
5
− 48y
4
− 21y
3
+ 3y
2
+ 5y + 1
c
2
, c
5
, c
6
c
11
y
10
− 4y
9
+ 10y
8
− 16y
7
+ 19y
6
− 19y
5
+ 16y
4
− 13y
3
+ 7y
2
− 3y + 1
c
3
, c
4
, c
8
c
9
(y
2
− 3y + 1)
5
c
7
y
10
− 4y
9
+ ··· − 23y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.501486 + 0.805060I
8.88264 10.0000
u = −0.501486 − 0.805060I
8.88264 10.0000
u = −0.974665 + 0.570706I
0.986960 10.0000
u = −0.974665 − 0.570706I
0.986960 10.0000
u = −1.14608
8.88264 10.0000
u = 0.802076
0.986960 10.0000
u = 0.573627 + 0.524384I
0.986960 10.0000
u = 0.573627 − 0.524384I
0.986960 10.0000
u = 1.074530 + 0.643996I
8.88264 10.0000
u = 1.074530 − 0.643996I
8.88264 10.0000
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
10
+ 4u
9
+ 10u
8
+ 16u
7
+ 19u
6
+ 19u
5
+ 16u
4
+ 13u
3
+ 7u
2
+ 3u + 1)
· (u
52
+ 23u
51
+ ··· + 2u + 1)
c
2
, c
6
(u
10
− 2u
8
+ ··· + u − 1)(u
52
− u
51
+ ··· − 2u + 1)
c
3
, c
4
, c
8
c
9
((u
2
+ u − 1)
5
)(u
52
− 4u
51
+ ··· + 36u + 4)
c
5
, c
11
(u
10
− 2u
8
+ ··· + u − 1)(u
52
− u
51
+ ··· − u
2
+ 1)
c
7
(u
10
− 2u
8
+ 2u
7
+ 9u
6
− 5u
5
− 12u
4
+ u
3
+ 13u
2
− 7u + 1)
· (u
52
− 3u
51
+ ··· − 8u + 5)
c
10
, c
12
(u
10
− 4u
9
+ 10u
8
− 16u
7
+ 19u
6
− 19u
5
+ 16u
4
− 13u
3
+ 7u
2
− 3u + 1)
· (u
52
− 19u
51
+ ··· − 2u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
10
+ 4y
9
+ 10y
8
+ 4y
7
− 17y
6
− 51y
5
− 48y
4
− 21y
3
+ 3y
2
+ 5y + 1)
· (y
52
+ 13y
51
+ ··· − 6y + 1)
c
2
, c
6
(y
10
− 4y
9
+ 10y
8
− 16y
7
+ 19y
6
− 19y
5
+ 16y
4
− 13y
3
+ 7y
2
− 3y + 1)
· (y
52
− 23y
51
+ ··· − 2y + 1)
c
3
, c
4
, c
8
c
9
((y
2
− 3y + 1)
5
)(y
52
− 60y
51
+ ··· − 696y + 16)
c
5
, c
11
(y
10
− 4y
9
+ 10y
8
− 16y
7
+ 19y
6
− 19y
5
+ 16y
4
− 13y
3
+ 7y
2
− 3y + 1)
· (y
52
− 19y
51
+ ··· − 2y + 1)
c
7
(y
10
− 4y
9
+ ··· − 23y + 1)(y
52
− 3y
51
+ ··· − 534y + 25)
c
10
, c
12
(y
10
+ 4y
9
+ 10y
8
+ 4y
7
− 17y
6
− 51y
5
− 48y
4
− 21y
3
+ 3y
2
+ 5y + 1)
· (y
52
+ 29y
51
+ ··· − 6y + 1)
12
