12n
0039
(K12n
0039
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 10 11 5 1 12 8 7
Solving Sequence
7,11 4,8
5 12 1 10 6 3 2 9
c
7
c
4
c
11
c
12
c
10
c
6
c
3
c
2
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
57
− u
56
+ ··· + 2b − 4, 13u
57
− 29u
56
+ ··· + 2a − 9, u
58
− 3u
57
+ ··· − 3u + 1i
I
u
2
= h−u
2
a + b, −u
4
a − u
3
a + u
2
a − u
3
+ a
2
+ au − u
2
+ 1, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 70 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
57
−u
56
+· · ·+2b−4, 13u
57
−29u
56
+· · ·+2a−9, u
58
−3u
57
+· · ·−3u+1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
−
13
2
u
57
+
29
2
u
56
+ ··· −
33
2
u +
9
2
1
2
u
57
+
1
2
u
56
+ ··· −
1
2
u + 2
a
8
=
1
u
2
a
5
=
−
7
2
u
57
+
15
2
u
56
+ ··· −
17
2
u +
3
2
3
2
u
57
−
5
2
u
56
+ ··· − 3u
2
+
5
2
u
a
12
=
−u
−u
3
+ u
a
1
=
−u
3
−u
3
+ u
a
10
=
u
3
u
5
− u
3
+ u
a
6
=
u
8
− u
6
+ u
4
+ 1
u
10
− 2u
8
+ 3u
6
− 2u
4
+ u
2
a
3
=
−5u
57
+ 11u
56
+ ··· −
23
2
u +
7
2
u
57
− u
56
+ ··· +
3
2
u + 1
a
2
=
−u
57
+ 2u
56
+ ··· −
3
2
u +
3
2
1
2
u
54
−
1
2
u
53
+ ··· +
1
2
u
2
+
1
2
u
a
9
=
−u
11
+ 2u
9
− 2u
7
+ u
3
−u
11
+ 3u
9
− 4u
7
+ 3u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 12u
57
−
51
2
u
56
+ ··· +
53
2
u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
58
+ 35u
57
+ ··· + 9u + 1
c
2
, c
5
u
58
+ 7u
57
+ ··· + 5u + 1
c
3
u
58
− 7u
57
+ ··· − 27u + 2
c
4
, c
8
u
58
− u
57
+ ··· + 8192u + 4096
c
6
u
58
− 3u
57
+ ··· + 2221u + 937
c
7
, c
11
u
58
+ 3u
57
+ ··· + 3u + 1
c
9
u
58
+ 3u
57
+ ··· + 3u + 1
c
10
u
58
+ 29u
57
+ ··· + 3u + 1
c
12
u
58
+ 9u
57
+ ··· + 689u + 176
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
58
− 17y
57
+ ··· + 213y + 1
c
2
, c
5
y
58
+ 35y
57
+ ··· + 9y + 1
c
3
y
58
− 69y
57
+ ··· − 217y + 4
c
4
, c
8
y
58
+ 65y
57
+ ··· + 234881024y + 16777216
c
6
y
58
+ 11y
57
+ ··· − 1936315y + 877969
c
7
, c
11
y
58
− 29y
57
+ ··· − 3y + 1
c
9
y
58
+ 71y
57
+ ··· − 3y + 1
c
10
y
58
+ 3y
57
+ ··· + 29y + 1
c
12
y
58
+ 23y
57
+ ··· + 427807y + 30976
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.742693 + 0.676776I
a = 1.10927 + 1.46827I
b = −1.018590 − 0.353888I
−6.05935 − 7.31192I 0.79047 + 6.14491I
u = 0.742693 − 0.676776I
a = 1.10927 − 1.46827I
b = −1.018590 + 0.353888I
−6.05935 + 7.31192I 0.79047 − 6.14491I
u = 0.766215 + 0.621161I
a = −1.04148 − 1.38838I
b = 1.014310 + 0.129064I
−2.46464 − 2.40510I 3.55783 + 3.26427I
u = 0.766215 − 0.621161I
a = −1.04148 + 1.38838I
b = 1.014310 − 0.129064I
−2.46464 + 2.40510I 3.55783 − 3.26427I
u = 0.965439 + 0.350364I
a = −0.543909 − 0.897747I
b = 0.404805 − 0.591191I
−1.64630 − 1.27469I −1.46430 + 0.39248I
u = 0.965439 − 0.350364I
a = −0.543909 + 0.897747I
b = 0.404805 + 0.591191I
−1.64630 + 1.27469I −1.46430 − 0.39248I
u = 0.830822 + 0.646667I
a = 1.15502 + 1.22562I
b = −1.220950 − 0.029193I
−6.32401 + 2.24227I 0
u = 0.830822 − 0.646667I
a = 1.15502 − 1.22562I
b = −1.220950 + 0.029193I
−6.32401 − 2.24227I 0
u = 1.067360 + 0.176258I
a = −1.04041 + 1.60217I
b = −1.155750 + 0.597676I
−3.60243 + 0.04887I −5.83111 + 0.I
u = 1.067360 − 0.176258I
a = −1.04041 − 1.60217I
b = −1.155750 − 0.597676I
−3.60243 − 0.04887I −5.83111 + 0.I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.282860 + 0.819808I
a = 0.45994 − 1.98856I
b = 1.20420 + 3.10728I
−8.53365 + 9.60020I 0.29238 − 5.04135I
u = 0.282860 − 0.819808I
a = 0.45994 + 1.98856I
b = 1.20420 − 3.10728I
−8.53365 − 9.60020I 0.29238 + 5.04135I
u = −1.066250 + 0.414950I
a = −0.25930 + 2.89782I
b = 1.64743 + 2.38773I
−2.23944 + 0.13129I 0
u = −1.066250 − 0.414950I
a = −0.25930 − 2.89782I
b = 1.64743 − 2.38773I
−2.23944 − 0.13129I 0
u = 1.051460 + 0.470149I
a = −0.92877 − 1.78383I
b = 1.42872 − 1.74233I
−0.78474 − 1.84538I 0
u = 1.051460 − 0.470149I
a = −0.92877 + 1.78383I
b = 1.42872 + 1.74233I
−0.78474 + 1.84538I 0
u = 0.213505 + 0.811297I
a = 0.52309 − 2.11889I
b = 0.41666 + 3.33148I
−9.55678 − 0.64630I −1.020161 + 0.779335I
u = 0.213505 − 0.811297I
a = 0.52309 + 2.11889I
b = 0.41666 − 3.33148I
−9.55678 + 0.64630I −1.020161 − 0.779335I
u = −0.413914 + 0.728124I
a = 0.200276 + 0.074948I
b = −0.458662 − 0.847775I
0.99825 − 2.10282I 0.620949 − 0.071334I
u = −0.413914 − 0.728124I
a = 0.200276 − 0.074948I
b = −0.458662 + 0.847775I
0.99825 + 2.10282I 0.620949 + 0.071334I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.258688 + 0.794593I
a = −0.52040 + 2.05773I
b = −0.84030 − 3.00010I
−4.98964 + 4.15775I 2.66640 − 2.12473I
u = 0.258688 − 0.794593I
a = −0.52040 − 2.05773I
b = −0.84030 + 3.00010I
−4.98964 − 4.15775I 2.66640 + 2.12473I
u = −1.060250 + 0.490356I
a = −0.10148 − 1.91006I
b = −1.30941 − 1.14927I
−0.61621 + 4.67232I 0
u = −1.060250 − 0.490356I
a = −0.10148 + 1.91006I
b = −1.30941 + 1.14927I
−0.61621 − 4.67232I 0
u = −0.501104 + 0.655716I
a = −0.0331531 − 0.0143983I
b = −0.466931 + 0.591950I
1.49411 − 0.06269I 3.02061 + 1.27535I
u = −0.501104 − 0.655716I
a = −0.0331531 + 0.0143983I
b = −0.466931 − 0.591950I
1.49411 + 0.06269I 3.02061 − 1.27535I
u = −1.046050 + 0.556065I
a = 0.395453 − 1.090660I
b = −0.524046 − 0.784867I
−0.12505 + 4.79073I 0
u = −1.046050 − 0.556065I
a = 0.395453 + 1.090660I
b = −0.524046 + 0.784867I
−0.12505 − 4.79073I 0
u = 1.135510 + 0.371005I
a = 2.45266 + 0.82876I
b = 0.97469 + 2.43445I
−5.61981 − 0.51916I 0
u = 1.135510 − 0.371005I
a = 2.45266 − 0.82876I
b = 0.97469 − 2.43445I
−5.61981 + 0.51916I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.094110 + 0.493111I
a = 1.25150 + 2.49948I
b = −1.88271 + 2.81236I
−1.62734 − 6.95830I 0
u = 1.094110 − 0.493111I
a = 1.25150 − 2.49948I
b = −1.88271 − 2.81236I
−1.62734 + 6.95830I 0
u = −1.185760 + 0.284132I
a = −0.65680 + 4.29093I
b = 2.21766 + 4.41254I
−9.47227 − 0.82173I 0
u = −1.185760 − 0.284132I
a = −0.65680 − 4.29093I
b = 2.21766 − 4.41254I
−9.47227 + 0.82173I 0
u = −1.199640 + 0.258578I
a = 1.04819 − 4.24993I
b = −1.69840 − 4.66322I
−13.2521 − 6.3035I 0
u = −1.199640 − 0.258578I
a = 1.04819 + 4.24993I
b = −1.69840 + 4.66322I
−13.2521 + 6.3035I 0
u = −1.087500 + 0.582128I
a = −1.317980 + 0.269553I
b = −0.386637 + 1.148330I
−0.97594 + 7.10425I 0
u = −1.087500 − 0.582128I
a = −1.317980 − 0.269553I
b = −0.386637 − 1.148330I
−0.97594 − 7.10425I 0
u = −1.136660 + 0.500162I
a = 1.47085 + 2.25651I
b = 2.79786 + 0.33790I
−4.72739 + 7.37382I 0
u = −1.136660 − 0.500162I
a = 1.47085 − 2.25651I
b = 2.79786 − 0.33790I
−4.72739 − 7.37382I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.203940 + 0.312053I
a = 0.40507 − 4.64412I
b = −2.83156 − 4.64587I
−13.9557 + 4.2640I 0
u = −1.203940 − 0.312053I
a = 0.40507 + 4.64412I
b = −2.83156 + 4.64587I
−13.9557 − 4.2640I 0
u = −0.660650 + 0.306406I
a = 0.914280 − 0.165501I
b = 0.997317 − 0.499930I
−0.62635 + 2.91588I 3.27340 − 4.97212I
u = −0.660650 − 0.306406I
a = 0.914280 + 0.165501I
b = 0.997317 + 0.499930I
−0.62635 − 2.91588I 3.27340 + 4.97212I
u = 1.157920 + 0.552647I
a = 1.67092 + 3.72168I
b = −2.68114 + 4.31965I
−7.63974 − 9.17351I 0
u = 1.157920 − 0.552647I
a = 1.67092 − 3.72168I
b = −2.68114 − 4.31965I
−7.63974 + 9.17351I 0
u = 1.160620 + 0.568042I
a = −1.47283 − 3.87780I
b = 2.98552 − 4.18011I
−11.1402 − 14.7500I 0
u = 1.160620 − 0.568042I
a = −1.47283 + 3.87780I
b = 2.98552 + 4.18011I
−11.1402 + 14.7500I 0
u = 1.174810 + 0.538366I
a = −2.03584 − 3.78686I
b = 2.49310 − 4.78492I
−12.40280 − 4.33791I 0
u = 1.174810 − 0.538366I
a = −2.03584 + 3.78686I
b = 2.49310 + 4.78492I
−12.40280 + 4.33791I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.172748 + 0.667933I
a = −0.405277 + 0.856827I
b = 1.48790 − 0.60667I
−2.02121 − 2.91480I −0.43838 + 3.64289I
u = −0.172748 − 0.667933I
a = −0.405277 − 0.856827I
b = 1.48790 + 0.60667I
−2.02121 + 2.91480I −0.43838 − 3.64289I
u = −0.406360 + 0.529126I
a = −0.238632 − 0.235635I
b = −0.732719 + 0.328694I
1.267410 − 0.491154I 7.40856 + 1.29163I
u = −0.406360 − 0.529126I
a = −0.238632 + 0.235635I
b = −0.732719 − 0.328694I
1.267410 + 0.491154I 7.40856 − 1.29163I
u = 0.447120 + 0.450989I
a = −0.61398 − 2.20773I
b = 0.687783 + 0.451795I
0.99990 − 2.08688I 2.42768 + 5.77119I
u = 0.447120 − 0.450989I
a = −0.61398 + 2.20773I
b = 0.687783 − 0.451795I
0.99990 + 2.08688I 2.42768 − 5.77119I
u = 0.291693 + 0.528701I
a = 0.15373 + 2.35394I
b = −0.550148 − 1.147150I
0.62843 + 2.75213I 0.46673 − 2.33667I
u = 0.291693 − 0.528701I
a = 0.15373 − 2.35394I
b = −0.550148 + 1.147150I
0.62843 − 2.75213I 0.46673 + 2.33667I
10
II. I
u
2
=
h−u
2
a+b, −u
4
a−u
3
a+u
2
a−u
3
+a
2
+au−u
2
+1, u
6
+u
5
−u
4
−2u
3
+u+1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
a
u
2
a
a
8
=
1
u
2
a
5
=
a
u
2
a
a
12
=
−u
−u
3
+ u
a
1
=
−u
3
−u
3
+ u
a
10
=
u
3
u
5
− u
3
+ u
a
6
=
u
3
u
3
− u
a
3
=
u
3
a
u
5
a − u
3
a + au
a
2
=
u
3
a − u
4
− u
3
+ u
2
+ u
u
5
a − u
3
a − u
3
+ au + u + 1
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
5
a − u
4
a + 4u
3
a + 4u
4
+ 2u
2
a − u
3
− 4au − 5u
2
− 4u + 3
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
6
c
2
(u
2
+ u + 1)
6
c
4
, c
8
u
12
c
6
, c
9
, c
11
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
c
7
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
c
10
, c
12
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
6
c
4
, c
8
y
12
c
6
, c
7
, c
9
c
11
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
c
10
, c
12
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002190 + 0.295542I
a = 0.578212 + 1.125030I
b = −0.136196 + 1.374220I
−1.89061 + 1.10558I −0.42156 − 3.46269I
u = 1.002190 + 0.295542I
a = −1.263410 − 0.061767I
b = −1.122010 − 0.805060I
−1.89061 − 2.95419I −3.93112 + 4.16322I
u = 1.002190 − 0.295542I
a = 0.578212 − 1.125030I
b = −0.136196 − 1.374220I
−1.89061 − 1.10558I −0.42156 + 3.46269I
u = 1.002190 − 0.295542I
a = −1.263410 + 0.061767I
b = −1.122010 + 0.805060I
−1.89061 + 2.95419I −3.93112 − 4.16322I
u = −0.428243 + 0.664531I
a = 0.224551 + 0.930349I
b = 0.471538 − 0.368031I
1.89061 + 1.10558I 5.61650 − 2.84542I
u = −0.428243 + 0.664531I
a = 0.693431 − 0.659641I
b = −0.554493 − 0.224349I
1.89061 − 2.95419I 7.50338 + 4.33850I
u = −0.428243 − 0.664531I
a = 0.224551 − 0.930349I
b = 0.471538 + 0.368031I
1.89061 − 1.10558I 5.61650 + 2.84542I
u = −0.428243 − 0.664531I
a = 0.693431 + 0.659641I
b = −0.554493 + 0.224349I
1.89061 + 2.95419I 7.50338 − 4.33850I
u = −1.073950 + 0.558752I
a = −0.036219 + 0.825237I
b = 0.959936 + 0.737627I
3.66314I 1.09315 − 1.33646I
u = −1.073950 + 0.558752I
a = −0.696567 − 0.443985I
b = −1.118770 + 0.462515I
7.72290I 4.13964 − 9.04329I
14
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.073950 − 0.558752I
a = −0.036219 − 0.825237I
b = 0.959936 − 0.737627I
− 3.66314I 1.09315 + 1.33646I
u = −1.073950 − 0.558752I
a = −0.696567 + 0.443985I
b = −1.118770 − 0.462515I
− 7.72290I 4.13964 + 9.04329I
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
6
)(u
58
+ 35u
57
+ ··· + 9u + 1)
c
2
((u
2
+ u + 1)
6
)(u
58
+ 7u
57
+ ··· + 5u + 1)
c
3
((u
2
− u + 1)
6
)(u
58
− 7u
57
+ ··· − 27u + 2)
c
4
, c
8
u
12
(u
58
− u
57
+ ··· + 8192u + 4096)
c
5
((u
2
− u + 1)
6
)(u
58
+ 7u
57
+ ··· + 5u + 1)
c
6
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
58
− 3u
57
+ ··· + 2221u + 937)
c
7
((u
6
+ u
5
− u
4
− 2u
3
+ u + 1)
2
)(u
58
+ 3u
57
+ ··· + 3u + 1)
c
9
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
58
+ 3u
57
+ ··· + 3u + 1)
c
10
((u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
)(u
58
+ 29u
57
+ ··· + 3u + 1)
c
11
((u
6
− u
5
− u
4
+ 2u
3
− u + 1)
2
)(u
58
+ 3u
57
+ ··· + 3u + 1)
c
12
((u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)
2
)(u
58
+ 9u
57
+ ··· + 689u + 176)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
6
)(y
58
− 17y
57
+ ··· + 213y + 1)
c
2
, c
5
((y
2
+ y + 1)
6
)(y
58
+ 35y
57
+ ··· + 9y + 1)
c
3
((y
2
+ y + 1)
6
)(y
58
− 69y
57
+ ··· − 217y + 4)
c
4
, c
8
y
12
(y
58
+ 65y
57
+ ··· + 2.34881 × 10
8
y + 1.67772 × 10
7
)
c
6
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
· (y
58
+ 11y
57
+ ··· − 1936315y + 877969)
c
7
, c
11
((y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
)(y
58
− 29y
57
+ ··· − 3y + 1)
c
9
((y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)
2
)(y
58
+ 71y
57
+ ··· − 3y + 1)
c
10
((y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
)(y
58
+ 3y
57
+ ··· + 29y + 1)
c
12
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
2
· (y
58
+ 23y
57
+ ··· + 427807y + 30976)
17
