12n
0523
(K12n
0523
)
A knot diagram
1
Linearized knot diagam
3 6 9 8 2 12 11 4 3 6 7 10
Solving Sequence
3,9
4
6,10
11 2 1 5 8 7 12
c
3
c
9
c
10
c
2
c
1
c
5
c
8
c
7
c
12
c
4
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−2237795u
14
− 2035326u
13
+ ··· + 214299626b − 45089178,
− 5439477u
14
− 13822761u
13
+ ··· + 857198504a − 455155360, u
15
+ u
14
+ ··· − 24u
2
+ 8i
I
u
2
= hb + 1, 4a
3
− 2a
2
u + 12a
2
− 4au + 12a − 3u + 4, u
2
+ 2i
I
v
1
= ha, b − 1, v
3
+ v
2
− 1i
* 3 irreducible components of dim
C
= 0, with total 24 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−2.24 × 10
6
u
14
− 2.04 × 10
6
u
13
+ · · · + 2.14 × 10
8
b − 4.51 × 10
7
, −5.44 ×
10
6
u
14
−1.38×10
7
u
13
+· · ·+8.57×10
8
a−4.55×10
8
, u
15
+u
14
+· · ·−24u
2
+8i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
−u
2
a
6
=
0.00634564u
14
+ 0.0161255u
13
+ ··· − 1.29993u + 0.530980
0.0104424u
14
+ 0.00949757u
13
+ ··· + 0.658438u + 0.210403
a
10
=
u
u
a
11
=
0.0534854u
14
+ 0.0891983u
13
+ ··· + 0.512158u − 1.09323
−0.0306520u
14
− 0.0138981u
13
+ ··· + 1.32388u + 0.760098
a
2
=
0.0409458u
14
+ 0.0558447u
13
+ ··· − 0.692253u + 0.663144
−0.0540665u
14
− 0.0698196u
13
+ ··· − 0.280107u − 0.0129726
a
1
=
−0.0131207u
14
− 0.0139749u
13
+ ··· − 0.972359u + 0.650171
−0.0540665u
14
− 0.0698196u
13
+ ··· − 0.280107u − 0.0129726
a
5
=
u
2
+ 1
−u
4
− 2u
2
a
8
=
−u
u
3
+ u
a
7
=
−0.0757597u
14
− 0.0957170u
13
+ ··· + 0.656898u + 1.36357
0.0170459u
14
+ 0.0203363u
13
+ ··· + 0.767262u + 0.281197
a
12
=
0.00634564u
14
+ 0.0161255u
13
+ ··· − 1.29993u + 0.530980
−0.0346001u
14
− 0.0397191u
13
+ ··· − 0.607673u − 0.132164
(ii) Obstruction class = −1
(iii) Cusp Shapes =
107735653
428599252
u
14
+
61824743
428599252
u
13
+ ··· −
778177628
107149813
u +
655579008
107149813
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
15
+ 32u
14
+ ··· + 6478u + 289
c
2
, c
5
u
15
+ 4u
14
+ ··· + 52u − 17
c
3
, c
4
, c
8
c
9
u
15
− u
14
+ ··· + 24u
2
− 8
c
6
, c
7
, c
11
u
15
− 2u
14
+ ··· + u − 3
c
10
u
15
+ 2u
14
+ ··· + 77u − 87
c
12
u
15
− 2u
14
+ ··· − 127u − 171
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
15
− 88y
14
+ ··· + 29457142y −83521
c
2
, c
5
y
15
− 32y
14
+ ··· + 6478y −289
c
3
, c
4
, c
8
c
9
y
15
+ 29y
14
+ ··· + 384y −64
c
6
, c
7
, c
11
y
15
+ 18y
14
+ ··· + 7y −9
c
10
y
15
+ 26y
14
+ ··· − 75329y −7569
c
12
y
15
+ 50y
14
+ ··· − 77237y −29241
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.218267 + 1.045140I
a = −0.313995 + 0.477157I
b = 0.671441 − 0.644775I
−6.94499 + 3.31343I −0.90808 − 3.10960I
u = −0.218267 − 1.045140I
a = −0.313995 − 0.477157I
b = 0.671441 + 0.644775I
−6.94499 − 3.31343I −0.90808 + 3.10960I
u = 0.097863 + 0.602652I
a = 0.107862 − 0.284601I
b = 0.722048 + 0.340057I
−1.30578 − 1.09993I −0.13148 + 4.47979I
u = 0.097863 − 0.602652I
a = 0.107862 + 0.284601I
b = 0.722048 − 0.340057I
−1.30578 + 1.09993I −0.13148 − 4.47979I
u = 0.585051 + 0.119200I
a = 1.13543 + 1.12186I
b = −0.446783 − 0.537153I
−3.77822 + 2.04196I 4.64535 − 1.53079I
u = 0.585051 − 0.119200I
a = 1.13543 − 1.12186I
b = −0.446783 + 0.537153I
−3.77822 − 2.04196I 4.64535 + 1.53079I
u = 0.18120 + 1.44491I
a = −0.956860 − 0.227359I
b = −1.212370 − 0.335064I
−7.73448 + 0.54824I −1.59243 − 0.48598I
u = 0.18120 − 1.44491I
a = −0.956860 + 0.227359I
b = −1.212370 + 0.335064I
−7.73448 − 0.54824I −1.59243 + 0.48598I
u = −0.321204
a = 1.40589
b = −0.228997
0.692564 15.0030
u = −0.71797 + 1.87258I
a = −0.566365 + 0.445862I
b = −2.21916 + 0.81697I
−16.8214 − 1.5623I −1.46630 + 0.66617I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.71797 − 1.87258I
a = −0.566365 − 0.445862I
b = −2.21916 − 0.81697I
−16.8214 + 1.5623I −1.46630 − 0.66617I
u = −0.39582 + 2.04334I
a = 1.144260 − 0.389222I
b = 2.23977 + 0.72183I
9.61396 − 8.62358I −1.03984 + 3.00810I
u = −0.39582 − 2.04334I
a = 1.144260 + 0.389222I
b = 2.23977 − 0.72183I
9.61396 + 8.62358I −1.03984 − 3.00810I
u = 0.12855 + 2.10555I
a = 1.246730 + 0.135922I
b = 2.35955 − 0.23642I
16.7551 + 3.3007I 0.99122 − 1.97685I
u = 0.12855 − 2.10555I
a = 1.246730 − 0.135922I
b = 2.35955 + 0.23642I
16.7551 − 3.3007I 0.99122 + 1.97685I
6
II. I
u
2
= hb + 1, 4a
3
− 2a
2
u + 12a
2
− 4au + 12a − 3u + 4, u
2
+ 2i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
4
=
1
2
a
6
=
a
−1
a
10
=
u
u
a
11
=
a
2
u + au + u
−au
a
2
=
a + 1
−1
a
1
=
a
−1
a
5
=
−1
0
a
8
=
−u
−u
a
7
=
−a
2
u − 2a
2
− 2au − 3a −
3
2
u − 2
−2a
2
u − 2a
2
− 4au − 4a − 3u − 3
a
12
=
−a − 2
−2a − 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4au + 4u
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
6
c
2
(u + 1)
6
c
3
, c
4
, c
8
c
9
(u
2
+ 2)
3
c
6
, c
7
(u
3
− u
2
+ 2u − 1)
2
c
10
(u
3
− u
2
+ 1)
2
c
11
(u
3
+ u
2
+ 2u + 1)
2
c
12
(u
3
+ u
2
− 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
8
c
9
(y + 2)
6
c
6
, c
7
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
10
, c
12
(y
3
− y
2
+ 2y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.414210I
a = −1.000000 − 0.533779I
b = −1.00000
−5.46628 3.01951 + 0.I
u = 1.414210I
a = −0.473303 + 0.620443I
b = −1.00000
−9.60386 − 2.82812I −3.50976 + 2.97945I
u = 1.414210I
a = −1.52670 + 0.62044I
b = −1.00000
−9.60386 + 2.82812I −3.50976 − 2.97945I
u = − 1.414210I
a = −1.000000 + 0.533779I
b = −1.00000
−5.46628 3.01951 + 0.I
u = − 1.414210I
a = −0.473303 − 0.620443I
b = −1.00000
−9.60386 + 2.82812I −3.50976 − 2.97945I
u = − 1.414210I
a = −1.52670 − 0.62044I
b = −1.00000
−9.60386 − 2.82812I −3.50976 + 2.97945I
10
III. I
v
1
= ha, b − 1, v
3
+ v
2
− 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
v
0
a
4
=
1
0
a
6
=
0
1
a
10
=
v
0
a
11
=
v
−v
a
2
=
1
−1
a
1
=
0
−1
a
5
=
1
0
a
8
=
v
0
a
7
=
v
2
+ v −1
−v
2
+ 1
a
12
=
v
2
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2v
2
+ 2v + 2
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
8
c
9
u
3
c
5
(u + 1)
3
c
6
, c
7
u
3
+ u
2
+ 2u + 1
c
10
, c
12
u
3
+ u
2
− 1
c
11
u
3
− u
2
+ 2u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
3
c
3
, c
4
, c
8
c
9
y
3
c
6
, c
7
, c
11
y
3
+ 3y
2
+ 2y −1
c
10
, c
12
y
3
− y
2
+ 2y −1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.877439 + 0.744862I
a = 0
b = 1.00000
−4.66906 − 2.82812I −0.18504 + 4.10401I
v = −0.877439 − 0.744862I
a = 0
b = 1.00000
−4.66906 + 2.82812I −0.18504 − 4.10401I
v = 0.754878
a = 0
b = 1.00000
−0.531480 2.37010
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
15
+ 32u
14
+ ··· + 6478u + 289)
c
2
((u − 1)
3
)(u + 1)
6
(u
15
+ 4u
14
+ ··· + 52u − 17)
c
3
, c
4
, c
8
c
9
u
3
(u
2
+ 2)
3
(u
15
− u
14
+ ··· + 24u
2
− 8)
c
5
((u − 1)
6
)(u + 1)
3
(u
15
+ 4u
14
+ ··· + 52u − 17)
c
6
, c
7
((u
3
− u
2
+ 2u − 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
15
− 2u
14
+ ··· + u − 3)
c
10
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
15
+ 2u
14
+ ··· + 77u − 87)
c
11
(u
3
− u
2
+ 2u − 1)(u
3
+ u
2
+ 2u + 1)
2
(u
15
− 2u
14
+ ··· + u − 3)
c
12
((u
3
+ u
2
− 1)
3
)(u
15
− 2u
14
+ ··· − 127u − 171)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
9
)(y
15
− 88y
14
+ ··· + 2.94571 × 10
7
y −83521)
c
2
, c
5
((y −1)
9
)(y
15
− 32y
14
+ ··· + 6478y −289)
c
3
, c
4
, c
8
c
9
y
3
(y + 2)
6
(y
15
+ 29y
14
+ ··· + 384y −64)
c
6
, c
7
, c
11
((y
3
+ 3y
2
+ 2y −1)
3
)(y
15
+ 18y
14
+ ··· + 7y −9)
c
10
((y
3
− y
2
+ 2y −1)
3
)(y
15
+ 26y
14
+ ··· − 75329y −7569)
c
12
((y
3
− y
2
+ 2y −1)
3
)(y
15
+ 50y
14
+ ··· − 77237y −29241)
16
