12a
0767
(K12a
0767
)
A knot diagram
1
Linearized knot diagam
3 8 10 11 1 12 9 2 7 5 4 6
Solving Sequence
2,9
8 3
1,6
5 7 10 4 12 11
c
8
c
2
c
1
c
5
c
7
c
9
c
3
c
12
c
11
c
4
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h2u
25
+ 3u
24
+ ··· + b − 3, 3u
26
+ 9u
25
+ ··· + 2a − 9, u
27
+ 3u
26
+ ··· + u − 2i
I
u
2
= h21u
19
a + 223u
19
+ ··· − 81a − 318, −2u
19
a + u
19
+ ··· + a
2
− 3, u
20
− u
19
+ ··· + u
2
+ 1i
I
u
3
= h−u
5
+ b − u, −u
4
− u
2
+ a + u − 2, u
6
+ u
4
+ 2u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 73 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
= h2u
25
+3u
24
+· · ·+b−3, 3u
26
+9u
25
+· · ·+2a−9, u
27
+3u
26
+· · ·+u−2i
(i) Arc colorings
a
2
=
0
u
a
9
=
1
0
a
8
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
6
=
−
3
2
u
26
−
9
2
u
25
+ ··· − 4u +
9
2
−2u
25
− 3u
24
+ ··· − 5u + 3
a
5
=
−
7
2
u
26
−
21
2
u
25
+ ··· − 8u +
17
2
−5u
25
− 7u
24
+ ··· − 12u + 7
a
7
=
u
2
+ 1
u
2
a
10
=
u
4
+ u
2
+ 1
u
4
a
4
=
−u
11
− 2u
9
− 4u
7
− 4u
5
− 3u
3
−u
11
− u
9
− 2u
7
− u
5
+ u
3
+ u
a
12
=
−
3
2
u
26
−
5
2
u
25
+ ··· + 2u +
1
2
−u
26
− 2u
25
+ ··· + u + 1
a
11
=
1
2
u
26
+
1
2
u
25
+ ··· +
1
2
u
2
+
1
2
u
26
+ 2u
25
+ ··· + u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
26
− 10u
25
− 26u
24
− 38u
23
− 78u
22
− 106u
21
− 180u
20
−
202u
19
− 298u
18
− 286u
17
− 366u
16
− 274u
15
− 332u
14
− 172u
13
− 200u
12
− 10u
11
−
70u
10
+ 84u
9
− 18u
8
+ 58u
7
− 62u
6
+ 10u
5
− 52u
4
+ 6u
3
− 18u
2
+ 14u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
9
u
27
+ 7u
26
+ ··· − 7u − 4
c
2
, c
8
u
27
− 3u
26
+ ··· + u + 2
c
3
u
27
− 3u
26
+ ··· + 192u + 128
c
4
, c
5
, c
6
c
10
, c
11
, c
12
u
27
+ 15u
25
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
y
27
+ 27y
26
+ ··· + 385y −16
c
2
, c
8
y
27
+ 7y
26
+ ··· − 7y −4
c
3
y
27
+ y
26
+ ··· − 323584y −16384
c
4
, c
5
, c
6
c
10
, c
11
, c
12
y
27
+ 30y
26
+ ··· − 7y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.277460 + 0.929850I
a = −0.226509 + 0.100294I
b = −0.350781 + 0.505562I
−3.57952 + 2.60716I −14.9352 − 5.3810I
u = 0.277460 − 0.929850I
a = −0.226509 − 0.100294I
b = −0.350781 − 0.505562I
−3.57952 − 2.60716I −14.9352 + 5.3810I
u = 0.126747 + 1.039010I
a = 1.75601 + 0.00499I
b = −0.485752 + 0.316127I
4.48046 − 3.61631I −5.50483 + 2.14074I
u = 0.126747 − 1.039010I
a = 1.75601 − 0.00499I
b = −0.485752 − 0.316127I
4.48046 + 3.61631I −5.50483 − 2.14074I
u = −0.752094 + 0.565194I
a = −0.248696 + 0.216030I
b = 1.22743 + 0.73470I
10.36450 − 3.20982I 1.89568 + 3.18066I
u = −0.752094 − 0.565194I
a = −0.248696 − 0.216030I
b = 1.22743 − 0.73470I
10.36450 + 3.20982I 1.89568 − 3.18066I
u = 0.387921 + 1.022180I
a = −0.396664 + 1.123690I
b = −0.35306 − 1.73889I
6.03730 + 9.94630I −4.12408 − 8.16397I
u = 0.387921 − 1.022180I
a = −0.396664 − 1.123690I
b = −0.35306 + 1.73889I
6.03730 − 9.94630I −4.12408 + 8.16397I
u = −0.827337 + 0.833435I
a = 0.269864 − 1.251390I
b = −0.725756 − 0.997287I
3.30382 + 0.39142I −7.35863 − 2.13067I
u = −0.827337 − 0.833435I
a = 0.269864 + 1.251390I
b = −0.725756 + 0.997287I
3.30382 − 0.39142I −7.35863 + 2.13067I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.649915 + 0.979302I
a = −0.176933 − 0.959042I
b = −0.517512 + 0.774284I
9.15980 − 1.98047I 0.08290 + 2.09302I
u = −0.649915 − 0.979302I
a = −0.176933 + 0.959042I
b = −0.517512 − 0.774284I
9.15980 + 1.98047I 0.08290 − 2.09302I
u = 0.780493 + 0.883681I
a = 0.480688 + 0.900878I
b = −0.045523 + 0.725234I
4.69939 + 2.93735I −5.31916 − 3.32522I
u = 0.780493 − 0.883681I
a = 0.480688 − 0.900878I
b = −0.045523 − 0.725234I
4.69939 − 2.93735I −5.31916 + 3.32522I
u = −0.899672 + 0.815653I
a = −0.58148 + 3.00390I
b = 1.13268 + 3.45473I
14.5319 + 8.1806I 0.62316 − 3.31406I
u = −0.899672 − 0.815653I
a = −0.58148 − 3.00390I
b = 1.13268 − 3.45473I
14.5319 − 8.1806I 0.62316 + 3.31406I
u = −0.794071 + 0.949068I
a = 1.009610 − 0.549638I
b = 0.536925 − 1.276200I
2.94749 − 6.45639I −8.12410 + 6.99903I
u = −0.794071 − 0.949068I
a = 1.009610 + 0.549638I
b = 0.536925 + 1.276200I
2.94749 + 6.45639I −8.12410 − 6.99903I
u = 0.724841 + 0.204931I
a = −0.333556 − 1.065420I
b = 1.20442 − 1.15144I
8.67788 − 5.98718I 1.18729 + 3.50832I
u = 0.724841 − 0.204931I
a = −0.333556 + 1.065420I
b = 1.20442 + 1.15144I
8.67788 + 5.98718I 1.18729 − 3.50832I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.884876 + 0.922327I
a = −2.17724 − 2.75480I
b = 0.33580 − 4.16119I
19.2754 + 3.2655I 2.41004 − 2.43597I
u = 0.884876 − 0.922327I
a = −2.17724 + 2.75480I
b = 0.33580 + 4.16119I
19.2754 − 3.2655I 2.41004 + 2.43597I
u = −0.823258 + 0.994644I
a = −3.10389 + 1.52970I
b = −0.65697 + 3.62118I
13.9657 − 14.5534I −0.35354 + 8.08275I
u = −0.823258 − 0.994644I
a = −3.10389 − 1.52970I
b = −0.65697 − 3.62118I
13.9657 + 14.5534I −0.35354 − 8.08275I
u = −0.168845 + 0.630303I
a = 0.528148 − 0.293642I
b = −0.263129 − 0.261083I
−0.403036 − 0.836917I −8.78276 + 7.97359I
u = −0.168845 − 0.630303I
a = 0.528148 + 0.293642I
b = −0.263129 + 0.261083I
−0.403036 + 0.836917I −8.78276 − 7.97359I
u = 0.465710
a = 0.901293
b = −0.0775306
−1.04458 −9.39350
7
II. I
u
2
= h21u
19
a + 223u
19
+ · · · − 81a − 318, −2u
19
a + u
19
+ · · · + a
2
−
3, u
20
− u
19
+ · · · + u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
9
=
1
0
a
8
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
6
=
a
−0.0830040au
19
− 0.881423u
19
+ ··· + 0.320158a + 1.25692
a
5
=
−0.0750988au
19
− 1.03557u
19
+ ··· + 1.00395a + 0.422925
0.573123au
19
− 1.67589u
19
+ ··· + 0.0750988a + 2.03557
a
7
=
u
2
+ 1
u
2
a
10
=
u
4
+ u
2
+ 1
u
4
a
4
=
−u
11
− 2u
9
− 4u
7
− 4u
5
− 3u
3
−u
11
− u
9
− 2u
7
− u
5
+ u
3
+ u
a
12
=
0.320158au
19
+ 1.25692u
19
+ ··· − 0.806324a − 0.276680
−0.422925au
19
− 0.252964u
19
+ ··· − 0.0830040a − 0.881423
a
11
=
0.320158au
19
+ 1.25692u
19
+ ··· − 0.806324a + 0.723320
0.150198au
19
+ 0.0711462u
19
+ ··· − 0.00790514a − 0.845850
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
19
− 8u
17
− 4u
16
− 28u
15
− 8u
14
− 40u
13
− 24u
12
− 64u
11
−
36u
10
− 64u
9
− 44u
8
− 60u
7
− 44u
6
− 36u
5
− 24u
4
− 24u
3
− 8u
2
− 8u − 6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
9
(u
20
+ 5u
19
+ ··· + 2u + 1)
2
c
2
, c
8
(u
20
+ u
19
+ ··· + u
2
+ 1)
2
c
3
(u
20
+ u
19
+ ··· + 4u + 1)
2
c
4
, c
5
, c
6
c
10
, c
11
, c
12
u
40
+ u
39
+ ··· + 6u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
(y
20
+ 21y
19
+ ··· + 10y + 1)
2
c
2
, c
8
(y
20
+ 5y
19
+ ··· + 2y + 1)
2
c
3
(y
20
+ y
19
+ ··· + 18y + 1)
2
c
4
, c
5
, c
6
c
10
, c
11
, c
12
y
40
+ 31y
39
+ ··· + 16y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.362805 + 0.953641I
a = −0.473240 − 1.154840I
b = −0.17255 + 1.80829I
0.79812 − 6.06247I −8.39660 + 7.82928I
u = −0.362805 + 0.953641I
a = −0.462437 − 0.239841I
b = −0.323095 − 0.392131I
0.79812 − 6.06247I −8.39660 + 7.82928I
u = −0.362805 − 0.953641I
a = −0.473240 + 1.154840I
b = −0.17255 − 1.80829I
0.79812 + 6.06247I −8.39660 − 7.82928I
u = −0.362805 − 0.953641I
a = −0.462437 + 0.239841I
b = −0.323095 + 0.392131I
0.79812 + 6.06247I −8.39660 − 7.82928I
u = −0.161278 + 0.924181I
a = 1.76129 + 0.00175I
b = −0.207836 − 0.314442I
−0.345495 + 0.748059I −11.88926 − 0.17223I
u = −0.161278 + 0.924181I
a = 0.0174489 + 0.1253060I
b = −0.507066 − 0.616738I
−0.345495 + 0.748059I −11.88926 − 0.17223I
u = −0.161278 − 0.924181I
a = 1.76129 − 0.00175I
b = −0.207836 + 0.314442I
−0.345495 − 0.748059I −11.88926 + 0.17223I
u = −0.161278 − 0.924181I
a = 0.0174489 − 0.1253060I
b = −0.507066 + 0.616738I
−0.345495 − 0.748059I −11.88926 + 0.17223I
u = 0.351156 + 0.820236I
a = −0.804851 + 1.144580I
b = 0.34299 − 1.91217I
2.95992 + 1.83292I −4.44386 − 4.26331I
u = 0.351156 + 0.820236I
a = 1.80458 − 0.05939I
b = 0.204203 + 0.494473I
2.95992 + 1.83292I −4.44386 − 4.26331I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.351156 − 0.820236I
a = −0.804851 − 1.144580I
b = 0.34299 + 1.91217I
2.95992 − 1.83292I −4.44386 + 4.26331I
u = 0.351156 − 0.820236I
a = 1.80458 + 0.05939I
b = 0.204203 − 0.494473I
2.95992 − 1.83292I −4.44386 + 4.26331I
u = 0.765553 + 0.891086I
a = 0.208269 + 0.849702I
b = −0.170011 + 0.227981I
4.71375 + 2.89577I −6.31229 − 2.74717I
u = 0.765553 + 0.891086I
a = 0.745886 + 1.016220I
b = −0.004581 + 1.103120I
4.71375 + 2.89577I −6.31229 − 2.74717I
u = 0.765553 − 0.891086I
a = 0.208269 − 0.849702I
b = −0.170011 − 0.227981I
4.71375 − 2.89577I −6.31229 + 2.74717I
u = 0.765553 − 0.891086I
a = 0.745886 − 1.016220I
b = −0.004581 − 1.103120I
4.71375 − 2.89577I −6.31229 + 2.74717I
u = 0.872273 + 0.832901I
a = 0.216187 + 1.258090I
b = −0.909321 + 1.037960I
8.70951 − 3.75485I −2.25682 + 2.44199I
u = 0.872273 + 0.832901I
a = −0.72234 − 3.52789I
b = 1.39922 − 3.79149I
8.70951 − 3.75485I −2.25682 + 2.44199I
u = 0.872273 − 0.832901I
a = 0.216187 − 1.258090I
b = −0.909321 − 1.037960I
8.70951 + 3.75485I −2.25682 − 2.44199I
u = 0.872273 − 0.832901I
a = −0.72234 + 3.52789I
b = 1.39922 + 3.79149I
8.70951 + 3.75485I −2.25682 − 2.44199I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.857922 + 0.867417I
a = 0.583226 − 0.288576I
b = 0.935046 − 0.649367I
10.37890 − 1.55876I 0.11661 + 2.37917I
u = −0.857922 + 0.867417I
a = −1.53336 + 3.87998I
b = 1.18542 + 4.43172I
10.37890 − 1.55876I 0.11661 + 2.37917I
u = −0.857922 − 0.867417I
a = 0.583226 + 0.288576I
b = 0.935046 + 0.649367I
10.37890 + 1.55876I 0.11661 − 2.37917I
u = −0.857922 − 0.867417I
a = −1.53336 − 3.87998I
b = 1.18542 − 4.43172I
10.37890 + 1.55876I 0.11661 − 2.37917I
u = −0.828456 + 0.942427I
a = 0.115226 − 1.123400I
b = −0.965278 − 0.339588I
10.14230 − 4.70967I −0.36261 + 2.80351I
u = −0.828456 + 0.942427I
a = −3.47576 + 2.62759I
b = −0.50717 + 4.57832I
10.14230 − 4.70967I −0.36261 + 2.80351I
u = −0.828456 − 0.942427I
a = 0.115226 + 1.123400I
b = −0.965278 + 0.339588I
10.14230 + 4.70967I −0.36261 − 2.80351I
u = −0.828456 − 0.942427I
a = −3.47576 − 2.62759I
b = −0.50717 − 4.57832I
10.14230 + 4.70967I −0.36261 − 2.80351I
u = 0.818606 + 0.971044I
a = 1.029030 + 0.400942I
b = 0.70511 + 1.34475I
8.27570 + 10.03250I −3.16919 − 7.28178I
u = 0.818606 + 0.971044I
a = −3.45786 − 1.83520I
b = −0.82034 − 4.00237I
8.27570 + 10.03250I −3.16919 − 7.28178I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.818606 − 0.971044I
a = 1.029030 − 0.400942I
b = 0.70511 − 1.34475I
8.27570 − 10.03250I −3.16919 + 7.28178I
u = 0.818606 − 0.971044I
a = −3.45786 + 1.83520I
b = −0.82034 + 4.00237I
8.27570 − 10.03250I −3.16919 + 7.28178I
u = 0.483351 + 0.483677I
a = −1.169990 − 0.258941I
b = 1.22095 − 1.10150I
3.96963 + 1.37271I −0.87985 − 4.43993I
u = 0.483351 + 0.483677I
a = 0.440357 + 1.167290I
b = −0.006838 + 0.783868I
3.96963 + 1.37271I −0.87985 − 4.43993I
u = 0.483351 − 0.483677I
a = −1.169990 + 0.258941I
b = 1.22095 + 1.10150I
3.96963 − 1.37271I −0.87985 + 4.43993I
u = 0.483351 − 0.483677I
a = 0.440357 − 1.167290I
b = −0.006838 − 0.783868I
3.96963 − 1.37271I −0.87985 + 4.43993I
u = −0.580477 + 0.222282I
a = 0.879103 + 0.241455I
b = −0.061684 + 0.219008I
3.03554 + 2.59904I −2.40613 − 3.16627I
u = −0.580477 + 0.222282I
a = −0.700755 + 1.190060I
b = 1.16284 + 1.01300I
3.03554 + 2.59904I −2.40613 − 3.16627I
u = −0.580477 − 0.222282I
a = 0.879103 − 0.241455I
b = −0.061684 − 0.219008I
3.03554 − 2.59904I −2.40613 + 3.16627I
u = −0.580477 − 0.222282I
a = −0.700755 − 1.190060I
b = 1.16284 − 1.01300I
3.03554 − 2.59904I −2.40613 + 3.16627I
14
III. I
u
3
= h−u
5
+ b − u, −u
4
− u
2
+ a + u − 2, u
6
+ u
4
+ 2u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
9
=
1
0
a
8
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
6
=
u
4
+ u
2
− u + 2
u
5
+ u
a
5
=
u
4
− u + 1
u
5
− u
2
+ u
a
7
=
u
2
+ 1
u
2
a
10
=
u
4
+ u
2
+ 1
u
4
a
4
=
u
u
3
+ u
a
12
=
u
5
+ u
4
+ u
3
+ u
2
+ 2u + 1
u
5
+ u
3
+ u − 1
a
11
=
u
5
+ 2u
4
+ u
3
+ 2u
2
+ 2u + 2
u
5
+ u
4
+ u
3
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
4
− 4u
2
− 8
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
8
u
6
+ u
4
+ 2u
2
+ 1
c
3
u
6
c
4
, c
5
, c
6
c
10
, c
11
, c
12
(u
2
+ 1)
3
c
9
(u
3
+ u
2
+ 2u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
8
(y
3
+ y
2
+ 2y + 1)
2
c
3
y
6
c
4
, c
5
, c
6
c
10
, c
11
, c
12
(y + 1)
6
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.744862 + 0.877439I
a = −0.622301 − 0.132577I
b = − 1.000000I
6.31400 + 2.82812I −0.49024 − 2.97945I
u = 0.744862 − 0.877439I
a = −0.622301 + 0.132577I
b = 1.000000I
6.31400 − 2.82812I −0.49024 + 2.97945I
u = −0.744862 + 0.877439I
a = 0.86742 − 1.62230I
b = − 1.000000I
6.31400 − 2.82812I −0.49024 + 2.97945I
u = −0.744862 − 0.877439I
a = 0.86742 + 1.62230I
b = 1.000000I
6.31400 + 2.82812I −0.49024 − 2.97945I
u = 0.754878I
a = 1.75488 − 0.75488I
b = 1.000000I
2.17641 −7.01950
u = − 0.754878I
a = 1.75488 + 0.75488I
b = − 1.000000I
2.17641 −7.01950
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
((u
3
− u
2
+ 2u − 1)
2
)(u
20
+ 5u
19
+ ··· + 2u + 1)
2
· (u
27
+ 7u
26
+ ··· − 7u − 4)
c
2
, c
8
(u
6
+ u
4
+ 2u
2
+ 1)(u
20
+ u
19
+ ··· + u
2
+ 1)
2
(u
27
− 3u
26
+ ··· + u + 2)
c
3
u
6
(u
20
+ u
19
+ ··· + 4u + 1)
2
(u
27
− 3u
26
+ ··· + 192u + 128)
c
4
, c
5
, c
6
c
10
, c
11
, c
12
((u
2
+ 1)
3
)(u
27
+ 15u
25
+ ··· + 3u + 1)(u
40
+ u
39
+ ··· + 6u + 1)
c
9
((u
3
+ u
2
+ 2u + 1)
2
)(u
20
+ 5u
19
+ ··· + 2u + 1)
2
· (u
27
+ 7u
26
+ ··· − 7u − 4)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
9
((y
3
+ 3y
2
+ 2y −1)
2
)(y
20
+ 21y
19
+ ··· + 10y + 1)
2
· (y
27
+ 27y
26
+ ··· + 385y −16)
c
2
, c
8
((y
3
+ y
2
+ 2y + 1)
2
)(y
20
+ 5y
19
+ ··· + 2y + 1)
2
· (y
27
+ 7y
26
+ ··· − 7y −4)
c
3
y
6
(y
20
+ y
19
+ ··· + 18y + 1)
2
(y
27
+ y
26
+ ··· − 323584y −16384)
c
4
, c
5
, c
6
c
10
, c
11
, c
12
((y + 1)
6
)(y
27
+ 30y
26
+ ··· − 7y −1)(y
40
+ 31y
39
+ ··· + 16y + 1)
20
