12a
0268
(K12a
0268
)
A knot diagram
1
Linearized knot diagam
3 6 8 7 2 11 10 12 4 5 1 9
Solving Sequence
2,5
6
3,11
7 1 4 10 8 9 12
c
5
c
2
c
6
c
1
c
4
c
10
c
7
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h6u
25
− 4u
24
+ ··· + b − 2, −9u
25
+ 23u
24
+ ··· + a + 31, u
26
− 2u
25
+ ··· − 4u + 1i
I
u
2
= h−u
2
a + au + u
2
+ b − u + 1, −u
2
a + a
2
+ 3au + u
2
− 2a − 3u + 2, u
3
− u
2
+ 1i
I
u
3
= h−50a
5
+ 47a
4
− 54a
3
+ 71a
2
+ 1503b − 998a + 328, a
6
+ 2a
4
+ 2a
3
+ 6a
2
+ 11a − 23, u + 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
=
h6u
25
−4u
24
+· · ·+b−2, −9u
25
+23u
24
+· · ·+a+31, u
26
−2u
25
+· · ·−4u+1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
3
+ u
a
11
=
9u
25
− 23u
24
+ ··· + 69u − 31
−6u
25
+ 4u
24
+ ··· − 8u + 2
a
7
=
−4u
25
+ 14u
24
+ ··· − 43u + 27
−u
25
+ u
24
+ ··· + 2u
2
− 4u
a
1
=
u
3
u
5
− u
3
+ u
a
4
=
−32u
25
+ 28u
24
+ ··· − 79u + 9
10u
25
− 11u
24
+ ··· + 34u − 10
a
10
=
15u
25
− 27u
24
+ ··· + 77u − 33
−6u
25
+ 4u
24
+ ··· − 8u + 2
a
8
=
37u
25
− 38u
24
+ ··· + 103u − 23
−16u
25
+ 5u
24
+ ··· − 19u + 1
a
9
=
21u
25
− 12u
24
+ ··· + 25u + 4
−25u
25
+ 29u
24
+ ··· − 80u + 22
a
12
=
17u
25
− 37u
24
+ ··· + 107u − 43
−6u
25
+ 12u
24
+ ··· − 29u + 12
(ii) Obstruction class = 1
(iii) Cusp Shapes = 61u
25
− 54u
24
− 343u
23
+ 372u
22
+ 878u
21
− 898u
20
− 1579u
19
+
1127u
18
+ 2535u
17
− 859u
16
− 3329u
15
+ 282u
14
+ 3349u
13
+ 450u
12
− 2660u
11
−
969u
10
+ 2067u
9
+ 348u
8
− 518u
7
− 770u
6
+ 524u
5
+ 209u
4
− 85u
3
− 178u
2
+ 111u − 12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
26
− 12u
25
+ ··· − 6u + 1
c
2
, c
12
u
26
+ 2u
25
+ ··· + 4u + 1
c
3
u
26
+ 2u
25
+ ··· − 2u
2
+ 1
c
4
u
26
− 3u
24
+ ··· + 167u + 85
c
5
, c
8
u
26
− 2u
25
+ ··· − 4u + 1
c
6
u
26
− 2u
25
+ ··· − 2u
2
+ 1
c
7
u
26
− 3u
24
+ ··· − 167u + 85
c
9
u
26
+ u
24
+ ··· + u
2
+ 1
c
10
u
26
+ u
24
+ ··· + u
2
+ 1
c
11
u
26
+ 12u
25
+ ··· + 6u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
11
y
26
− 8y
24
+ ··· + 22y + 1
c
2
, c
5
, c
8
c
12
y
26
− 12y
25
+ ··· − 6y + 1
c
3
, c
6
y
26
− 20y
25
+ ··· − 4y + 1
c
4
, c
7
y
26
− 6y
25
+ ··· + 51331y + 7225
c
9
, c
10
y
26
+ 2y
25
+ ··· + 2y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.461384 + 0.912562I
a = −0.080894 − 0.407248I
b = 0.575155 + 0.340795I
1.12657 + 3.32893I 5.69551 − 1.13200I
u = −0.461384 − 0.912562I
a = −0.080894 + 0.407248I
b = 0.575155 − 0.340795I
1.12657 − 3.32893I 5.69551 + 1.13200I
u = 0.849370 + 0.605420I
a = −1.25013 − 1.25835I
b = −0.14769 − 1.58072I
3.52283 − 2.38934I 15.7660 + 2.5226I
u = 0.849370 − 0.605420I
a = −1.25013 + 1.25835I
b = −0.14769 + 1.58072I
3.52283 + 2.38934I 15.7660 − 2.5226I
u = −0.762938 + 0.732586I
a = −0.154380 + 0.405636I
b = 0.965729 + 0.259552I
5.79972I 0. − 8.96426I
u = −0.762938 − 0.732586I
a = −0.154380 − 0.405636I
b = 0.965729 − 0.259552I
− 5.79972I 0. + 8.96426I
u = 0.998814 + 0.491144I
a = 0.29293 + 2.27648I
b = −1.11561 + 0.93756I
−2.06063 − 7.20438I −4.87470 + 7.46295I
u = 0.998814 − 0.491144I
a = 0.29293 − 2.27648I
b = −1.11561 − 0.93756I
−2.06063 + 7.20438I −4.87470 − 7.46295I
u = 0.742293 + 0.445659I
a = 0.246099 − 0.102462I
b = 1.28686 + 0.76250I
−1.12657 + 3.32893I −5.69551 − 1.13200I
u = 0.742293 − 0.445659I
a = 0.246099 + 0.102462I
b = 1.28686 − 0.76250I
−1.12657 − 3.32893I −5.69551 + 1.13200I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.316384 + 0.759790I
a = 0.202465 + 1.071890I
b = −0.525339 − 0.441494I
2.06063 + 7.20438I 4.87470 − 7.46295I
u = −0.316384 − 0.759790I
a = 0.202465 − 1.071890I
b = −0.525339 + 0.441494I
2.06063 − 7.20438I 4.87470 + 7.46295I
u = 1.081650 + 0.521444I
a = 0.05036 + 1.94989I
b = −0.811958 + 0.761507I
−1.82571 − 6.70629I −2.75982 + 7.40474I
u = 1.081650 − 0.521444I
a = 0.05036 − 1.94989I
b = −0.811958 − 0.761507I
−1.82571 + 6.70629I −2.75982 − 7.40474I
u = 0.516527 + 0.544349I
a = −0.92356 − 1.49397I
b = −0.655244 − 0.614530I
1.82571 + 6.70629I 2.75982 − 7.40474I
u = 0.516527 − 0.544349I
a = −0.92356 + 1.49397I
b = −0.655244 + 0.614530I
1.82571 − 6.70629I 2.75982 + 7.40474I
u = 1.089350 + 0.612332I
a = −0.17797 − 1.62097I
b = 0.581197 − 0.813763I
− 11.5593I 0. + 10.56005I
u = 1.089350 − 0.612332I
a = −0.17797 + 1.62097I
b = 0.581197 + 0.813763I
11.5593I 0. − 10.56005I
u = −0.632233 + 0.402098I
a = 2.06829 − 1.29149I
b = −0.531949 − 0.961805I
2.02248 + 0.44868I 1.93644 − 3.28945I
u = −0.632233 − 0.402098I
a = 2.06829 + 1.29149I
b = −0.531949 + 0.961805I
2.02248 − 0.44868I 1.93644 + 3.28945I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.193370 + 0.413519I
a = −0.631044 + 0.511481I
b = −0.440341 + 0.796171I
−2.02248 − 0.44868I −1.93644 + 3.28945I
u = −1.193370 − 0.413519I
a = −0.631044 − 0.511481I
b = −0.440341 − 0.796171I
−2.02248 + 0.44868I −1.93644 − 3.28945I
u = 0.583202 + 0.387797I
a = −0.56568 + 1.58491I
b = 0.877784 + 0.479057I
2.72365I 0. − 5.95098I
u = 0.583202 − 0.387797I
a = −0.56568 − 1.58491I
b = 0.877784 − 0.479057I
− 2.72365I 0. + 5.95098I
u = −1.49489 + 0.08893I
a = −0.076473 + 0.557618I
b = −0.058595 + 0.627148I
−3.52283 + 2.38934I −15.7660 − 2.5226I
u = −1.49489 − 0.08893I
a = −0.076473 − 0.557618I
b = −0.058595 − 0.627148I
−3.52283 − 2.38934I −15.7660 + 2.5226I
7
II.
I
u
2
= h−u
2
a+au+u
2
+b−u+1, −u
2
a+a
2
+3au+u
2
−2a−3u+2, u
3
−u
2
+1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
11
=
a
u
2
a − au − u
2
+ u − 1
a
7
=
−u
2
a + au + 2u
2
− 2u + 2
0
a
1
=
u
2
− 1
−u
2
a
4
=
1
0
a
10
=
−u
2
a + au + u
2
+ a − u + 1
u
2
a − au − u
2
+ u − 1
a
8
=
−u
2
+ 1
u
2
a
9
=
a
u
2
a − au − u
2
+ u − 1
a
12
=
u
2
+ a − 1
u
2
a − au − 2u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −10u
2
a + 17au + 11u
2
− 14a − 17u + 18
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
u
6
c
5
(u
3
− u
2
+ 1)
2
c
6
, c
7
u
6
+ 3u
5
+ 6u
4
+ 7u
3
+ 5u
2
+ 2u − 1
c
8
, c
11
(u + 1)
6
c
9
, c
10
u
6
+ 2u
5
− 2u
4
− 3u
3
+ 2u
2
+ 2u − 1
c
12
(u − 1)
6
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
4
y
6
c
6
, c
7
y
6
+ 3y
5
+ 4y
4
− 3y
3
− 15y
2
− 14y + 1
c
8
, c
11
, c
12
(y −1)
6
c
9
, c
10
y
6
− 8y
5
+ 20y
4
− 27y
3
+ 20y
2
− 8y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.626026 + 0.207777I
b = −0.869124 − 0.347901I
4.66906 − 2.82812I 4.76162 + 1.20354I
u = 0.877439 + 0.744862I
a = −1.04326 − 1.13522I
b = 0.991685 − 0.396961I
4.66906 − 2.82812I 6.27312 + 3.54360I
u = 0.877439 − 0.744862I
a = 0.626026 − 0.207777I
b = −0.869124 + 0.347901I
4.66906 + 2.82812I 4.76162 − 1.20354I
u = 0.877439 − 0.744862I
a = −1.04326 + 1.13522I
b = 0.991685 + 0.396961I
4.66906 + 2.82812I 6.27312 − 3.54360I
u = −0.754878
a = 1.41297
b = −0.452937
0.531480 −8.86450
u = −0.754878
a = 3.42151
b = 2.20781
0.531480 −74.2050
11
III.
I
u
3
= h−50a
5
+1503b + · · · −998a + 328, a
6
+2a
4
+2a
3
+6a
2
+11a − 23, u+1i
(i) Arc colorings
a
2
=
0
−1
a
5
=
1
0
a
6
=
1
1
a
3
=
1
0
a
11
=
a
0.0332668a
5
− 0.0312708a
4
+ ··· + 0.664005a − 0.218230
a
7
=
0.0312708a
5
+ 0.0306055a
4
+ ··· + 0.584165a + 0.234864
0.0232868a
5
− 0.0552229a
4
+ ··· + 0.264804a + 0.713906
a
1
=
−1
−1
a
4
=
0.0372588a
5
+ 0.178310a
4
+ ··· + 0.823686a + 1.20892
0.0312708a
5
+ 0.0306055a
4
+ ··· + 0.584165a + 0.234864
a
10
=
−0.0332668a
5
+ 0.0312708a
4
+ ··· + 0.335995a + 0.218230
0.0332668a
5
− 0.0312708a
4
+ ··· + 0.664005a − 0.218230
a
8
=
0.0312708a
5
+ 0.0306055a
4
+ ··· + 0.584165a + 0.234864
0.0232868a
5
− 0.0552229a
4
+ ··· + 0.264804a + 0.713906
a
9
=
0.0525615a
5
+ 0.0372588a
4
+ ··· + 0.769128a + 0.401863
0.0113107a
5
− 0.0172987a
4
+ ··· + 0.785762a + 0.0991351
a
12
=
0.0332668a
5
− 0.0312708a
4
+ ··· + 0.664005a − 0.218230
0.0665336a
5
− 0.0625416a
4
+ ··· + 0.328011a − 0.436460
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
625
1503
a
5
+
2341
1503
a
4
−
1394
501
a
3
+
9655
1503
a
2
−
13978
1503
a +
20132
1503
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
4
u
6
− 3u
5
+ 6u
4
− 7u
3
+ 5u
2
− 2u − 1
c
5
(u + 1)
6
c
6
, c
11
(u
3
+ u
2
+ 2u + 1)
2
c
7
u
6
c
8
(u
3
− u
2
+ 1)
2
c
9
, c
10
u
6
− 2u
5
− 2u
4
+ 3u
3
+ 2u
2
− 2u − 1
c
12
(u
3
+ u
2
− 1)
2
13
(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
y
6
+ 3y
5
+ 4y
4
− 3y
3
− 15y
2
− 14y + 1
c
6
, c
11
(y
3
+ 3y
2
+ 2y −1)
2
c
7
y
6
c
8
, c
12
(y
3
− y
2
+ 2y −1)
2
c
9
, c
10
y
6
− 8y
5
+ 20y
4
− 27y
3
+ 20y
2
− 8y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.02278
b = 0.452937
−0.531480 8.86450
u = −1.00000
a = −1.63797
b = −2.20781
−0.531480 74.2050
u = −1.00000
a = −0.77661 + 1.70410I
b = −0.991685 + 0.396961I
−4.66906 + 2.82812I −6.27312 − 3.54360I
u = −1.00000
a = −0.77661 − 1.70410I
b = −0.991685 − 0.396961I
−4.66906 − 2.82812I −6.27312 + 3.54360I
u = −1.00000
a = 1.08420 + 1.65504I
b = 0.869124 + 0.347901I
−4.66906 + 2.82812I −4.76162 − 1.20354I
u = −1.00000
a = 1.08420 − 1.65504I
b = 0.869124 − 0.347901I
−4.66906 − 2.82812I −4.76162 + 1.20354I
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
3
− u
2
+ 2u − 1)
2
(u
26
− 12u
25
+ ··· − 6u + 1)
c
2
, c
12
((u − 1)
6
)(u
3
+ u
2
− 1)
2
(u
26
+ 2u
25
+ ··· + 4u + 1)
c
3
(u
3
− u
2
+ 2u − 1)
2
(u
6
− 3u
5
+ 6u
4
− 7u
3
+ 5u
2
− 2u − 1)
· (u
26
+ 2u
25
+ ··· − 2u
2
+ 1)
c
4
u
6
(u
6
− 3u
5
+ ··· − 2u − 1)(u
26
− 3u
24
+ ··· + 167u + 85)
c
5
, c
8
((u + 1)
6
)(u
3
− u
2
+ 1)
2
(u
26
− 2u
25
+ ··· − 4u + 1)
c
6
(u
3
+ u
2
+ 2u + 1)
2
(u
6
+ 3u
5
+ 6u
4
+ 7u
3
+ 5u
2
+ 2u − 1)
· (u
26
− 2u
25
+ ··· − 2u
2
+ 1)
c
7
u
6
(u
6
+ 3u
5
+ ··· + 2u − 1)(u
26
− 3u
24
+ ··· − 167u + 85)
c
9
(u
6
− 2u
5
− 2u
4
+ 3u
3
+ 2u
2
− 2u − 1)
· (u
6
+ 2u
5
+ ··· + 2u − 1)(u
26
+ u
24
+ ··· + u
2
+ 1)
c
10
(u
6
− 2u
5
− 2u
4
+ 3u
3
+ 2u
2
− 2u − 1)
· (u
6
+ 2u
5
+ ··· + 2u − 1)(u
26
+ u
24
+ ··· + u
2
+ 1)
c
11
((u + 1)
6
)(u
3
+ u
2
+ 2u + 1)
2
(u
26
+ 12u
25
+ ··· + 6u + 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
11
((y −1)
6
)(y
3
+ 3y
2
+ 2y −1)
2
(y
26
− 8y
24
+ ··· + 22y + 1)
c
2
, c
5
, c
8
c
12
((y −1)
6
)(y
3
− y
2
+ 2y −1)
2
(y
26
− 12y
25
+ ··· − 6y + 1)
c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
(y
6
+ 3y
5
+ 4y
4
− 3y
3
− 15y
2
− 14y + 1)
· (y
26
− 20y
25
+ ··· − 4y + 1)
c
4
, c
7
y
6
(y
6
+ 3y
5
+ 4y
4
− 3y
3
− 15y
2
− 14y + 1)
· (y
26
− 6y
25
+ ··· + 51331y + 7225)
c
9
, c
10
((y
6
− 8y
5
+ ··· − 8y + 1)
2
)(y
26
+ 2y
25
+ ··· + 2y + 1)
17
