12a
0937
(K12a
0937
)
A knot diagram
1
Linearized knot diagam
4 6 9 1 2 10 11 12 3 7 8 5
Solving Sequence
5,12
1 4 2
6,9
3 8 11 7 10
c
12
c
4
c
1
c
5
c
3
c
8
c
11
c
7
c
10
c
2
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
44
+ 4u
43
+ ··· + 2b + 1, −4u
44
− 14u
43
+ ··· + 2a + 11, u
45
+ 3u
44
+ ··· − 4u − 1i
I
u
2
= h−u
2
a + b − a, −u
2
a + a
2
+ au − a − u, u
3
− u
2
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 51 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
44
+4u
43
+· · ·+2b+1, −4u
44
−14u
43
+· · ·+2a+11, u
45
+3u
44
+· · ·−4u−1i
(i) Arc colorings
a
5
=
0
u
a
12
=
1
0
a
1
=
1
u
2
a
4
=
u
u
3
+ u
a
2
=
u
2
+ 1
u
4
+ 2u
2
a
6
=
−u
5
− 2u
3
− u
−u
7
− 3u
5
− 2u
3
+ u
a
9
=
2u
44
+ 7u
43
+ ··· − 2u −
11
2
−
1
2
u
44
− 2u
43
+ ··· + 3u −
1
2
a
3
=
−u
8
− 3u
6
− 3u
4
+ 1
−u
10
− 4u
8
− 5u
6
+ 3u
2
a
8
=
3
2
u
44
+ 5u
43
+ ··· + u − 6
−
1
2
u
44
− 2u
43
+ ··· + 3u −
1
2
a
11
=
−
1
2
u
44
− u
43
+ ··· + 7u + 1
−
1
2
u
44
− u
43
+ ··· + 2u +
1
2
a
7
=
−
1
2
u
42
− u
41
+ ··· + 7u −
1
2
−
1
2
u
44
− u
43
+ ··· + u +
1
2
a
10
=
−u
44
− u
43
+ ··· + 7u −
7
2
−
3
2
u
44
− 5u
43
+ ··· + 7u +
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
44
+
5
2
u
43
+ ··· −
15
2
u −
21
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
12
u
45
− 3u
44
+ ··· − 4u + 1
c
2
, c
5
u
45
+ 3u
44
+ ··· − 2u + 41
c
3
, c
9
u
45
− u
44
+ ··· − 32u + 64
c
6
, c
7
, c
8
c
10
, c
11
u
45
+ 4u
44
+ ··· + u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
12
y
45
+ 37y
44
+ ··· + 40y −1
c
2
, c
5
y
45
− 39y
44
+ ··· + 54616y −1681
c
3
, c
9
y
45
+ 35y
44
+ ··· + 29696y −4096
c
6
, c
7
, c
8
c
10
, c
11
y
45
− 62y
44
+ ··· + 33y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.890413 + 0.122697I
a = −1.10889 − 1.11126I
b = −1.76542 + 0.09202I
18.6493 + 7.3520I −17.3944 − 3.6030I
u = −0.890413 − 0.122697I
a = −1.10889 + 1.11126I
b = −1.76542 − 0.09202I
18.6493 − 7.3520I −17.3944 + 3.6030I
u = −0.857781 + 0.085739I
a = 0.773971 + 0.850892I
b = 1.146650 − 0.350828I
−10.38620 + 5.44992I −17.1451 − 4.4258I
u = −0.857781 − 0.085739I
a = 0.773971 − 0.850892I
b = 1.146650 + 0.350828I
−10.38620 − 5.44992I −17.1451 + 4.4258I
u = 0.851384
a = 1.98833
b = 1.74823
−15.7842 −16.5970
u = −0.826935 + 0.030025I
a = −0.276934 − 0.575320I
b = −0.381901 + 0.636194I
−5.58348 + 2.07140I −14.3541 − 3.3642I
u = −0.826935 − 0.030025I
a = −0.276934 + 0.575320I
b = −0.381901 − 0.636194I
−5.58348 − 2.07140I −14.3541 + 3.3642I
u = 0.606365 + 0.531100I
a = 0.104984 + 1.236330I
b = −1.74792 − 0.01574I
−14.7030 − 2.1539I −15.6545 + 3.1266I
u = 0.606365 − 0.531100I
a = 0.104984 − 1.236330I
b = −1.74792 + 0.01574I
−14.7030 + 2.1539I −15.6545 − 3.1266I
u = 0.778750
a = −1.54037
b = −1.06860
−5.57462 −16.4290
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.110795 + 1.218650I
a = −1.31939 + 1.71047I
b = 1.61924 − 0.05616I
−5.93702 + 1.70967I −11.27232 + 1.90310I
u = −0.110795 − 1.218650I
a = −1.31939 − 1.71047I
b = 1.61924 + 0.05616I
−5.93702 − 1.70967I −11.27232 − 1.90310I
u = −0.002801 + 1.227720I
a = 0.43866 − 1.57194I
b = −0.738328 + 0.291820I
2.19010 + 0.49707I −9.03459 − 1.32514I
u = −0.002801 − 1.227720I
a = 0.43866 + 1.57194I
b = −0.738328 − 0.291820I
2.19010 − 0.49707I −9.03459 + 1.32514I
u = −0.463428 + 1.150540I
a = 0.206323 − 0.367607I
b = −1.77236 − 0.07475I
−17.6778 − 2.5277I −14.7181 + 0.I
u = −0.463428 − 1.150540I
a = 0.206323 + 0.367607I
b = −1.77236 + 0.07475I
−17.6778 + 2.5277I −14.7181 + 0.I
u = −0.407555 + 1.183180I
a = −0.449308 − 0.082403I
b = 1.184170 + 0.302548I
−7.01503 − 0.90790I −14.2280 + 0.I
u = −0.407555 − 1.183180I
a = −0.449308 + 0.082403I
b = 1.184170 − 0.302548I
−7.01503 + 0.90790I −14.2280 + 0.I
u = −0.370756 + 1.245550I
a = 0.615468 + 0.827180I
b = −0.445891 − 0.613389I
−1.82707 + 2.23489I 0
u = −0.370756 − 1.245550I
a = 0.615468 − 0.827180I
b = −0.445891 + 0.613389I
−1.82707 − 2.23489I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.085876 + 1.300140I
a = 0.31567 + 1.38001I
b = −0.051237 − 0.435616I
4.23150 − 2.03860I 0
u = 0.085876 − 1.300140I
a = 0.31567 − 1.38001I
b = −0.051237 + 0.435616I
4.23150 + 2.03860I 0
u = 0.334888 + 1.264580I
a = −0.392426 + 1.134500I
b = −1.068570 − 0.086970I
−1.65436 − 4.01424I 0
u = 0.334888 − 1.264580I
a = −0.392426 − 1.134500I
b = −1.068570 + 0.086970I
−1.65436 + 4.01424I 0
u = 0.242153 + 1.287270I
a = 0.318641 − 0.571138I
b = 0.280859 + 0.154657I
2.59980 − 3.13937I 0
u = 0.242153 − 1.287270I
a = 0.318641 + 0.571138I
b = 0.280859 − 0.154657I
2.59980 + 3.13937I 0
u = 0.503331 + 0.451082I
a = −0.360474 − 1.020470I
b = 1.066120 + 0.076462I
−4.50790 − 1.79515I −15.5943 + 4.2867I
u = 0.503331 − 0.451082I
a = −0.360474 + 1.020470I
b = 1.066120 − 0.076462I
−4.50790 + 1.79515I −15.5943 − 4.2867I
u = 0.391513 + 1.271870I
a = 0.32209 − 1.48073I
b = 1.74641 + 0.02116I
−11.83510 − 4.45927I 0
u = 0.391513 − 1.271870I
a = 0.32209 + 1.48073I
b = 1.74641 − 0.02116I
−11.83510 + 4.45927I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.371472 + 1.293170I
a = −0.36103 − 1.45695I
b = −0.326465 + 0.656457I
−1.45817 + 6.38100I 0
u = −0.371472 − 1.293170I
a = −0.36103 + 1.45695I
b = −0.326465 − 0.656457I
−1.45817 − 6.38100I 0
u = 0.627430
a = 0.675315
b = 0.234560
−1.43225 −5.16570
u = −0.385838 + 1.331430I
a = −0.13299 + 1.85543I
b = 1.112000 − 0.380680I
−5.94556 + 9.91745I 0
u = −0.385838 − 1.331430I
a = −0.13299 − 1.85543I
b = 1.112000 + 0.380680I
−5.94556 − 9.91745I 0
u = 0.145397 + 1.390720I
a = −1.28933 − 1.01084I
b = 0.939446 + 0.150253I
1.30405 − 3.95914I 0
u = 0.145397 − 1.390720I
a = −1.28933 + 1.01084I
b = 0.939446 − 0.150253I
1.30405 + 3.95914I 0
u = −0.39751 + 1.35949I
a = 0.53384 − 2.10202I
b = −1.75547 + 0.10280I
−16.1695 + 11.9726I 0
u = −0.39751 − 1.35949I
a = 0.53384 + 2.10202I
b = −1.75547 − 0.10280I
−16.1695 − 11.9726I 0
u = 0.15162 + 1.44483I
a = 1.95312 + 1.07952I
b = −1.72058 − 0.03402I
−8.28166 − 4.65852I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.15162 − 1.44483I
a = 1.95312 − 1.07952I
b = −1.72058 + 0.03402I
−8.28166 + 4.65852I 0
u = −0.369157
a = 2.62661
b = 1.63816
−9.52192 −4.22190
u = 0.271798 + 0.224791I
a = 0.661349 + 1.168710I
b = −0.244731 − 0.245078I
−0.383394 − 0.784998I −9.02382 + 8.78053I
u = 0.271798 − 0.224791I
a = 0.661349 − 1.168710I
b = −0.244731 + 0.245078I
−0.383394 + 0.784998I −9.02382 − 8.78053I
u = −0.183739
a = −2.85656
b = −0.704396
−1.23322 −6.78810
9
II. I
u
2
= h−u
2
a + b − a, −u
2
a + a
2
+ au − a − u, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
5
=
0
u
a
12
=
1
0
a
1
=
1
u
2
a
4
=
u
u
2
− u + 1
a
2
=
u
2
+ 1
u
2
− u + 1
a
6
=
−1
0
a
9
=
a
u
2
a + a
a
3
=
u
u
2
− u + 1
a
8
=
u
2
a + 2a
u
2
a + a
a
11
=
−u
2
a − u
2
− 2a + u − 1
−u
2
a − a − 1
a
7
=
−u
2
− a + u − 2
−u
2
a − a − 1
a
10
=
a
u
2
a + a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −au − 5u
2
− a + 3u − 20
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
12
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
3
, c
9
u
6
c
4
(u
3
+ u
2
+ 2u + 1)
2
c
5
(u
3
− u
2
+ 1)
2
c
6
, c
7
, c
8
(u
2
+ u − 1)
3
c
10
, c
11
(u
2
− u − 1)
3
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
5
(y
3
− y
2
+ 2y −1)
2
c
3
, c
9
y
6
c
6
, c
7
, c
8
c
10
, c
11
(y
2
− 3y + 1)
3
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = 0.542287 + 0.460350I
b = −0.618034
2.03717 − 2.82812I −11.10015 − 0.15818I
u = 0.215080 + 1.307140I
a = −1.41973 − 1.20521I
b = 1.61803
−5.85852 − 2.82812I −10.89327 + 4.43024I
u = 0.215080 − 1.307140I
a = 0.542287 − 0.460350I
b = −0.618034
2.03717 + 2.82812I −11.10015 + 0.15818I
u = 0.215080 − 1.307140I
a = −1.41973 + 1.20521I
b = 1.61803
−5.85852 + 2.82812I −10.89327 − 4.43024I
u = 0.569840
a = 1.22142
b = 1.61803
−9.99610 −21.8310
u = 0.569840
a = −0.466540
b = −0.618034
−2.10041 −19.1820
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
12
((u
3
− u
2
+ 2u − 1)
2
)(u
45
− 3u
44
+ ··· − 4u + 1)
c
2
((u
3
+ u
2
− 1)
2
)(u
45
+ 3u
44
+ ··· − 2u + 41)
c
3
, c
9
u
6
(u
45
− u
44
+ ··· − 32u + 64)
c
4
((u
3
+ u
2
+ 2u + 1)
2
)(u
45
− 3u
44
+ ··· − 4u + 1)
c
5
((u
3
− u
2
+ 1)
2
)(u
45
+ 3u
44
+ ··· − 2u + 41)
c
6
, c
7
, c
8
((u
2
+ u − 1)
3
)(u
45
+ 4u
44
+ ··· + u − 1)
c
10
, c
11
((u
2
− u − 1)
3
)(u
45
+ 4u
44
+ ··· + u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
12
((y
3
+ 3y
2
+ 2y −1)
2
)(y
45
+ 37y
44
+ ··· + 40y −1)
c
2
, c
5
((y
3
− y
2
+ 2y −1)
2
)(y
45
− 39y
44
+ ··· + 54616y −1681)
c
3
, c
9
y
6
(y
45
+ 35y
44
+ ··· + 29696y −4096)
c
6
, c
7
, c
8
c
10
, c
11
((y
2
− 3y + 1)
3
)(y
45
− 62y
44
+ ··· + 33y −1)
15
