12a
0141
(K12a
0141
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 10 11 12 4 8 6 1 7
Solving Sequence
1,7
12
4,8
9 3 2 11 6 10 5
c
12
c
7
c
8
c
3
c
1
c
11
c
6
c
10
c
5
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
83
− u
82
+ ··· + b − 3u, −u
83
− u
82
+ ··· + a + 1, u
85
+ 2u
84
+ ··· + 3u + 1i
I
u
2
= hb − 1, u
3
− u
2
+ a + u, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 90 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
83
−u
82
+· · ·+b−3u, −u
83
−u
82
+· · ·+a+1, u
85
+2u
84
+· · ·+3u+1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
12
=
1
−u
2
a
4
=
u
83
+ u
82
+ ··· − 2u
2
− 1
u
83
+ u
82
+ ··· − u
3
+ 3u
a
8
=
−u
u
3
+ u
a
9
=
u
12
+ 3u
10
+ 3u
8
− 2u
6
− 4u
4
− u
2
+ 1
−u
14
− 4u
12
− 7u
10
− 4u
8
+ 2u
6
+ 4u
4
+ u
2
a
3
=
−u
83
− u
82
+ ··· − u − 2
u
83
+ u
82
+ ··· − 2u
2
+ 2u
a
2
=
u
81
+ u
80
+ ··· + 2u
2
+ 2
−u
83
− u
82
+ ··· − u
3
− 2u
a
11
=
u
2
+ 1
−u
2
a
6
=
−u
5
− 2u
3
− u
u
5
+ u
3
+ u
a
10
=
−u
8
− 3u
6
− 3u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
5
=
u
11
+ 4u
9
+ 6u
7
+ 2u
5
− 3u
3
− 2u
−u
11
− 3u
9
− 4u
7
− u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
84
− 3u
83
+ ··· − 19u
2
− 7
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
85
+ 44u
84
+ ··· + 5u + 1
c
2
, c
4
u
85
− 6u
84
+ ··· − 7u + 1
c
3
, c
8
u
85
− u
84
+ ··· + 280u
2
+ 32
c
5
, c
6
, c
10
u
85
− 2u
84
+ ··· + 69u + 9
c
7
, c
12
u
85
+ 2u
84
+ ··· + 3u + 1
c
9
u
85
− 33u
84
+ ··· − 17920u + 1024
c
11
u
85
− 48u
84
+ ··· + 11u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
85
+ 68y
83
+ ··· + 29y −1
c
2
, c
4
y
85
− 44y
84
+ ··· + 5y −1
c
3
, c
8
y
85
+ 33y
84
+ ··· − 17920y −1024
c
5
, c
6
, c
10
y
85
− 88y
84
+ ··· + 5643y −81
c
7
, c
12
y
85
+ 48y
84
+ ··· + 11y −1
c
9
y
85
+ 29y
84
+ ··· + 17432576y −1048576
c
11
y
85
− 20y
84
+ ··· + 283y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.427067 + 0.903288I
a = −1.45806 + 0.54783I
b = −0.093443 − 0.592929I
0.15850 + 2.06285I 0
u = −0.427067 − 0.903288I
a = −1.45806 − 0.54783I
b = −0.093443 + 0.592929I
0.15850 − 2.06285I 0
u = −0.299920 + 0.963143I
a = −0.70616 + 1.28928I
b = −0.444864 + 0.101720I
0.80832 + 2.36710I 0
u = −0.299920 − 0.963143I
a = −0.70616 − 1.28928I
b = −0.444864 − 0.101720I
0.80832 − 2.36710I 0
u = −0.515060 + 0.884059I
a = 2.20223 − 0.11337I
b = −0.489025 + 1.097400I
−2.47171 − 1.64225I 0
u = −0.515060 − 0.884059I
a = 2.20223 + 0.11337I
b = −0.489025 − 1.097400I
−2.47171 + 1.64225I 0
u = −0.127090 + 0.966533I
a = 0.14971 − 1.50302I
b = −0.162922 − 0.575892I
−0.189918 − 0.769631I 0
u = −0.127090 − 0.966533I
a = 0.14971 + 1.50302I
b = −0.162922 + 0.575892I
−0.189918 + 0.769631I 0
u = 0.472646 + 0.937491I
a = −1.30201 − 2.40692I
b = −0.75331 + 1.77642I
−2.91281 − 3.22612I 0
u = 0.472646 − 0.937491I
a = −1.30201 + 2.40692I
b = −0.75331 − 1.77642I
−2.91281 + 3.22612I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.148743 + 1.053900I
a = 0.031322 + 0.766541I
b = 1.145100 + 0.559030I
3.86682 + 1.00502I 0
u = 0.148743 − 1.053900I
a = 0.031322 − 0.766541I
b = 1.145100 − 0.559030I
3.86682 − 1.00502I 0
u = −0.482726 + 0.958907I
a = 2.09417 − 0.93018I
b = 0.361332 + 1.235000I
−2.60128 + 5.81097I 0
u = −0.482726 − 0.958907I
a = 2.09417 + 0.93018I
b = 0.361332 − 1.235000I
−2.60128 − 5.81097I 0
u = 0.094327 + 1.083490I
a = −0.282344 − 1.027780I
b = −1.192310 − 0.747964I
1.72458 + 5.87992I 0
u = 0.094327 − 1.083490I
a = −0.282344 + 1.027780I
b = −1.192310 + 0.747964I
1.72458 − 5.87992I 0
u = 0.221914 + 0.883158I
a = 1.20820 − 1.44716I
b = −1.45625 − 0.47093I
−0.855582 − 1.085530I 4.15462 − 1.86707I
u = 0.221914 − 0.883158I
a = 1.20820 + 1.44716I
b = −1.45625 + 0.47093I
−0.855582 + 1.085530I 4.15462 + 1.86707I
u = 0.488530 + 0.986861I
a = 0.70570 + 2.33378I
b = 1.01921 − 1.22307I
1.47835 − 6.73642I 0
u = 0.488530 − 0.986861I
a = 0.70570 − 2.33378I
b = 1.01921 + 1.22307I
1.47835 + 6.73642I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.517691 + 0.981760I
a = −0.62999 − 2.67424I
b = −1.38554 + 1.33578I
−1.25752 − 11.69060I 0
u = 0.517691 − 0.981760I
a = −0.62999 + 2.67424I
b = −1.38554 − 1.33578I
−1.25752 + 11.69060I 0
u = −0.429195 + 0.777445I
a = −1.174550 − 0.530055I
b = 0.415792 − 0.088181I
−0.06950 + 1.83366I 0.66649 − 5.14350I
u = −0.429195 − 0.777445I
a = −1.174550 + 0.530055I
b = 0.415792 + 0.088181I
−0.06950 − 1.83366I 0.66649 + 5.14350I
u = 0.290125 + 1.079680I
a = −0.170641 − 0.295666I
b = 0.738944 + 0.278480I
5.05386 − 1.05004I 0
u = 0.290125 − 1.079680I
a = −0.170641 + 0.295666I
b = 0.738944 − 0.278480I
5.05386 + 1.05004I 0
u = −0.872592 + 0.011682I
a = −0.13799 + 1.46426I
b = 0.319385 − 1.125930I
9.16696 − 2.75702I 3.57442 + 3.01021I
u = −0.872592 − 0.011682I
a = −0.13799 − 1.46426I
b = 0.319385 + 1.125930I
9.16696 + 2.75702I 3.57442 − 3.01021I
u = −0.863233 + 0.080625I
a = 0.81138 − 1.42675I
b = −1.91866 + 1.20507I
3.48230 − 11.13990I −1.12808 + 7.06717I
u = −0.863233 − 0.080625I
a = 0.81138 + 1.42675I
b = −1.91866 − 1.20507I
3.48230 + 11.13990I −1.12808 − 7.06717I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.860540 + 0.064517I
a = −0.70023 + 1.48083I
b = 1.62593 − 1.28684I
6.12444 − 5.75564I 2.12476 + 3.50801I
u = −0.860540 − 0.064517I
a = −0.70023 − 1.48083I
b = 1.62593 + 1.28684I
6.12444 + 5.75564I 2.12476 − 3.50801I
u = 0.844152 + 0.062878I
a = 0.268712 − 0.119606I
b = 0.752344 + 0.745016I
1.62409 + 4.88126I −2.48263 − 3.81093I
u = 0.844152 − 0.062878I
a = 0.268712 + 0.119606I
b = 0.752344 − 0.745016I
1.62409 − 4.88126I −2.48263 + 3.81093I
u = 0.388889 + 1.096620I
a = 0.099797 + 0.999827I
b = −0.0317490 − 0.1256150I
4.33398 − 5.66355I 0
u = 0.388889 − 1.096620I
a = 0.099797 − 0.999827I
b = −0.0317490 + 0.1256150I
4.33398 + 5.66355I 0
u = −0.831465 + 0.054925I
a = 0.75430 − 1.75204I
b = −1.62680 + 1.91695I
0.96832 − 2.23146I −2.03853 + 3.04442I
u = −0.831465 − 0.054925I
a = 0.75430 + 1.75204I
b = −1.62680 − 1.91695I
0.96832 + 2.23146I −2.03853 − 3.04442I
u = 0.829680 + 0.029626I
a = −0.212091 + 0.030369I
b = −0.707910 − 0.329097I
3.58523 + 0.53976I 0.236036 + 1.036399I
u = 0.829680 − 0.029626I
a = −0.212091 − 0.030369I
b = −0.707910 + 0.329097I
3.58523 − 0.53976I 0.236036 − 1.036399I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.549150 + 0.588273I
a = 1.10721 + 1.72614I
b = −0.755841 − 1.170040I
−3.29788 + 5.93867I −5.94323 − 6.90497I
u = −0.549150 − 0.588273I
a = 1.10721 − 1.72614I
b = −0.755841 + 1.170040I
−3.29788 − 5.93867I −5.94323 + 6.90497I
u = 0.767403 + 0.105507I
a = 0.514337 + 0.150139I
b = −0.047203 + 0.669447I
0.74936 − 1.67557I −1.83419 + 4.19037I
u = 0.767403 − 0.105507I
a = 0.514337 − 0.150139I
b = −0.047203 − 0.669447I
0.74936 + 1.67557I −1.83419 − 4.19037I
u = 0.418759 + 1.172800I
a = −0.159145 + 0.894091I
b = 0.135953 + 0.365111I
4.44095 − 5.64736I 0
u = 0.418759 − 1.172800I
a = −0.159145 − 0.894091I
b = 0.135953 − 0.365111I
4.44095 + 5.64736I 0
u = −0.457326 + 0.591105I
a = −0.94785 − 1.41236I
b = 0.377468 + 0.777654I
−0.68556 + 1.70417I −2.97415 − 3.79810I
u = −0.457326 − 0.591105I
a = −0.94785 + 1.41236I
b = 0.377468 − 0.777654I
−0.68556 − 1.70417I −2.97415 + 3.79810I
u = 0.595689 + 0.441663I
a = 1.83391 + 1.66748I
b = −1.15659 − 1.23971I
−2.76956 + 7.29426I −5.22729 − 5.96862I
u = 0.595689 − 0.441663I
a = 1.83391 − 1.66748I
b = −1.15659 + 1.23971I
−2.76956 − 7.29426I −5.22729 + 5.96862I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.487621 + 1.191850I
a = 1.067100 − 0.424092I
b = 0.121066 − 0.748933I
3.93138 − 2.95603I 0
u = 0.487621 − 1.191850I
a = 1.067100 + 0.424092I
b = 0.121066 + 0.748933I
3.93138 + 2.95603I 0
u = 0.490907 + 0.508135I
a = 1.56014 + 2.20056I
b = −0.40694 − 1.54213I
−4.10374 − 0.78290I −8.23718 + 0.26691I
u = 0.490907 − 0.508135I
a = 1.56014 − 2.20056I
b = −0.40694 + 1.54213I
−4.10374 + 0.78290I −8.23718 − 0.26691I
u = −0.432592 + 1.231440I
a = 0.76495 + 1.44194I
b = −1.82893 + 1.84485I
4.81532 + 2.20135I 0
u = −0.432592 − 1.231440I
a = 0.76495 − 1.44194I
b = −1.82893 − 1.84485I
4.81532 − 2.20135I 0
u = 0.445663 + 1.230960I
a = −0.581637 − 0.898159I
b = −0.761101 − 0.242276I
7.34340 − 3.97545I 0
u = 0.445663 − 1.230960I
a = −0.581637 + 0.898159I
b = −0.761101 + 0.242276I
7.34340 + 3.97545I 0
u = −0.512165 + 0.463031I
a = 0.63934 + 1.63821I
b = 0.036905 − 1.325170I
−3.96910 − 1.72223I −7.92428 + 1.16968I
u = −0.512165 − 0.463031I
a = 0.63934 − 1.63821I
b = 0.036905 + 1.325170I
−3.96910 + 1.72223I −7.92428 − 1.16968I
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.427548 + 1.238760I
a = 0.382640 + 1.176150I
b = 0.746159 + 0.605982I
5.54728 + 0.43357I 0
u = 0.427548 − 1.238760I
a = 0.382640 − 1.176150I
b = 0.746159 − 0.605982I
5.54728 − 0.43357I 0
u = 0.474420 + 1.224890I
a = −1.072410 − 0.442407I
b = −0.780878 + 0.397385I
7.13458 − 5.22782I 0
u = 0.474420 − 1.224890I
a = −1.072410 + 0.442407I
b = −0.780878 − 0.397385I
7.13458 + 5.22782I 0
u = −0.484834 + 1.222640I
a = −2.19593 + 1.58462I
b = −1.65391 − 2.09380I
4.43882 + 6.98841I 0
u = −0.484834 − 1.222640I
a = −2.19593 − 1.58462I
b = −1.65391 + 2.09380I
4.43882 − 6.98841I 0
u = −0.416514 + 1.251400I
a = −0.163665 + 1.195830I
b = −1.97265 + 1.08672I
7.54683 − 6.68500I 0
u = −0.416514 − 1.251400I
a = −0.163665 − 1.195830I
b = −1.97265 − 1.08672I
7.54683 + 6.68500I 0
u = −0.426785 + 1.249060I
a = −0.122637 − 1.014800I
b = 1.72547 − 1.20199I
10.11510 − 1.25157I 0
u = −0.426785 − 1.249060I
a = −0.122637 + 1.014800I
b = 1.72547 + 1.20199I
10.11510 + 1.25157I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.490124 + 1.226340I
a = 1.43483 + 0.30890I
b = 0.866884 − 0.767430I
5.09598 − 9.70048I 0
u = 0.490124 − 1.226340I
a = 1.43483 − 0.30890I
b = 0.866884 + 0.767430I
5.09598 + 9.70048I 0
u = 0.543800 + 0.402118I
a = −1.51318 − 1.68366I
b = 0.806831 + 1.004280I
−0.12612 + 2.57133I −1.97535 − 2.97016I
u = 0.543800 − 0.402118I
a = −1.51318 + 1.68366I
b = 0.806831 − 1.004280I
−0.12612 − 2.57133I −1.97535 + 2.97016I
u = −0.493919 + 1.232880I
a = 1.62810 − 1.79732I
b = 1.66947 + 1.38660I
9.6280 + 10.6382I 0
u = −0.493919 − 1.232880I
a = 1.62810 + 1.79732I
b = 1.66947 − 1.38660I
9.6280 − 10.6382I 0
u = −0.501415 + 1.230750I
a = −1.62921 + 2.08576I
b = −1.99524 − 1.27465I
6.9320 + 16.0693I 0
u = −0.501415 − 1.230750I
a = −1.62921 − 2.08576I
b = −1.99524 + 1.27465I
6.9320 − 16.0693I 0
u = −0.458122 + 1.250850I
a = −0.747325 + 0.157441I
b = 0.438672 − 1.193640I
12.99140 + 1.96517I 0
u = −0.458122 − 1.250850I
a = −0.747325 − 0.157441I
b = 0.438672 + 1.193640I
12.99140 − 1.96517I 0
12
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.470369 + 1.247810I
a = 1.022100 − 0.693142I
b = 0.219562 + 1.214950I
12.9023 + 7.5473I 0
u = −0.470369 − 1.247810I
a = 1.022100 + 0.693142I
b = 0.219562 − 1.214950I
12.9023 − 7.5473I 0
u = 0.595116 + 0.162032I
a = −0.699007 − 0.996568I
b = 0.426595 + 0.028304I
1.64828 + 1.80340I 1.26340 − 4.31445I
u = 0.595116 − 0.162032I
a = −0.699007 + 0.996568I
b = 0.426595 − 0.028304I
1.64828 − 1.80340I 1.26340 + 4.31445I
u = −0.243333
a = −1.34823
b = −0.653998
−1.20251 −8.85310
13
II. I
u
2
= hb − 1, u
3
− u
2
+ a + u, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
1
=
1
0
a
7
=
0
u
a
12
=
1
−u
2
a
4
=
−u
3
+ u
2
− u
1
a
8
=
−u
u
3
+ u
a
9
=
−u
u
3
+ u
a
3
=
−u
3
+ u
2
− u
1
a
2
=
−u
3
+ u
2
− u + 1
1
a
11
=
u
2
+ 1
−u
2
a
6
=
−u
4
− u
2
− 1
u
4
− u
3
+ u
2
+ 1
a
10
=
−u
u
3
+ u
a
5
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2u
4
− 7u
3
+ 8u
2
− 6u
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
8
, c
9
u
5
c
4
(u + 1)
5
c
5
, c
6
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
7
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
c
10
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
11
u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1
c
12
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
8
, c
9
y
5
c
5
, c
6
, c
10
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
7
, c
12
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
11
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.339110 + 0.822375I
a = −0.871221 − 1.107660I
b = 1.00000
−1.31583 + 1.53058I −6.99101 − 6.23673I
u = −0.339110 − 0.822375I
a = −0.871221 + 1.107660I
b = 1.00000
−1.31583 − 1.53058I −6.99101 + 6.23673I
u = 0.766826
a = −0.629714
b = 1.00000
0.756147 −2.36160
u = 0.455697 + 1.200150I
a = 0.186078 + 0.874646I
b = 1.00000
4.22763 − 4.40083I 1.17182 + 3.02310I
u = 0.455697 − 1.200150I
a = 0.186078 − 0.874646I
b = 1.00000
4.22763 + 4.40083I 1.17182 − 3.02310I
17
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
85
+ 44u
84
+ ··· + 5u + 1)
c
2
((u − 1)
5
)(u
85
− 6u
84
+ ··· − 7u + 1)
c
3
, c
8
u
5
(u
85
− u
84
+ ··· + 280u
2
+ 32)
c
4
((u + 1)
5
)(u
85
− 6u
84
+ ··· − 7u + 1)
c
5
, c
6
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
85
− 2u
84
+ ··· + 69u + 9)
c
7
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
85
+ 2u
84
+ ··· + 3u + 1)
c
9
u
5
(u
85
− 33u
84
+ ··· − 17920u + 1024)
c
10
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
85
− 2u
84
+ ··· + 69u + 9)
c
11
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)(u
85
− 48u
84
+ ··· + 11u + 1)
c
12
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
85
+ 2u
84
+ ··· + 3u + 1)
18
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
85
+ 68y
83
+ ··· + 29y −1)
c
2
, c
4
((y −1)
5
)(y
85
− 44y
84
+ ··· + 5y −1)
c
3
, c
8
y
5
(y
85
+ 33y
84
+ ··· − 17920y −1024)
c
5
, c
6
, c
10
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
85
− 88y
84
+ ··· + 5643y −81)
c
7
, c
12
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
85
+ 48y
84
+ ··· + 11y −1)
c
9
y
5
(y
85
+ 29y
84
+ ··· + 1.74326 × 10
7
y −1048576)
c
11
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
85
− 20y
84
+ ··· + 283y −1)
19
