12a
0789
(K12a
0789
)
A knot diagram
1
Linearized knot diagam
3 8 11 12 9 10 2 7 6 1 4 5
Solving Sequence
3,11
4 12
5,8
2 1 7 9 10 6
c
3
c
11
c
4
c
2
c
1
c
7
c
8
c
10
c
6
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
55
+ 32u
53
+ ··· + b + 1, u
55
+ u
54
+ ··· + a − 3, u
56
− 2u
55
+ ··· + 4u + 1i
I
u
2
= hb, a − u − 1, u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 58 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
55
+32u
53
+· · ·+b+1, u
55
+u
54
+· · ·+a−3, u
56
−2u
55
+· · ·+4u+1i
(i) Arc colorings
a
3
=
1
0
a
11
=
0
u
a
4
=
1
−u
2
a
12
=
u
−u
3
+ u
a
5
=
−u
2
+ 1
u
4
− 2u
2
a
8
=
−u
55
− u
54
+ ··· + 3u + 3
u
55
− 32u
53
+ ··· − 6u − 1
a
2
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
a
1
=
−u
3
+ 2u
u
5
− 3u
3
+ u
a
7
=
u
55
− u
54
+ ··· − 2u + 2
−u
55
+ 32u
53
+ ··· + 3u + 1
a
9
=
−u
55
− u
54
+ ··· + 5u + 3
u
55
− 32u
53
+ ··· − 6u − 1
a
10
=
−u
7
+ 4u
5
− 4u
3
u
9
− 5u
7
+ 7u
5
− 2u
3
+ u
a
6
=
−u
54
+ u
53
+ ··· + 2u + 3
−u
29
+ 17u
27
+ ··· − 2u
2
− u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
55
− u
54
+ ··· + 5u − 11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
56
+ 15u
55
+ ··· + 216u + 16
c
2
, c
7
u
56
+ u
55
+ ··· − 4u − 4
c
3
, c
4
, c
11
c
12
u
56
− 2u
55
+ ··· + 4u + 1
c
5
, c
6
, c
9
u
56
− 3u
55
+ ··· − u − 1
c
10
u
56
+ 18u
55
+ ··· + 6542u + 1153
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
56
+ 49y
55
+ ··· − 800y + 256
c
2
, c
7
y
56
− 15y
55
+ ··· − 216y + 16
c
3
, c
4
, c
11
c
12
y
56
− 66y
55
+ ··· − 16y + 1
c
5
, c
6
, c
9
y
56
− 45y
55
+ ··· + 29y + 1
c
10
y
56
− 30y
55
+ ··· − 62348032y + 1329409
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.861614 + 0.380828I
a = 1.202800 + 0.472116I
b = −0.915762 − 0.737386I
2.07733 − 3.75955I 0. + 2.06314I
u = 0.861614 − 0.380828I
a = 1.202800 − 0.472116I
b = −0.915762 + 0.737386I
2.07733 + 3.75955I 0. − 2.06314I
u = −0.776749 + 0.490571I
a = −0.50179 + 2.16195I
b = −1.024790 − 0.785571I
1.24745 − 10.95200I −0.88355 + 9.10486I
u = −0.776749 − 0.490571I
a = −0.50179 − 2.16195I
b = −1.024790 + 0.785571I
1.24745 + 10.95200I −0.88355 − 9.10486I
u = 0.917254
a = −0.937794
b = 0.865247
−2.32133 −4.40840
u = 0.813380 + 0.406723I
a = −1.172270 − 0.567248I
b = 0.827652 + 0.819827I
6.04894 + 0.47222I 4.63325 − 1.43874I
u = 0.813380 − 0.406723I
a = −1.172270 + 0.567248I
b = 0.827652 − 0.819827I
6.04894 − 0.47222I 4.63325 + 1.43874I
u = −0.782454 + 0.456203I
a = 0.58253 − 2.23050I
b = 0.943810 + 0.783646I
5.68928 − 6.47510I 3.47548 + 7.00475I
u = −0.782454 − 0.456203I
a = 0.58253 + 2.23050I
b = 0.943810 − 0.783646I
5.68928 + 6.47510I 3.47548 − 7.00475I
u = 0.768228 + 0.434814I
a = 1.144950 + 0.658680I
b = −0.742662 − 0.902492I
2.13106 + 4.71480I 0.47050 − 5.02848I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.768228 − 0.434814I
a = 1.144950 − 0.658680I
b = −0.742662 + 0.902492I
2.13106 − 4.71480I 0.47050 + 5.02848I
u = −0.778193 + 0.407597I
a = −0.71065 + 2.31160I
b = −0.838906 − 0.752287I
2.31622 − 1.89869I 0.70275 + 3.94226I
u = −0.778193 − 0.407597I
a = −0.71065 − 2.31160I
b = −0.838906 + 0.752287I
2.31622 + 1.89869I 0.70275 − 3.94226I
u = −0.570014 + 0.501355I
a = 0.76909 − 1.45994I
b = 1.087280 + 0.321819I
−6.02729 − 5.28844I −6.96026 + 7.61800I
u = −0.570014 − 0.501355I
a = 0.76909 + 1.45994I
b = 1.087280 − 0.321819I
−6.02729 + 5.28844I −6.96026 − 7.61800I
u = −0.544488 + 0.378983I
a = −1.32009 + 1.52978I
b = −0.842302 − 0.270646I
−0.55880 − 2.99317I −2.43682 + 9.49735I
u = −0.544488 − 0.378983I
a = −1.32009 − 1.52978I
b = −0.842302 + 0.270646I
−0.55880 + 2.99317I −2.43682 − 9.49735I
u = −0.079318 + 0.638620I
a = 0.556233 + 0.364082I
b = 0.992202 − 0.742680I
−0.82232 + 7.15155I −4.60706 − 5.03276I
u = −0.079318 − 0.638620I
a = 0.556233 − 0.364082I
b = 0.992202 + 0.742680I
−0.82232 − 7.15155I −4.60706 + 5.03276I
u = −0.316519 + 0.540275I
a = −0.814732 + 0.436386I
b = −1.068200 + 0.206607I
−6.75768 + 1.69595I −9.62058 − 0.39515I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.316519 − 0.540275I
a = −0.814732 − 0.436386I
b = −1.068200 − 0.206607I
−6.75768 − 1.69595I −9.62058 + 0.39515I
u = 0.605204 + 0.139914I
a = 0.523878 + 0.514590I
b = −0.281922 − 0.406522I
1.104200 + 0.351253I 7.66165 − 1.23198I
u = 0.605204 − 0.139914I
a = 0.523878 − 0.514590I
b = −0.281922 + 0.406522I
1.104200 − 0.351253I 7.66165 + 1.23198I
u = −0.038185 + 0.607819I
a = −0.646423 − 0.466088I
b = −0.882448 + 0.766731I
3.48917 + 2.89621I −0.22897 − 2.80424I
u = −0.038185 − 0.607819I
a = −0.646423 + 0.466088I
b = −0.882448 − 0.766731I
3.48917 − 2.89621I −0.22897 + 2.80424I
u = 0.457801 + 0.362756I
a = −0.775748 − 0.907568I
b = 0.118428 + 0.812194I
−2.68266 + 1.34854I −4.20838 − 4.89529I
u = 0.457801 − 0.362756I
a = −0.775748 + 0.907568I
b = 0.118428 − 0.812194I
−2.68266 − 1.34854I −4.20838 + 4.89529I
u = 0.024733 + 0.567150I
a = 0.743934 + 0.615367I
b = 0.729699 − 0.800933I
−0.035156 − 1.335270I −3.76168 + 0.44055I
u = 0.024733 − 0.567150I
a = 0.743934 − 0.615367I
b = 0.729699 + 0.800933I
−0.035156 + 1.335270I −3.76168 − 0.44055I
u = 1.44775
a = −0.137478
b = 1.13818
−1.53547 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.316942 + 0.333544I
a = 1.67144 − 0.45362I
b = 0.730562 − 0.086920I
−1.184330 + 0.293211I −7.24661 − 0.11134I
u = −0.316942 − 0.333544I
a = 1.67144 + 0.45362I
b = 0.730562 + 0.086920I
−1.184330 − 0.293211I −7.24661 + 0.11134I
u = −1.54283 + 0.06225I
a = 0.45736 − 1.45447I
b = −0.258482 + 0.942392I
4.08661 − 2.66620I 0
u = −1.54283 − 0.06225I
a = 0.45736 + 1.45447I
b = −0.258482 − 0.942392I
4.08661 + 2.66620I 0
u = 1.54469 + 0.03373I
a = −0.851909 − 0.654429I
b = −0.876218 + 0.153382I
5.30389 + 0.47425I 0
u = 1.54469 − 0.03373I
a = −0.851909 + 0.654429I
b = −0.876218 − 0.153382I
5.30389 − 0.47425I 0
u = 1.54394 + 0.12483I
a = 0.126116 − 1.372580I
b = −1.122220 + 0.424521I
1.01324 + 7.50643I 0
u = 1.54394 − 0.12483I
a = 0.126116 + 1.372580I
b = −1.122220 − 0.424521I
1.01324 − 7.50643I 0
u = 1.55716 + 0.08171I
a = 0.395590 + 1.327910I
b = 0.950907 − 0.362288I
6.54610 + 4.55343I 0
u = 1.55716 − 0.08171I
a = 0.395590 − 1.327910I
b = 0.950907 + 0.362288I
6.54610 − 4.55343I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.59291 + 0.03318I
a = −0.437806 + 0.931798I
b = 0.277160 − 0.618435I
8.71978 − 0.96955I 0
u = −1.59291 − 0.03318I
a = −0.437806 − 0.931798I
b = 0.277160 + 0.618435I
8.71978 + 0.96955I 0
u = −1.62812 + 0.12417I
a = −1.23139 + 1.34321I
b = 0.773399 − 0.956394I
10.33130 − 6.82915I 0
u = −1.62812 − 0.12417I
a = −1.23139 − 1.34321I
b = 0.773399 + 0.956394I
10.33130 + 6.82915I 0
u = 1.62976 + 0.11604I
a = 0.00161 + 2.29347I
b = 0.911680 − 0.767070I
10.56670 + 3.88406I 0
u = 1.62976 − 0.11604I
a = 0.00161 − 2.29347I
b = 0.911680 + 0.767070I
10.56670 − 3.88406I 0
u = 1.63214 + 0.14225I
a = −0.26342 + 2.25155I
b = 1.041030 − 0.821782I
9.4693 + 13.3538I 0
u = 1.63214 − 0.14225I
a = −0.26342 − 2.25155I
b = 1.041030 + 0.821782I
9.4693 − 13.3538I 0
u = 1.63313 + 0.13047I
a = 0.15431 − 2.28786I
b = −0.980165 + 0.809362I
13.9539 + 8.7007I 0
u = 1.63313 − 0.13047I
a = 0.15431 + 2.28786I
b = −0.980165 − 0.809362I
13.9539 − 8.7007I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.63993 + 0.11295I
a = 1.27230 − 1.22415I
b = −0.815467 + 0.876391I
14.4734 − 2.4457I 0
u = −1.63993 − 0.11295I
a = 1.27230 + 1.22415I
b = −0.815467 − 0.876391I
14.4734 + 2.4457I 0
u = −1.65226
a = 0.971661
b = −0.658927
6.52621 0
u = −1.65143 + 0.09936I
a = −1.30952 + 1.08207I
b = 0.855453 − 0.778423I
10.73980 + 1.94649I 0
u = −1.65143 − 0.09936I
a = −1.30952 − 1.08207I
b = 0.855453 + 0.778423I
10.73980 − 1.94649I 0
u = −0.340130
a = 2.97078
b = 0.476062
−1.17659 −11.8220
10
II. I
u
2
= hb, a − u − 1, u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
11
=
0
u
a
4
=
1
u − 1
a
12
=
u
−u + 1
a
5
=
u
−u
a
8
=
u + 1
0
a
2
=
1
0
a
1
=
1
0
a
7
=
u + 1
0
a
9
=
u + 1
0
a
10
=
−u
u
a
6
=
2u + 1
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
8
u
2
c
3
, c
4
u
2
+ u − 1
c
5
, c
6
(u − 1)
2
c
9
(u + 1)
2
c
10
, c
11
, c
12
u
2
− u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
c
8
y
2
c
3
, c
4
, c
10
c
11
, c
12
y
2
− 3y + 1
c
5
, c
6
, c
9
(y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.61803
b = 0
−0.657974 5.00000
u = −1.61803
a = −0.618034
b = 0
7.23771 5.00000
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
8
u
2
(u
56
+ 15u
55
+ ··· + 216u + 16)
c
2
, c
7
u
2
(u
56
+ u
55
+ ··· − 4u − 4)
c
3
, c
4
(u
2
+ u − 1)(u
56
− 2u
55
+ ··· + 4u + 1)
c
5
, c
6
((u − 1)
2
)(u
56
− 3u
55
+ ··· − u − 1)
c
9
((u + 1)
2
)(u
56
− 3u
55
+ ··· − u − 1)
c
10
(u
2
− u − 1)(u
56
+ 18u
55
+ ··· + 6542u + 1153)
c
11
, c
12
(u
2
− u − 1)(u
56
− 2u
55
+ ··· + 4u + 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
2
(y
56
+ 49y
55
+ ··· − 800y + 256)
c
2
, c
7
y
2
(y
56
− 15y
55
+ ··· − 216y + 16)
c
3
, c
4
, c
11
c
12
(y
2
− 3y + 1)(y
56
− 66y
55
+ ··· − 16y + 1)
c
5
, c
6
, c
9
((y −1)
2
)(y
56
− 45y
55
+ ··· + 29y + 1)
c
10
(y
2
− 3y + 1)(y
56
− 30y
55
+ ··· − 6.23480 × 10
7
y + 1329409)
16
