12n
0250
(K12n
0250
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 12 11 10 4 5 7 6 9
Solving Sequence
6,11
7 12
2,5
3 1 4 10 8 9
c
6
c
11
c
5
c
2
c
1
c
4
c
10
c
7
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
21
− 2u
20
+ ··· + b − 1, u
24
− 2u
23
+ ··· + a + 3, u
25
− 2u
24
+ ··· + 5u − 1i
I
u
2
= hu
2
+ b + u + 1, −u
4
− u
3
− 3u
2
+ a − 2u − 1, u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1i
* 2 irreducible components of dim
C
= 0, with total 30 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
21
−2u
20
+· · ·+b −1, u
24
−2u
23
+· · ·+a +3, u
25
−2u
24
+· · ·+5u −1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
12
=
u
u
a
2
=
−u
24
+ 2u
23
+ ··· + 2u − 3
−u
21
+ 2u
20
+ ··· − u + 1
a
5
=
u
2
+ 1
u
2
a
3
=
−u
24
+ 2u
23
+ ··· + 2u − 4
−u
22
− 11u
20
+ ··· − u + 1
a
1
=
u
13
+ 8u
11
+ 23u
9
+ 30u
7
+ 20u
5
+ 6u
3
+ u
u
13
+ 7u
11
+ 15u
9
+ 8u
7
− 4u
5
− 3u
3
+ u
a
4
=
u
24
− 2u
23
+ ··· + u + 3
−u
22
+ 2u
21
+ ··· + 2u − 1
a
10
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
9
=
−u
7
− 4u
5
− 4u
3
− 2u
−u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
24
+ 2u
23
− 18u
22
+ 32u
21
− 142u
20
+ 222u
19
− 644u
18
+
875u
17
− 1848u
16
+ 2155u
15
− 3472u
14
+ 3429u
13
− 4258u
12
+ 3506u
11
− 3270u
10
+
2195u
9
− 1387u
8
+ 732u
7
− 164u
6
+ 62u
5
+ 90u
4
− 28u
3
+ 16u
2
− 3u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
25
+ 4u
24
+ ··· − 5u + 1
c
2
, c
4
u
25
− 6u
24
+ ··· − 5u + 1
c
3
, c
8
u
25
− u
24
+ ··· + 32u + 32
c
5
, c
6
, c
7
c
10
, c
11
u
25
+ 2u
24
+ ··· + 5u + 1
c
9
u
25
+ 2u
24
+ ··· + 3u + 1
c
12
u
25
− 8u
24
+ ··· + 14437u − 1751
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
25
+ 40y
24
+ ··· − 5y −1
c
2
, c
4
y
25
− 4y
24
+ ··· − 5y −1
c
3
, c
8
y
25
+ 33y
24
+ ··· − 3584y −1024
c
5
, c
6
, c
7
c
10
, c
11
y
25
+ 32y
24
+ ··· + 25y −1
c
9
y
25
− 32y
24
+ ··· + 25y −1
c
12
y
25
− 44y
24
+ ··· + 245499141y −3066001
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.459758 + 0.889622I
a = 0.076243 − 0.510379I
b = 0.96095 + 1.37863I
7.04309 + 7.69753I 0.69223 − 5.87020I
u = 0.459758 − 0.889622I
a = 0.076243 + 0.510379I
b = 0.96095 − 1.37863I
7.04309 − 7.69753I 0.69223 + 5.87020I
u = −0.233597 + 0.958426I
a = −0.400131 + 0.212647I
b = 0.719444 − 0.275653I
−2.25524 − 2.86318I 0.67832 + 5.18628I
u = −0.233597 − 0.958426I
a = −0.400131 − 0.212647I
b = 0.719444 + 0.275653I
−2.25524 + 2.86318I 0.67832 − 5.18628I
u = 0.480913 + 0.815098I
a = −0.540781 − 0.132936I
b = −1.59979 − 0.36621I
7.50954 + 0.03310I 1.50006 − 1.57887I
u = 0.480913 − 0.815098I
a = −0.540781 + 0.132936I
b = −1.59979 + 0.36621I
7.50954 − 0.03310I 1.50006 + 1.57887I
u = −0.277299 + 0.715967I
a = −0.206006 + 0.120919I
b = −0.106133 + 1.044080I
−0.57147 − 2.01138I 1.23672 + 5.35998I
u = −0.277299 − 0.715967I
a = −0.206006 − 0.120919I
b = −0.106133 − 1.044080I
−0.57147 + 2.01138I 1.23672 − 5.35998I
u = 0.085005 + 0.759627I
a = 1.31404 + 0.60640I
b = 0.296392 − 1.022520I
−3.36592 + 0.96368I −4.01560 + 0.59686I
u = 0.085005 − 0.759627I
a = 1.31404 − 0.60640I
b = 0.296392 + 1.022520I
−3.36592 − 0.96368I −4.01560 − 0.59686I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.683438 + 0.037064I
a = 0.47321 − 1.98307I
b = −0.484746 + 0.125338I
9.85760 + 3.86250I 4.92075 − 2.49747I
u = 0.683438 − 0.037064I
a = 0.47321 + 1.98307I
b = −0.484746 − 0.125338I
9.85760 − 3.86250I 4.92075 + 2.49747I
u = −0.454087 + 0.131996I
a = −0.930347 + 0.905066I
b = −0.170032 − 0.148405I
1.125730 − 0.539778I 7.59664 + 2.57696I
u = −0.454087 − 0.131996I
a = −0.930347 − 0.905066I
b = −0.170032 + 0.148405I
1.125730 + 0.539778I 7.59664 − 2.57696I
u = −0.05917 + 1.63615I
a = 0.26489 + 2.09173I
b = 0.28511 + 2.85242I
−8.78516 − 3.17139I −0.87361 + 3.19607I
u = −0.05917 − 1.63615I
a = 0.26489 − 2.09173I
b = 0.28511 − 2.85242I
−8.78516 + 3.17139I −0.87361 − 3.19607I
u = 0.12897 + 1.64332I
a = −1.43354 − 1.14853I
b = −1.74726 − 1.22919I
−0.90994 + 2.33809I −0.195140 − 0.570968I
u = 0.12897 − 1.64332I
a = −1.43354 + 1.14853I
b = −1.74726 + 1.22919I
−0.90994 − 2.33809I −0.195140 + 0.570968I
u = 0.01776 + 1.65047I
a = −0.47939 − 1.88937I
b = −1.43279 − 2.83187I
−11.86300 + 1.31883I −3.44425 + 0.I
u = 0.01776 − 1.65047I
a = −0.47939 + 1.88937I
b = −1.43279 + 2.83187I
−11.86300 − 1.31883I −3.44425 + 0.I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.12689 + 1.67737I
a = 1.19016 + 2.59619I
b = 2.22016 + 3.74930I
−1.85042 + 9.98114I −1.18698 − 4.78469I
u = 0.12689 − 1.67737I
a = 1.19016 − 2.59619I
b = 2.22016 − 3.74930I
−1.85042 − 9.98114I −1.18698 + 4.78469I
u = −0.05535 + 1.70735I
a = 1.56322 − 0.66579I
b = 2.67420 − 0.95840I
−11.72920 − 3.98033I 0. + 4.58718I
u = −0.05535 − 1.70735I
a = 1.56322 + 0.66579I
b = 2.67420 + 0.95840I
−11.72920 + 3.98033I 0. − 4.58718I
u = 0.193537
a = −2.78315
b = 0.768975
−1.31003 −10.0440
7
II.
I
u
2
= hu
2
+ b + u + 1, −u
4
− u
3
− 3u
2
+ a − 2u − 1, u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
12
=
u
u
a
2
=
u
4
+ u
3
+ 3u
2
+ 2u + 1
−u
2
− u − 1
a
5
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
3
+ 4u
2
+ 2u + 2
−u − 1
a
1
=
−u
2
− 1
−u
2
a
4
=
u
4
+ u
3
+ 4u
2
+ 2u + 2
−u − 1
a
10
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
9
=
u
2
+ 1
−u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
4
+ 5u
3
+ 20u
2
+ 14u + 9
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
8
u
5
c
4
(u + 1)
5
c
5
, c
6
, c
7
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
9
, c
12
u
5
− u
4
+ u
2
+ u − 1
c
10
, c
11
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
8
y
5
c
5
, c
6
, c
7
c
10
, c
11
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
c
9
, c
12
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.233677 + 0.885557I
a = −0.758138 + 0.584034I
b = −0.036717 − 0.471689I
−3.46474 − 2.21397I −4.37343 + 4.39306I
u = −0.233677 − 0.885557I
a = −0.758138 − 0.584034I
b = −0.036717 + 0.471689I
−3.46474 + 2.21397I −4.37343 − 4.39306I
u = −0.416284
a = 0.645200
b = −0.757008
−0.762751 6.42730
u = −0.05818 + 1.69128I
a = 0.935538 − 0.903908I
b = 1.91522 − 1.49448I
−12.60320 − 3.33174I −5.84024 + 1.26157I
u = −0.05818 − 1.69128I
a = 0.935538 + 0.903908I
b = 1.91522 + 1.49448I
−12.60320 + 3.33174I −5.84024 − 1.26157I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
25
+ 4u
24
+ ··· − 5u + 1)
c
2
((u − 1)
5
)(u
25
− 6u
24
+ ··· − 5u + 1)
c
3
, c
8
u
5
(u
25
− u
24
+ ··· + 32u + 32)
c
4
((u + 1)
5
)(u
25
− 6u
24
+ ··· − 5u + 1)
c
5
, c
6
, c
7
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
25
+ 2u
24
+ ··· + 5u + 1)
c
9
(u
5
− u
4
+ u
2
+ u − 1)(u
25
+ 2u
24
+ ··· + 3u + 1)
c
10
, c
11
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
25
+ 2u
24
+ ··· + 5u + 1)
c
12
(u
5
− u
4
+ u
2
+ u − 1)(u
25
− 8u
24
+ ··· + 14437u − 1751)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
25
+ 40y
24
+ ··· − 5y −1)
c
2
, c
4
((y −1)
5
)(y
25
− 4y
24
+ ··· − 5y −1)
c
3
, c
8
y
5
(y
25
+ 33y
24
+ ··· − 3584y −1024)
c
5
, c
6
, c
7
c
10
, c
11
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
25
+ 32y
24
+ ··· + 25y −1)
c
9
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
25
− 32y
24
+ ··· + 25y −1)
c
12
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)
· (y
25
− 44y
24
+ ··· + 245499141y −3066001)
13
