12n
0502
(K12n
0502
)
A knot diagram
1
Linearized knot diagam
3 6 12 9 2 10 11 3 6 7 4 8
Solving Sequence
6,9
10 7
3,11
2 1 5 4 8 12
c
9
c
6
c
10
c
2
c
1
c
5
c
4
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
9
− 2u
8
+ 5u
7
+ 11u
6
− 2u
5
− 14u
4
− 15u
3
− 6u
2
+ 2b − u − 1,
− u
9
− 2u
8
+ 7u
7
+ 15u
6
− 14u
5
− 38u
4
+ u
3
+ 32u
2
+ 4a + 11u − 3,
u
10
+ 4u
9
− 2u
8
− 24u
7
− 16u
6
+ 35u
5
+ 44u
4
+ 11u
3
− u
2
+ 3u + 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 16 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−u
9
−2u
8
+· · ·+2b−1, −u
9
−2u
8
+· · ·+4a−3, u
10
+4u
9
+· · ·+3u+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
9
+
1
2
u
8
+ ··· −
11
4
u +
3
4
1
2
u
9
+ u
8
+ ··· +
1
2
u +
1
2
a
11
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
1
4
u
9
+
1
2
u
8
+ ··· −
11
4
u +
3
4
−
1
4
u
9
−
1
4
u
8
+ ··· +
7
2
u
2
−
3
4
u
a
1
=
−
7
4
u
9
−
13
4
u
8
+ ··· −
7
4
u +
1
2
−
25
4
u
9
−
45
4
u
8
+ ··· −
35
4
u − 3
a
5
=
−
1
4
u
8
−
3
4
u
7
+ ··· − 3u −
3
4
−
1
4
u
9
−
1
2
u
8
+ ··· +
5
4
u −
1
4
a
4
=
−
1
4
u
9
−
3
4
u
8
+ ··· −
7
4
u − 1
−
1
4
u
9
−
1
2
u
8
+ ··· +
5
4
u −
1
4
a
8
=
u
3
− 2u
u
5
− 3u
3
+ u
a
12
=
−
1
2
u
8
− u
7
+ ··· +
1
2
u +
3
2
3
4
u
9
+
7
4
u
8
+ ··· +
5
2
u
2
−
3
4
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
9
− 5u
8
+
61
2
u
6
+ 29u
5
−
91
2
u
4
− 68u
3
− 15u
2
+
15
2
u −
29
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
10
+ 29u
9
+ ··· − 19u + 4
c
2
, c
5
u
10
+ 3u
9
− 10u
8
− 34u
7
− 17u
6
+ 7u
5
+ 59u
4
− 55u
3
− 11u
2
− 5u − 2
c
3
, c
11
u
10
− 3u
9
+ 8u
8
− 13u
7
+ 18u
6
− 20u
5
+ 16u
4
− 11u
3
+ 3u
2
− 2u − 1
c
4
u
10
− 25u
9
+ ··· − 1689u − 389
c
6
, c
7
, c
9
c
10
u
10
+ 4u
9
− 2u
8
− 24u
7
− 16u
6
+ 35u
5
+ 44u
4
+ 11u
3
− u
2
+ 3u + 1
c
8
u
10
+ u
9
+ ··· − 160u − 64
c
12
u
10
− 2u
9
− 11u
8
+ 16u
7
+ 23u
6
+ 10u
5
− u
4
+ 5u
3
+ 6u
2
+ 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
10
− 301y
9
+ ··· − 5681y + 16
c
2
, c
5
y
10
− 29y
9
+ ··· + 19y + 4
c
3
, c
11
y
10
+ 7y
9
+ 22y
8
+ 31y
7
− 76y
5
− 144y
4
− 141y
3
− 67y
2
− 10y + 1
c
4
y
10
− 169y
9
+ ··· − 1357405y + 151321
c
6
, c
7
, c
9
c
10
y
10
− 20y
9
+ ··· − 11y + 1
c
8
y
10
− 35y
9
+ ··· − 5120y + 4096
c
12
y
10
− 26y
9
+ ··· − 4y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.770282 + 0.350441I
a = −0.175699 + 0.338848I
b = 0.722401 + 0.709455I
−0.295773 + 0.495817I −13.91306 − 1.36095I
u = −0.770282 − 0.350441I
a = −0.175699 − 0.338848I
b = 0.722401 − 0.709455I
−0.295773 − 0.495817I −13.91306 + 1.36095I
u = 0.238689 + 0.328179I
a = 0.29003 − 2.34120I
b = −0.187083 + 0.681344I
2.64731 + 2.28565I −6.52151 − 0.97940I
u = 0.238689 − 0.328179I
a = 0.29003 + 2.34120I
b = −0.187083 − 0.681344I
2.64731 − 2.28565I −6.52151 + 0.97940I
u = −0.293390
a = 0.935254
b = 0.468829
−0.595897 −16.6550
u = 1.78423 + 0.22622I
a = −0.335255 + 0.782997I
b = −1.44386 + 1.68806I
−9.27245 − 2.54510I −15.8897 + 2.0711I
u = 1.78423 − 0.22622I
a = −0.335255 − 0.782997I
b = −1.44386 − 1.68806I
−9.27245 + 2.54510I −15.8897 − 2.0711I
u = −2.03029 + 0.17685I
a = 1.52970 + 0.03504I
b = 3.31663 − 1.27082I
15.1518 + 6.6636I −15.1690 − 2.5369I
u = −2.03029 − 0.17685I
a = 1.52970 − 0.03504I
b = 3.31663 + 1.27082I
15.1518 − 6.6636I −15.1690 + 2.5369I
u = −2.15130
a = −1.55281
b = −4.28499
10.4531 −17.3580
5
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
4
=
−a
2
u − a
2
+ u
−2a
2
u − a
2
+ u
a
8
=
1
0
a
12
=
a
2
u − au − u + 1
−au − a
(ii) Obstruction class = 1
(iii) Cusp Shapes = a
2
− 2au + a + u − 17
6
(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
7
(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
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −1.22142
b = 0
−2.10041 −18.8570
u = −0.618034
a = 1.41973 + 1.20521I
b = 0
2.03717 − 2.82812I −13.8803 + 6.1171I
u = −0.618034
a = 1.41973 − 1.20521I
b = 0
2.03717 + 2.82812I −13.8803 − 6.1171I
u = 1.61803
a = −0.542287 + 0.460350I
b = 0
−5.85852 + 2.82812I −14.0872 − 1.5287I
u = 1.61803
a = −0.542287 − 0.460350I
b = 0
−5.85852 − 2.82812I −14.0872 + 1.5287I
u = 1.61803
a = 0.466540
b = 0
−9.99610 −16.2080
9
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
− u
2
+ 2u − 1)
2
)(u
10
+ 29u
9
+ ··· − 19u + 4)
c
2
(u
3
+ u
2
− 1)
2
· (u
10
+ 3u
9
− 10u
8
− 34u
7
− 17u
6
+ 7u
5
+ 59u
4
− 55u
3
− 11u
2
− 5u − 2)
c
3
(u
3
+ u
2
+ 2u + 1)
2
· (u
10
− 3u
9
+ 8u
8
− 13u
7
+ 18u
6
− 20u
5
+ 16u
4
− 11u
3
+ 3u
2
− 2u − 1)
c
4
(u
6
+ 2u
5
+ ··· + 3u − 1)(u
10
− 25u
9
+ ··· − 1689u − 389)
c
5
(u
3
− u
2
+ 1)
2
· (u
10
+ 3u
9
− 10u
8
− 34u
7
− 17u
6
+ 7u
5
+ 59u
4
− 55u
3
− 11u
2
− 5u − 2)
c
6
, c
7
(u
2
+ u − 1)
3
· (u
10
+ 4u
9
− 2u
8
− 24u
7
− 16u
6
+ 35u
5
+ 44u
4
+ 11u
3
− u
2
+ 3u + 1)
c
8
u
6
(u
10
+ u
9
+ ··· − 160u − 64)
c
9
, c
10
(u
2
− u − 1)
3
· (u
10
+ 4u
9
− 2u
8
− 24u
7
− 16u
6
+ 35u
5
+ 44u
4
+ 11u
3
− u
2
+ 3u + 1)
c
11
(u
3
− u
2
+ 2u − 1)
2
· (u
10
− 3u
9
+ 8u
8
− 13u
7
+ 18u
6
− 20u
5
+ 16u
4
− 11u
3
+ 3u
2
− 2u − 1)
c
12
(u
6
+ u
5
− u
4
− 4u
3
+ 3u
2
− 1)
· (u
10
− 2u
9
− 11u
8
+ 16u
7
+ 23u
6
+ 10u
5
− u
4
+ 5u
3
+ 6u
2
+ 4u + 1)
10
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
3
+ 3y
2
+ 2y −1)
2
)(y
10
− 301y
9
+ ··· − 5681y + 16)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
2
)(y
10
− 29y
9
+ ··· + 19y + 4)
c
3
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
· (y
10
+ 7y
9
+ 22y
8
+ 31y
7
− 76y
5
− 144y
4
− 141y
3
− 67y
2
− 10y + 1)
c
4
(y
6
+ 6y
5
+ 39y
4
+ 12y
3
+ 11y
2
− 15y + 1)
· (y
10
− 169y
9
+ ··· − 1357405y + 151321)
c
6
, c
7
, c
9
c
10
((y
2
− 3y + 1)
3
)(y
10
− 20y
9
+ ··· − 11y + 1)
c
8
y
6
(y
10
− 35y
9
+ ··· − 5120y + 4096)
c
12
(y
6
− 3y
5
+ ··· − 6y + 1)(y
10
− 26y
9
+ ··· − 4y + 1)
11
