12a
1220
(K12a
1220
)
A knot diagram
1
Linearized knot diagam
5 6 7 11 10 3 4 12 1 2 8 9
Solving Sequence
8,12
9 1
4,10
7 3 6 11 5 2
c
8
c
12
c
9
c
7
c
3
c
6
c
11
c
4
c
1
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hb + u, −2u
8
− 3u
7
+ 9u
6
+ 11u
5
− 14u
4
− 8u
3
+ 11u
2
+ a + 2,
u
9
+ 2u
8
− 4u
7
− 8u
6
+ 5u
5
+ 8u
4
− 4u
3
− 3u
2
− u − 1i
I
u
2
= h−2.05889 × 10
21
u
39
− 4.20027 × 10
20
u
38
+ ··· + 5.82584 × 10
21
b + 1.03593 × 10
20
,
2.35540 × 10
21
u
39
+ 3.66482 × 10
21
u
38
+ ··· + 2.91292 × 10
21
a + 2.62000 × 10
21
, u
40
+ 2u
39
+ ··· + 11u + 1i
I
u
3
= hb − 1, a + 2, u
2
− u − 1i
I
u
4
= h2b + a, a
2
− 2a − 4, u + 1i
* 4 irreducible components of dim
C
= 0, with total 53 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
= hb + u, −2u
8
− 3u
7
+ · · · + a + 2, u
9
+ 2u
8
+ · · · − u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
2u
8
+ 3u
7
− 9u
6
− 11u
5
+ 14u
4
+ 8u
3
− 11u
2
− 2
−u
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
7
=
u
8
+ u
7
− 5u
6
− 4u
5
+ 8u
4
+ 3u
3
− 6u
2
− 1
u
2
a
3
=
u
8
+ 2u
7
− 5u
6
− 8u
5
+ 9u
4
+ 6u
3
− 8u
2
− 1
u
3
− u
a
6
=
u
8
+ 2u
7
− 5u
6
− 8u
5
+ 10u
4
+ 7u
3
− 9u
2
− 2
−u
4
+ 2u
2
a
11
=
−u
u
a
5
=
u
8
+ 2u
7
− 5u
6
− 8u
5
+ 9u
4
+ 7u
3
− 8u
2
− u − 1
u
8
+ u
7
− 4u
6
− 3u
5
+ 5u
4
+ u
3
− 3u
2
− 1
a
2
=
−u
8
− u
7
+ 5u
6
+ 3u
5
− 8u
4
− u
3
+ 5u
2
+ u
u
5
− 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 16u
8
+ 24u
7
− 76u
6
− 84u
5
+ 124u
4
+ 40u
3
− 88u
2
+ 20u − 22
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
9
− 2u
6
+ 5u
5
− 2u
4
− 4u
3
− 3u
2
+ 3u + 1
c
2
, c
3
, c
6
c
7
, c
8
, c
9
c
11
, c
12
u
9
− 2u
8
− 4u
7
+ 8u
6
+ 5u
5
− 8u
4
− 4u
3
+ 3u
2
− u + 1
c
4
u
9
+ 13u
8
+ ··· + 320u + 64
c
5
u
9
+ 13u
8
+ 71u
7
+ 214u
6
+ 390u
5
+ 435u
4
+ 279u
3
+ 78u
2
− 8u − 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
9
+ 10y
7
− 12y
6
+ 23y
5
− 56y
4
+ 38y
3
− 29y
2
+ 15y −1
c
2
, c
3
, c
6
c
7
, c
8
, c
9
c
11
, c
12
y
9
− 12y
8
+ 58y
7
− 144y
6
+ 195y
5
− 140y
4
+ 38y
3
+ 15y
2
− 5y −1
c
4
y
9
− 21y
8
+ ··· + 8192y −4096
c
5
y
9
− 27y
8
+ ··· + 1312y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.927341 + 0.453196I
a = 0.991720 − 0.824614I
b = −0.927341 − 0.453196I
4.81531 + 7.88365I 13.1197 − 8.5237I
u = 0.927341 − 0.453196I
a = 0.991720 + 0.824614I
b = −0.927341 + 0.453196I
4.81531 − 7.88365I 13.1197 + 8.5237I
u = −0.659939
a = −5.89270
b = 0.659939
2.11613 −57.9970
u = −1.43521
a = −2.20604
b = 1.43521
8.30534 10.1970
u = 0.002669 + 0.448114I
a = 0.830042 − 0.971880I
b = −0.002669 − 0.448114I
−0.78700 − 1.41074I 1.25059 + 3.40619I
u = 0.002669 − 0.448114I
a = 0.830042 + 0.971880I
b = −0.002669 + 0.448114I
−0.78700 + 1.41074I 1.25059 − 3.40619I
u = 1.66419
a = 3.91567
b = −1.66419
18.8023 6.12260
u = −1.71453 + 0.16075I
a = −2.23022 − 0.99164I
b = 1.71453 − 0.16075I
−16.1728 − 13.0673I 15.4687 + 5.5944I
u = −1.71453 − 0.16075I
a = −2.23022 + 0.99164I
b = 1.71453 + 0.16075I
−16.1728 + 13.0673I 15.4687 − 5.5944I
5
II. I
u
2
= h−2.06 × 10
21
u
39
− 4.20 × 10
20
u
38
+ · · · + 5.83 × 10
21
b + 1.04 ×
10
20
, 2.36 × 10
21
u
39
+ 3.66 × 10
21
u
38
+ · · · + 2.91 × 10
21
a + 2.62 ×
10
21
, u
40
+ 2u
39
+ · · · + 11u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
−0.808605u
39
− 1.25813u
38
+ ··· − 36.2164u − 0.899441
0.353407u
39
+ 0.0720972u
38
+ ··· − 5.91849u − 0.0177816
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
7
=
−2.15713u
39
− 2.78559u
38
+ ··· − 51.4085u − 3.21686
1.21823u
39
+ 1.13143u
38
+ ··· + 2.86886u + 0.489002
a
3
=
−0.342472u
39
− 0.586708u
38
+ ··· − 13.6913u + 0.476673
−0.105307u
39
+ 0.107577u
38
+ ··· + 3.48164u + 0.766785
a
6
=
−1.72446u
39
− 2.57322u
38
+ ··· − 46.0801u − 1.47304
0.444440u
39
+ 0.320947u
38
+ ··· − 9.97662u − 0.424757
a
11
=
−u
u
a
5
=
−2.06730u
39
− 2.12671u
38
+ ··· − 39.7036u − 1.17508
1.61210u
39
+ 0.940683u
38
+ ··· − 2.43132u + 0.257852
a
2
=
−3.62228u
39
− 3.22253u
38
+ ··· − 46.8489u − 5.52801
3.45055u
39
+ 2.41041u
38
+ ··· + 6.30294u + 0.399746
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
2825773251700395997919
728229534914329869523
u
39
−
2538801677611388911625
728229534914329869523
u
38
+ ··· −
40264361670734015524833
728229534914329869523
u +
2631129163027084440997
728229534914329869523
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
u
40
+ 3u
39
+ ··· + 11u
2
+ 4
c
2
, c
3
, c
6
c
7
, c
8
, c
9
c
11
, c
12
u
40
− 2u
39
+ ··· − 11u + 1
c
4
(u
20
− 7u
19
+ ··· − 191u + 47)
2
c
5
(u
20
− 6u
19
+ ··· − 16u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
40
+ 13y
39
+ ··· + 88y + 16
c
2
, c
3
, c
6
c
7
, c
8
, c
9
c
11
, c
12
y
40
− 50y
39
+ ··· − 49y + 1
c
4
(y
20
+ 3y
19
+ ··· + 16629y + 2209)
2
c
5
(y
20
− 24y
19
+ ··· − 210y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.964193 + 0.183114I
a = 1.81237 − 1.07233I
b = −1.68373 − 0.13054I
12.94220 + 2.92572I 16.8415 − 2.9709I
u = 0.964193 − 0.183114I
a = 1.81237 + 1.07233I
b = −1.68373 + 0.13054I
12.94220 − 2.92572I 16.8415 + 2.9709I
u = −0.871135 + 0.550509I
a = 0.457497 + 0.668616I
b = −0.858752 − 0.047670I
4.04379 − 0.34594I 19.0261 − 0.3312I
u = −0.871135 − 0.550509I
a = 0.457497 − 0.668616I
b = −0.858752 + 0.047670I
4.04379 + 0.34594I 19.0261 + 0.3312I
u = −0.094224 + 0.891903I
a = −0.324167 − 0.354300I
b = 1.67359 − 0.07029I
10.57610 − 5.30216I 12.68744 + 4.85316I
u = −0.094224 − 0.891903I
a = −0.324167 + 0.354300I
b = 1.67359 + 0.07029I
10.57610 + 5.30216I 12.68744 − 4.85316I
u = 0.841679 + 0.285962I
a = −0.249728 + 0.281271I
b = 0.079408 + 0.721551I
1.73170 + 3.96676I 10.64355 − 7.18805I
u = 0.841679 − 0.285962I
a = −0.249728 − 0.281271I
b = 0.079408 − 0.721551I
1.73170 − 3.96676I 10.64355 + 7.18805I
u = 0.858752 + 0.047670I
a = −0.911277 − 0.334390I
b = 0.871135 − 0.550509I
4.04379 − 0.34594I 19.0261 − 0.3312I
u = 0.858752 − 0.047670I
a = −0.911277 + 0.334390I
b = 0.871135 + 0.550509I
4.04379 + 0.34594I 19.0261 + 0.3312I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.996751 + 0.567865I
a = −1.56536 + 1.35978I
b = 1.69022 + 0.12129I
13.9204 + 10.1318I 14.3223 − 6.7026I
u = 0.996751 − 0.567865I
a = −1.56536 − 1.35978I
b = 1.69022 − 0.12129I
13.9204 − 10.1318I 14.3223 + 6.7026I
u = −1.15373
a = 0.932910
b = −0.386843
2.16630 −1.22340
u = −0.952961 + 0.719204I
a = −1.31275 − 0.94519I
b = 1.68108 + 0.01576I
13.06170 − 0.07749I 17.5894 + 0.I
u = −0.952961 − 0.719204I
a = −1.31275 + 0.94519I
b = 1.68108 − 0.01576I
13.06170 + 0.07749I 17.5894 + 0.I
u = −0.745297
a = 6.94927
b = −1.62727
10.1903 −24.0970
u = −0.079408 + 0.721551I
a = −0.182590 + 0.422871I
b = −0.841679 + 0.285962I
1.73170 − 3.96676I 10.64355 + 7.18805I
u = −0.079408 − 0.721551I
a = −0.182590 − 0.422871I
b = −0.841679 − 0.285962I
1.73170 + 3.96676I 10.64355 − 7.18805I
u = −0.651290 + 0.157168I
a = 0.836528 − 0.648564I
b = 0.154389 − 0.087035I
1.239960 − 0.397086I 8.64524 + 0.31446I
u = −0.651290 − 0.157168I
a = 0.836528 + 0.648564I
b = 0.154389 + 0.087035I
1.239960 + 0.397086I 8.64524 − 0.31446I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.224632 + 0.357420I
a = −2.05994 + 0.87263I
b = −1.63217 + 0.03725I
9.29517 − 1.07904I 8.55163 − 1.79539I
u = −0.224632 − 0.357420I
a = −2.05994 − 0.87263I
b = −1.63217 − 0.03725I
9.29517 + 1.07904I 8.55163 + 1.79539I
u = 0.386843
a = −2.78232
b = 1.15373
2.16630 −1.22340
u = 1.62727
a = −3.18279
b = 0.745297
10.1903 0
u = 1.63217 + 0.03725I
a = 0.105394 + 0.568790I
b = 0.224632 + 0.357420I
9.29517 + 1.07904I 0
u = 1.63217 − 0.03725I
a = 0.105394 − 0.568790I
b = 0.224632 − 0.357420I
9.29517 − 1.07904I 0
u = −1.67359 + 0.07029I
a = −0.213109 + 0.143861I
b = 0.094224 − 0.891903I
10.57610 − 5.30216I 0
u = −1.67359 − 0.07029I
a = −0.213109 − 0.143861I
b = 0.094224 + 0.891903I
10.57610 + 5.30216I 0
u = −1.68108 + 0.01576I
a = −1.148200 − 0.036582I
b = 0.952961 + 0.719204I
13.06170 + 0.07749I 0
u = −1.68108 − 0.01576I
a = −1.148200 + 0.036582I
b = 0.952961 − 0.719204I
13.06170 − 0.07749I 0
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.68373 + 0.13054I
a = 1.115450 − 0.503449I
b = −0.964193 − 0.183114I
12.94220 + 2.92572I 0
u = 1.68373 − 0.13054I
a = 1.115450 + 0.503449I
b = −0.964193 + 0.183114I
12.94220 − 2.92572I 0
u = −1.69022 + 0.12129I
a = 1.353200 + 0.373072I
b = −0.996751 + 0.567865I
13.9204 − 10.1318I 0
u = −1.69022 − 0.12129I
a = 1.353200 − 0.373072I
b = −0.996751 − 0.567865I
13.9204 + 10.1318I 0
u = −1.70429 + 0.04276I
a = 2.28176 + 0.53396I
b = −1.74240 + 0.19593I
−17.0616 − 3.7959I 0
u = −1.70429 − 0.04276I
a = 2.28176 − 0.53396I
b = −1.74240 − 0.19593I
−17.0616 + 3.7959I 0
u = 1.74240 + 0.19593I
a = −2.16516 + 0.70975I
b = 1.70429 + 0.04276I
−17.0616 + 3.7959I 0
u = 1.74240 − 0.19593I
a = −2.16516 − 0.70975I
b = 1.70429 − 0.04276I
−17.0616 − 3.7959I 0
u = −0.154389 + 0.087035I
a = 3.71156 − 1.49520I
b = 0.651290 − 0.157168I
1.239960 − 0.397086I 8.64524 + 0.31446I
u = −0.154389 − 0.087035I
a = 3.71156 + 1.49520I
b = 0.651290 + 0.157168I
1.239960 + 0.397086I 8.64524 − 0.31446I
12
III. I
u
3
= hb − 1, a + 2, u
2
− u − 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u − 1
a
1
=
u
−u − 1
a
4
=
−2
1
a
10
=
−u
u
a
7
=
−1
1
a
3
=
−1
0
a
6
=
−2
1
a
11
=
−u
u
a
5
=
u − 1
−u
a
2
=
1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 17
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
(u + 1)
2
c
4
, c
5
, c
11
c
12
u
2
+ u − 1
c
6
, c
7
(u − 1)
2
c
8
, c
9
u
2
− u − 1
c
10
u
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
6
, c
7
(y −1)
2
c
4
, c
5
, c
8
c
9
, c
11
, c
12
y
2
− 3y + 1
c
10
y
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −2.00000
b = 1.00000
2.63189 17.0000
u = 1.61803
a = −2.00000
b = 1.00000
10.5276 17.0000
16
IV. I
u
4
= h2b + a, a
2
− 2a − 4, u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
−1
a
9
=
1
−1
a
1
=
−1
0
a
4
=
a
−
1
2
a
a
10
=
0
−1
a
7
=
−a − 1
1
2
a + 1
a
3
=
−
1
2
a − 2
1
2
a + 1
a
6
=
1
2
a
−
1
2
a
a
11
=
1
−1
a
5
=
1
2
a
0
a
2
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 17
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
2
c
2
, c
3
u
2
− u − 1
c
4
, c
5
, c
6
c
7
u
2
+ u − 1
c
8
, c
9
, c
10
(u + 1)
2
c
11
, c
12
(u − 1)
2
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
2
c
2
, c
3
, c
4
c
5
, c
6
, c
7
y
2
− 3y + 1
c
8
, c
9
, c
10
c
11
, c
12
(y −1)
2
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.23607
b = 0.618034
2.63189 17.0000
u = −1.00000
a = 3.23607
b = −1.61803
10.5276 17.0000
20
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
10
u
2
(u + 1)
2
(u
9
− 2u
6
+ 5u
5
− 2u
4
− 4u
3
− 3u
2
+ 3u + 1)
· (u
40
+ 3u
39
+ ··· + 11u
2
+ 4)
c
2
, c
3
, c
8
c
9
(u + 1)
2
(u
2
− u − 1)
· (u
9
− 2u
8
− 4u
7
+ 8u
6
+ 5u
5
− 8u
4
− 4u
3
+ 3u
2
− u + 1)
· (u
40
− 2u
39
+ ··· − 11u + 1)
c
4
((u
2
+ u − 1)
2
)(u
9
+ 13u
8
+ ··· + 320u + 64)
· (u
20
− 7u
19
+ ··· − 191u + 47)
2
c
5
(u
2
+ u − 1)
2
· (u
9
+ 13u
8
+ 71u
7
+ 214u
6
+ 390u
5
+ 435u
4
+ 279u
3
+ 78u
2
− 8u − 8)
· (u
20
− 6u
19
+ ··· − 16u + 1)
2
c
6
, c
7
, c
11
c
12
(u − 1)
2
(u
2
+ u − 1)
· (u
9
− 2u
8
− 4u
7
+ 8u
6
+ 5u
5
− 8u
4
− 4u
3
+ 3u
2
− u + 1)
· (u
40
− 2u
39
+ ··· − 11u + 1)
21
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
10
y
2
(y −1)
2
(y
9
+ 10y
7
+ ··· + 15y −1)
· (y
40
+ 13y
39
+ ··· + 88y + 16)
c
2
, c
3
, c
6
c
7
, c
8
, c
9
c
11
, c
12
(y −1)
2
(y
2
− 3y + 1)
· (y
9
− 12y
8
+ 58y
7
− 144y
6
+ 195y
5
− 140y
4
+ 38y
3
+ 15y
2
− 5y −1)
· (y
40
− 50y
39
+ ··· − 49y + 1)
c
4
((y
2
− 3y + 1)
2
)(y
9
− 21y
8
+ ··· + 8192y −4096)
· (y
20
+ 3y
19
+ ··· + 16629y + 2209)
2
c
5
((y
2
− 3y + 1)
2
)(y
9
− 27y
8
+ ··· + 1312y −64)
· (y
20
− 24y
19
+ ··· − 210y + 1)
2
22
