12n
0500
(K12n
0500
)
A knot diagram
1
Linearized knot diagam
3 6 12 8 2 10 11 3 6 7 4 9
Solving Sequence
6,9
10 7
3,11
2 1 5 8 4 12
c
9
c
6
c
10
c
2
c
1
c
5
c
8
c
4
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h5u
30
− 14u
29
+ ··· + 2b + 5, −u
30
+ 2u
29
+ ··· + 4a + 5, u
31
− 4u
30
+ ··· − 2u − 1i
I
u
2
= hb, a
3
+ a
2
u + a
2
− 2u − 3, u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 37 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
=
h5u
30
−14u
29
+· · ·+2b+5, −u
30
+2u
29
+· · ·+4a+5, u
31
−4u
30
+· · ·−2u−1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
7
=
u
−u
3
+ u
a
3
=
1
4
u
30
−
1
2
u
29
+ ··· −
5
2
u −
5
4
−
5
2
u
30
+ 7u
29
+ ··· − 6u −
5
2
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
1
4
u
30
−
1
2
u
29
+ ··· −
5
2
u −
5
4
−
13
4
u
30
+
33
4
u
29
+ ··· −
29
4
u − 3
a
1
=
9
4
u
30
−
15
4
u
29
+ ··· +
21
4
u +
5
2
−
5
4
u
30
+
13
4
u
29
+ ··· −
9
4
u − 1
a
5
=
−
1
4
u
29
+
3
4
u
28
+ ··· −
17
4
u +
3
4
1
4
u
30
−
1
2
u
29
+ ··· +
1
2
u +
1
4
a
8
=
−u
3
+ 2u
u
5
− 3u
3
+ u
a
4
=
−
1
4
u
30
+
1
4
u
29
+ ··· −
19
4
u +
1
2
1
4
u
30
−
1
2
u
29
+ ··· +
1
2
u +
1
4
a
12
=
7
2
u
30
− 7u
29
+ ··· +
15
2
u +
7
2
−
5
4
u
30
+
13
4
u
29
+ ··· −
9
4
u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −11u
30
+ 29u
29
+ ··· − 16u
2
−
15
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
31
+ 35u
30
+ ··· + 177u + 4
c
2
, c
5
u
31
+ 3u
30
+ ··· − 15u + 2
c
3
, c
11
u
31
− 3u
30
+ ··· − 9u + 1
c
4
u
31
− 3u
30
+ ··· − 2916u + 243
c
6
, c
7
, c
9
c
10
u
31
− 4u
30
+ ··· − 2u − 1
c
8
u
31
+ u
30
+ ··· − 32u + 64
c
12
u
31
+ 22u
29
+ ··· − 2095u + 2071
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
31
− 75y
30
+ ··· + 52609y −16
c
2
, c
5
y
31
− 35y
30
+ ··· + 177y −4
c
3
, c
11
y
31
+ 25y
30
+ ··· + 65y −1
c
4
y
31
+ 5y
30
+ ··· + 3359232y −59049
c
6
, c
7
, c
9
c
10
y
31
− 34y
30
+ ··· − 2y −1
c
8
y
31
+ 35y
30
+ ··· + 1024y −4096
c
12
y
31
+ 44y
30
+ ··· − 9225729y −4289041
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.602350 + 0.764489I
a = −0.96517 − 1.53953I
b = 0.35606 − 1.75288I
−5.83199 − 7.50635I 2.66894 + 5.59456I
u = −0.602350 − 0.764489I
a = −0.96517 + 1.53953I
b = 0.35606 + 1.75288I
−5.83199 + 7.50635I 2.66894 − 5.59456I
u = −0.543360 + 0.792514I
a = 0.91165 + 1.54215I
b = −0.17449 + 1.80984I
−10.03040 − 2.62343I −1.03684 + 2.68072I
u = −0.543360 − 0.792514I
a = 0.91165 − 1.54215I
b = −0.17449 − 1.80984I
−10.03040 + 2.62343I −1.03684 − 2.68072I
u = −0.473452 + 0.802011I
a = −0.85502 − 1.55001I
b = −0.03027 − 1.79370I
−6.21873 + 2.31735I 1.87364 − 0.58111I
u = −0.473452 − 0.802011I
a = −0.85502 + 1.55001I
b = −0.03027 + 1.79370I
−6.21873 − 2.31735I 1.87364 + 0.58111I
u = 0.751448 + 0.311977I
a = −0.131566 − 0.324454I
b = −0.652866 + 0.653689I
3.50508 + 0.49905I 6.93091 − 1.38994I
u = 0.751448 − 0.311977I
a = −0.131566 + 0.324454I
b = −0.652866 − 0.653689I
3.50508 − 0.49905I 6.93091 + 1.38994I
u = −1.34946
a = −0.997928
b = 1.24408
2.46733 3.43490
u = 1.366810 + 0.074394I
a = 0.234215 − 0.622051I
b = −0.045603 − 1.264260I
3.50947 + 1.97376I 4.13605 − 3.59471I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.366810 − 0.074394I
a = 0.234215 + 0.622051I
b = −0.045603 + 1.264260I
3.50947 − 1.97376I 4.13605 + 3.59471I
u = −1.375110 + 0.092603I
a = 0.939498 + 0.219836I
b = −1.276320 − 0.062520I
6.34649 − 4.23203I 7.34906 + 3.43500I
u = −1.375110 − 0.092603I
a = 0.939498 − 0.219836I
b = −1.276320 + 0.062520I
6.34649 + 4.23203I 7.34906 − 3.43500I
u = 0.527133
a = −0.257315
b = 0.358345
0.784642 13.1720
u = 1.47433 + 0.08812I
a = −0.497026 + 0.970443I
b = 0.186128 + 1.163550I
9.65588 + 4.45983I 7.97346 − 3.71466I
u = 1.47433 − 0.08812I
a = −0.497026 − 0.970443I
b = 0.186128 − 1.163550I
9.65588 − 4.45983I 7.97346 + 3.71466I
u = 0.146677 + 0.492597I
a = −0.018556 + 1.292810I
b = 0.836447 + 0.454443I
1.62298 + 2.35384I 1.82470 − 4.53214I
u = 0.146677 − 0.492597I
a = −0.018556 − 1.292810I
b = 0.836447 − 0.454443I
1.62298 − 2.35384I 1.82470 + 4.53214I
u = 1.49116 + 0.29858I
a = 0.877997 − 0.367679I
b = −0.35046 − 1.70481I
0.10746 + 1.69492I 0
u = 1.49116 − 0.29858I
a = 0.877997 + 0.367679I
b = −0.35046 + 1.70481I
0.10746 − 1.69492I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.54074 + 0.28653I
a = −0.983728 + 0.406676I
b = 0.52058 + 1.68308I
−3.24464 + 6.59865I 0
u = 1.54074 − 0.28653I
a = −0.983728 − 0.406676I
b = 0.52058 − 1.68308I
−3.24464 − 6.59865I 0
u = −0.369500 + 0.215346I
a = 0.63729 + 2.88917I
b = −0.010339 + 0.599954I
3.53058 − 3.26263I −0.32878 + 7.06957I
u = −0.369500 − 0.215346I
a = 0.63729 − 2.88917I
b = −0.010339 − 0.599954I
3.53058 + 3.26263I −0.32878 − 7.06957I
u = 1.57119 + 0.26571I
a = 1.058660 − 0.452456I
b = −0.63708 − 1.61140I
1.31297 + 11.33500I 0
u = 1.57119 − 0.26571I
a = 1.058660 + 0.452456I
b = −0.63708 + 1.61140I
1.31297 − 11.33500I 0
u = −1.59573
a = 0.257600
b = −0.497580
8.24985 15.5880
u = −1.63289 + 0.05265I
a = −0.234203 − 0.321554I
b = 0.531375 + 0.618506I
11.78470 − 1.72102I 0
u = −1.63289 − 0.05265I
a = −0.234203 + 0.321554I
b = 0.531375 − 0.618506I
11.78470 + 1.72102I 0
u = −0.136671 + 0.320613I
a = 0.02478 − 2.19557I
b = −0.305592 − 0.594543I
−1.239080 − 0.576153I −5.00436 + 2.78236I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.136671 − 0.320613I
a = 0.02478 + 2.19557I
b = −0.305592 + 0.594543I
−1.239080 + 0.576153I −5.00436 − 2.78236I
8
II. I
u
2
= hb, a
3
+ a
2
u + a
2
− 2u − 3, u
2
+ u − 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
u − 1
a
7
=
u
−u + 1
a
3
=
a
0
a
11
=
u
−u
a
2
=
a
au − a
a
1
=
a
2
u + a − u − 1
au − a
a
5
=
a
2
u
−2a
2
u + a
2
+ u
a
8
=
1
0
a
4
=
3a
2
u − a
2
− u
−2a
2
u + a
2
+ u
a
12
=
a
2
u − au + 2a − u − 1
au − a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −a
2
− 6au + a + u + 5
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
11
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
3
(u
3
+ u
2
+ 2u + 1)
2
c
4
u
6
− 2u
5
+ 5u
4
+ 2u
3
+ 3u
2
− 3u − 1
c
5
(u
3
− u
2
+ 1)
2
c
6
, c
7
(u
2
− u − 1)
3
c
8
u
6
c
9
, c
10
(u
2
+ u − 1)
3
c
12
u
6
− u
5
− u
4
+ 4u
3
+ 3u
2
− 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
4
y
6
+ 6y
5
+ 39y
4
+ 12y
3
+ 11y
2
− 15y + 1
c
6
, c
7
, c
9
c
10
(y
2
− 3y + 1)
3
c
8
y
6
c
12
y
6
− 3y
5
+ 15y
4
− 24y
3
+ 11y
2
− 6y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.22142
b = 0
−0.126494 0.818320
u = 0.618034
a = −1.41973 + 1.20521I
b = 0
4.01109 + 2.82812I 8.89985 + 0.15818I
u = 0.618034
a = −1.41973 − 1.20521I
b = 0
4.01109 − 2.82812I 8.89985 − 0.15818I
u = −1.61803
a = 0.542287 + 0.460350I
b = 0
11.90680 − 2.82812I 9.10673 + 4.43024I
u = −1.61803
a = 0.542287 − 0.460350I
b = 0
11.90680 + 2.82812I 9.10673 − 4.43024I
u = −1.61803
a = −0.466540
b = 0
7.76919 −1.83150
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
2
)(u
31
+ 35u
30
+ ··· + 177u + 4)
c
2
((u
3
+ u
2
− 1)
2
)(u
31
+ 3u
30
+ ··· − 15u + 2)
c
3
((u
3
+ u
2
+ 2u + 1)
2
)(u
31
− 3u
30
+ ··· − 9u + 1)
c
4
(u
6
− 2u
5
+ ··· − 3u − 1)(u
31
− 3u
30
+ ··· − 2916u + 243)
c
5
((u
3
− u
2
+ 1)
2
)(u
31
+ 3u
30
+ ··· − 15u + 2)
c
6
, c
7
((u
2
− u − 1)
3
)(u
31
− 4u
30
+ ··· − 2u − 1)
c
8
u
6
(u
31
+ u
30
+ ··· − 32u + 64)
c
9
, c
10
((u
2
+ u − 1)
3
)(u
31
− 4u
30
+ ··· − 2u − 1)
c
11
((u
3
− u
2
+ 2u − 1)
2
)(u
31
− 3u
30
+ ··· − 9u + 1)
c
12
(u
6
− u
5
− u
4
+ 4u
3
+ 3u
2
− 1)(u
31
+ 22u
29
+ ··· − 2095u + 2071)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
+ 3y
2
+ 2y −1)
2
)(y
31
− 75y
30
+ ··· + 52609y −16)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
2
)(y
31
− 35y
30
+ ··· + 177y −4)
c
3
, c
11
((y
3
+ 3y
2
+ 2y −1)
2
)(y
31
+ 25y
30
+ ··· + 65y −1)
c
4
(y
6
+ 6y
5
+ 39y
4
+ 12y
3
+ 11y
2
− 15y + 1)
· (y
31
+ 5y
30
+ ··· + 3359232y −59049)
c
6
, c
7
, c
9
c
10
((y
2
− 3y + 1)
3
)(y
31
− 34y
30
+ ··· − 2y −1)
c
8
y
6
(y
31
+ 35y
30
+ ··· + 1024y −4096)
c
12
(y
6
− 3y
5
+ 15y
4
− 24y
3
+ 11y
2
− 6y + 1)
· (y
31
+ 44y
30
+ ··· − 9225729y −4289041)
14
