12n
0333
(K12n
0333
)
A knot diagram
1
Linearized knot diagam
3 5 7 11 2 4 3 1 12 4 10 9
Solving Sequence
4,11 2,5
6 7 3 1 10 12 9 8
c
4
c
5
c
6
c
3
c
1
c
10
c
11
c
9
c
8
c
2
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
14
− u
13
− u
12
+ 3u
11
+ 3u
10
− 5u
9
− 2u
8
+ 7u
7
+ 3u
6
− 6u
5
− 2u
4
+ 4u
3
+ b − u + 1,
− u
15
+ u
14
+ u
13
− 2u
12
− 3u
11
+ 4u
10
+ 4u
9
− 4u
8
− 3u
7
+ 4u
6
+ 6u
5
− 2u
4
+ 2a − 1,
u
16
− 3u
15
+ 3u
14
+ 2u
13
− 3u
12
− 6u
11
+ 12u
10
− 9u
8
− 2u
7
+ 12u
6
− 2u
5
− 8u
4
+ 4u
3
+ 2u
2
− 3u + 2i
I
u
2
= hb + 1, 2u
5
a + 4u
6
+ 2u
4
a + 7u
5
+ 3u
4
− 2u
2
a − 2u
3
+ a
2
+ 4au + 7u
2
+ 3a + 14u + 5,
u
7
+ u
6
− u
4
+ 2u
3
+ 2u
2
− 1i
I
u
3
= hu
7
− u
5
+ 2u
3
+ b − u + 1, u
7
− u
6
+ u
5
+ u
4
+ 2u
3
− 3u
2
+ a + 2u + 2, u
8
− u
6
+ 3u
4
− 2u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 38 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
14
−u
13
+· · ·+b+1, −u
15
+u
14
+· · ·+2a−1, u
16
−3u
15
+· · ·−3u+2i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
2
=
1
2
u
15
−
1
2
u
14
+ ··· + u
4
+
1
2
−u
14
+ u
13
+ ··· + u − 1
a
5
=
1
u
2
a
6
=
−
1
2
u
15
+
3
2
u
14
+ ··· − u +
3
2
u
15
− 2u
14
+ 3u
12
+ u
11
− 7u
10
+ u
9
+ 6u
8
− 7u
6
+ 4u
4
− u
3
+ u − 1
a
7
=
1
2
u
15
−
1
2
u
14
+ ··· + u
4
+
1
2
u
15
− 2u
14
+ 3u
12
+ u
11
− 7u
10
+ u
9
+ 6u
8
− 7u
6
+ 4u
4
− u
3
+ u − 1
a
3
=
−
1
2
u
15
+
3
2
u
14
+ ··· − u +
3
2
u
14
− u
13
+ ··· + u
3
+ 1
a
1
=
−u
7
− 2u
3
−u
7
+ u
5
− 2u
3
+ u
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
a
8
=
u
9
+ 3u
5
+ u
u
9
− u
7
+ 3u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= 2u
15
−4u
14
+10u
12
−18u
10
+10u
9
+24u
8
−8u
7
−22u
6
+12u
5
+20u
4
−8u
3
−8u
2
+6u−2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
+ 2u
15
+ ··· + 7u + 1
c
2
, c
3
, c
5
c
6
, c
7
u
16
+ u
14
+ ··· + u + 1
c
4
, c
10
u
16
− 3u
15
+ ··· − 3u + 2
c
8
, c
9
, c
11
c
12
u
16
+ 3u
15
+ ··· + u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
+ 34y
15
+ ··· + 3y + 1
c
2
, c
3
, c
5
c
6
, c
7
y
16
+ 2y
15
+ ··· + 7y + 1
c
4
, c
10
y
16
− 3y
15
+ ··· − y + 4
c
8
, c
9
, c
11
c
12
y
16
+ 21y
15
+ ··· − 33y + 16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.928405 + 0.260033I
a = 0.35481 − 2.30363I
b = 0.773479 − 1.013430I
−1.18138 − 3.90571I −4.06432 + 8.02120I
u = 0.928405 − 0.260033I
a = 0.35481 + 2.30363I
b = 0.773479 + 1.013430I
−1.18138 + 3.90571I −4.06432 − 8.02120I
u = −0.650391 + 0.833172I
a = −0.136050 + 0.366350I
b = 1.39607 − 0.78392I
5.41244 − 2.99211I 2.13069 + 2.15940I
u = −0.650391 − 0.833172I
a = −0.136050 − 0.366350I
b = 1.39607 + 0.78392I
5.41244 + 2.99211I 2.13069 − 2.15940I
u = −0.816725 + 0.248973I
a = 0.91106 − 1.21568I
b = −0.198116 − 0.632117I
−1.43484 + 0.76137I −4.76909 − 0.41867I
u = −0.816725 − 0.248973I
a = 0.91106 + 1.21568I
b = −0.198116 + 0.632117I
−1.43484 − 0.76137I −4.76909 + 0.41867I
u = −0.970812 + 0.659855I
a = −0.66898 + 2.08400I
b = 1.47947 + 0.97775I
4.31472 + 8.49137I −0.41000 − 7.95274I
u = −0.970812 − 0.659855I
a = −0.66898 − 2.08400I
b = 1.47947 − 0.97775I
4.31472 − 8.49137I −0.41000 + 7.95274I
u = 0.905631 + 0.833459I
a = 0.431652 + 0.930570I
b = −1.368560 + 0.130086I
4.87488 − 3.10725I 2.83461 + 2.27885I
u = 0.905631 − 0.833459I
a = 0.431652 − 0.930570I
b = −1.368560 − 0.130086I
4.87488 + 3.10725I 2.83461 − 2.27885I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.915125 + 0.963980I
a = −0.269698 − 0.434629I
b = 1.81390 + 0.76762I
15.6936 + 4.6465I 1.09085 − 1.72729I
u = 0.915125 − 0.963980I
a = −0.269698 + 0.434629I
b = 1.81390 − 0.76762I
15.6936 − 4.6465I 1.09085 + 1.72729I
u = 0.993185 + 0.916284I
a = −0.97982 − 1.63330I
b = 1.84398 − 0.79885I
15.4310 − 11.5459I 0.63569 + 6.14734I
u = 0.993185 − 0.916284I
a = −0.97982 + 1.63330I
b = 1.84398 + 0.79885I
15.4310 + 11.5459I 0.63569 − 6.14734I
u = 0.195584 + 0.595042I
a = 0.107025 − 0.207653I
b = 0.759770 + 0.417859I
1.30282 + 0.89270I 4.55158 − 1.98152I
u = 0.195584 − 0.595042I
a = 0.107025 + 0.207653I
b = 0.759770 − 0.417859I
1.30282 − 0.89270I 4.55158 + 1.98152I
6
II. I
u
2
= hb + 1, 2u
5
a + 4u
6
+ · · · + 3a + 5, u
7
+ u
6
− u
4
+ 2u
3
+ 2u
2
− 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
2
=
a
−1
a
5
=
1
u
2
a
6
=
−2u
5
− 2u
4
+ u
2
a + u
2
− a − 4u − 2
−u
2
a + u
2
− 1
a
7
=
−2u
5
− 2u
4
+ 2u
2
− a − 4u − 3
−u
2
a + u
2
− 1
a
3
=
−u
2
a + a − 1
−u
4
a − u
2
− 1
a
1
=
u
6
− u
4
+ 2u
2
− 1
u
6
+ u
5
− u
4
+ 2u
2
+ u − 1
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
a
8
=
u
4
− u
2
+ 1
u
6
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
5
− 4u
4
+ 4u
2
− 8u − 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
14
+ 3u
13
+ ··· + 27u + 4
c
2
, c
3
, c
5
c
6
, c
7
u
14
+ u
13
+ ··· + 5u + 2
c
4
, c
10
(u
7
+ u
6
− u
4
+ 2u
3
+ 2u
2
− 1)
2
c
8
, c
9
, c
11
c
12
(u
7
+ u
6
+ 6u
5
+ 5u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
+ 15y
13
+ ··· + 95y + 16
c
2
, c
3
, c
5
c
6
, c
7
y
14
+ 3y
13
+ ··· + 27y + 4
c
4
, c
10
(y
7
− y
6
+ 6y
5
− 5y
4
+ 10y
3
− 6y
2
+ 4y −1)
2
c
8
, c
9
, c
11
c
12
(y
7
+ 11y
6
+ 46y
5
+ 91y
4
+ 86y
3
+ 34y
2
+ 4y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.850452 + 0.793787I
a = 0.529865 + 1.201600I
b = −1.00000
4.70993 − 2.92126I 1.79653 + 2.94858I
u = 0.850452 + 0.793787I
a = 0.368398 + 0.272687I
b = −1.00000
4.70993 − 2.92126I 1.79653 + 2.94858I
u = 0.850452 − 0.793787I
a = 0.529865 − 1.201600I
b = −1.00000
4.70993 + 2.92126I 1.79653 − 2.94858I
u = 0.850452 − 0.793787I
a = 0.368398 − 0.272687I
b = −1.00000
4.70993 + 2.92126I 1.79653 − 2.94858I
u = −0.676751 + 0.491075I
a = −1.041030 − 0.810129I
b = −1.00000
−2.02205 + 1.83261I 0.22558 − 5.43914I
u = −0.676751 + 0.491075I
a = 1.15382 − 1.90944I
b = −1.00000
−2.02205 + 1.83261I 0.22558 − 5.43914I
u = −0.676751 − 0.491075I
a = −1.041030 + 0.810129I
b = −1.00000
−2.02205 − 1.83261I 0.22558 + 5.43914I
u = −0.676751 − 0.491075I
a = 1.15382 + 1.90944I
b = −1.00000
−2.02205 − 1.83261I 0.22558 + 5.43914I
u = −0.962510 + 0.950397I
a = 0.498708 − 1.218380I
b = −1.00000
16.6015 + 3.4867I 1.97231 − 2.18600I
u = −0.962510 + 0.950397I
a = 0.487449 + 0.125376I
b = −1.00000
16.6015 + 3.4867I 1.97231 − 2.18600I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.962510 − 0.950397I
a = 0.498708 + 1.218380I
b = −1.00000
16.6015 − 3.4867I 1.97231 + 2.18600I
u = −0.962510 − 0.950397I
a = 0.487449 − 0.125376I
b = −1.00000
16.6015 − 3.4867I 1.97231 + 2.18600I
u = 0.577619
a = −2.49721 + 3.11982I
b = −1.00000
−4.03510 −9.98880
u = 0.577619
a = −2.49721 − 3.11982I
b = −1.00000
−4.03510 −9.98880
11
III. I
u
3
= hu
7
− u
5
+ 2u
3
+ b − u + 1, u
7
− u
6
+ u
5
+ u
4
+ 2u
3
− 3u
2
+ a +
2u + 2, u
8
− u
6
+ 3u
4
− 2u
2
+ 1i
(i) Arc colorings
a
4
=
1
0
a
11
=
0
u
a
2
=
−u
7
+ u
6
− u
5
− u
4
− 2u
3
+ 3u
2
− 2u − 2
−u
7
+ u
5
− 2u
3
+ u − 1
a
5
=
1
u
2
a
6
=
u
6
+ u
5
− u
4
+ 3u
2
+ 2u − 2
u
7
+ 2u
3
a
7
=
u
7
+ u
6
+ u
5
− u
4
+ 2u
3
+ 3u
2
+ 2u − 2
u
7
+ 2u
3
a
3
=
u
6
− u
5
− u
4
+ 3u
2
− 2u − 2
−1
a
1
=
−u
7
− 2u
3
−u
7
+ u
5
− 2u
3
+ u
a
10
=
u
u
a
12
=
−u
3
−u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
a
8
=
u
7
+ 2u
3
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
6
− 4u
4
+ 12u
2
− 12
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
8
c
2
, c
3
, c
5
c
6
, c
7
(u
2
+ 1)
4
c
4
, c
10
u
8
− u
6
+ 3u
4
− 2u
2
+ 1
c
8
, c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
11
, c
12
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y −1)
8
c
2
, c
3
, c
5
c
6
, c
7
(y + 1)
8
c
4
, c
10
(y
4
− y
3
+ 3y
2
− 2y + 1)
2
c
8
, c
9
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.720342 + 0.351808I
a = −2.18387 − 0.72950I
b = −0.493156 − 0.395123I
−3.50087 − 1.41510I −7.82674 + 4.90874I
u = 0.720342 − 0.351808I
a = −2.18387 + 0.72950I
b = −0.493156 + 0.395123I
−3.50087 + 1.41510I −7.82674 − 4.90874I
u = −0.720342 + 0.351808I
a = 0.27050 − 3.18387I
b = −1.50684 − 0.39512I
−3.50087 + 1.41510I −7.82674 − 4.90874I
u = −0.720342 − 0.351808I
a = 0.27050 + 3.18387I
b = −1.50684 + 0.39512I
−3.50087 − 1.41510I −7.82674 + 4.90874I
u = 0.911292 + 0.851808I
a = 0.59788 + 1.68452I
b = −2.55249 + 0.10488I
3.50087 − 3.16396I −4.17326 + 2.56480I
u = 0.911292 − 0.851808I
a = 0.59788 − 1.68452I
b = −2.55249 − 0.10488I
3.50087 + 3.16396I −4.17326 − 2.56480I
u = −0.911292 + 0.851808I
a = −0.684515 + 0.402116I
b = 0.552492 + 0.104877I
3.50087 + 3.16396I −4.17326 − 2.56480I
u = −0.911292 − 0.851808I
a = −0.684515 − 0.402116I
b = 0.552492 − 0.104877I
3.50087 − 3.16396I −4.17326 + 2.56480I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
14
+ 3u
13
+ ··· + 27u + 4)(u
16
+ 2u
15
+ ··· + 7u + 1)
c
2
, c
3
, c
5
c
6
, c
7
((u
2
+ 1)
4
)(u
14
+ u
13
+ ··· + 5u + 2)(u
16
+ u
14
+ ··· + u + 1)
c
4
, c
10
(u
7
+ u
6
− u
4
+ 2u
3
+ 2u
2
− 1)
2
(u
8
− u
6
+ 3u
4
− 2u
2
+ 1)
· (u
16
− 3u
15
+ ··· − 3u + 2)
c
8
, c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
· ((u
7
+ u
6
+ ··· + 4u + 1)
2
)(u
16
+ 3u
15
+ ··· + u + 4)
c
11
, c
12
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
· ((u
7
+ u
6
+ ··· + 4u + 1)
2
)(u
16
+ 3u
15
+ ··· + u + 4)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
14
+ 15y
13
+ ··· + 95y + 16)(y
16
+ 34y
15
+ ··· + 3y + 1)
c
2
, c
3
, c
5
c
6
, c
7
((y + 1)
8
)(y
14
+ 3y
13
+ ··· + 27y + 4)(y
16
+ 2y
15
+ ··· + 7y + 1)
c
4
, c
10
(y
4
− y
3
+ 3y
2
− 2y + 1)
2
· ((y
7
− y
6
+ ··· + 4y −1)
2
)(y
16
− 3y
15
+ ··· − y + 4)
c
8
, c
9
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
· (y
7
+ 11y
6
+ 46y
5
+ 91y
4
+ 86y
3
+ 34y
2
+ 4y −1)
2
· (y
16
+ 21y
15
+ ··· − 33y + 16)
17
