12a
0259
(K12a
0259
)
A knot diagram
1
Linearized knot diagam
3 6 7 10 1 2 5 12 11 4 9 8
Solving Sequence
4,11
10 5 9 12 8 1 6 7 3 2
c
10
c
4
c
9
c
11
c
8
c
12
c
5
c
7
c
3
c
2
c
1
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
55
+ 5u
53
+ ··· + 2u − 1i
I
u
2
= hu
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 57 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
55
+ 5u
53
+ · · · + 2u − 1i
(i) Arc colorings
a
4
=
0
u
a
11
=
1
0
a
10
=
1
−u
2
a
5
=
−u
u
3
+ u
a
9
=
u
2
+ 1
−u
2
a
12
=
u
4
+ u
2
+ 1
−u
4
a
8
=
u
6
+ u
4
+ 2u
2
+ 1
−u
6
− u
2
a
1
=
u
8
+ u
6
+ 3u
4
+ 2u
2
+ 1
−u
8
− 2u
4
a
6
=
−u
19
− 2u
17
− 8u
15
− 12u
13
− 21u
11
− 22u
9
− 20u
7
− 12u
5
− 5u
3
− 2u
u
19
+ u
17
+ 6u
15
+ 5u
13
+ 11u
11
+ 7u
9
+ 6u
7
+ 2u
5
+ u
3
+ u
a
7
=
u
10
+ u
8
+ 4u
6
+ 3u
4
+ 3u
2
+ 1
−u
12
− 2u
10
− 4u
8
− 6u
6
− 3u
4
− 2u
2
a
3
=
u
21
+ 2u
19
+ ··· + 6u
3
+ u
−u
23
− 3u
21
+ ··· − 2u
3
+ u
a
2
=
−u
52
− 5u
50
+ ··· + 3u
2
+ 1
u
54
+ 6u
52
+ ··· − 4u
4
+ u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
54
− 4u
53
+ ··· + 16u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
55
+ 30u
54
+ ··· − 2u − 1
c
2
, c
6
u
55
− 2u
54
+ ··· − 4u + 1
c
3
, c
5
u
55
+ 2u
54
+ ··· + 20u + 1
c
4
, c
10
u
55
+ 5u
53
+ ··· + 2u + 1
c
7
u
55
− 10u
54
+ ··· − 10716u + 797
c
8
, c
9
, c
11
c
12
u
55
− 10u
54
+ ··· − 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
55
− 10y
54
+ ··· − 10y −1
c
2
, c
6
y
55
+ 30y
54
+ ··· − 2y −1
c
3
, c
5
y
55
− 50y
54
+ ··· + 94y −1
c
4
, c
10
y
55
+ 10y
54
+ ··· − 2y −1
c
7
y
55
− 30y
54
+ ··· + 41338098y −635209
c
8
, c
9
, c
11
c
12
y
55
+ 70y
54
+ ··· − 26y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.592575 + 0.744780I
−3.48431 + 2.23232I −9.07962 − 4.25588I
u = −0.592575 − 0.744780I
−3.48431 − 2.23232I −9.07962 + 4.25588I
u = 0.459054 + 0.825274I
0.68501 − 1.99360I 2.38430 + 3.67911I
u = 0.459054 − 0.825274I
0.68501 + 1.99360I 2.38430 − 3.67911I
u = 0.707571 + 0.613437I
−8.01052 − 2.80329I −9.21279 + 3.18913I
u = 0.707571 − 0.613437I
−8.01052 + 2.80329I −9.21279 − 3.18913I
u = −0.571994 + 0.897415I
−3.20861 + 6.06664I −2.24668 − 6.88488I
u = −0.571994 − 0.897415I
−3.20861 − 6.06664I −2.24668 + 6.88488I
u = 0.597490 + 0.892631I
−7.10023 − 2.03443I −6.83549 + 3.30973I
u = 0.597490 − 0.892631I
−7.10023 + 2.03443I −6.83549 − 3.30973I
u = 0.573656 + 0.915528I
−6.37071 − 10.79630I −5.21230 + 9.86505I
u = 0.573656 − 0.915528I
−6.37071 + 10.79630I −5.21230 − 9.86505I
u = 0.709473 + 0.569448I
−7.49807 + 6.03817I −8.34541 − 3.60218I
u = 0.709473 − 0.569448I
−7.49807 − 6.03817I −8.34541 + 3.60218I
u = −0.163111 + 0.892318I
−2.29512 + 6.47172I 0.57317 − 7.45825I
u = −0.163111 − 0.892318I
−2.29512 − 6.47172I 0.57317 + 7.45825I
u = −0.688302 + 0.586086I
−4.22010 − 1.37659I −5.30101 + 0.34335I
u = −0.688302 − 0.586086I
−4.22010 + 1.37659I −5.30101 − 0.34335I
u = −0.226650 + 0.864103I
−2.66284 − 1.79951I −0.604898 − 0.931152I
u = −0.226650 − 0.864103I
−2.66284 + 1.79951I −0.604898 + 0.931152I
u = 0.158391 + 0.848063I
0.73193 − 2.06229I 4.26089 + 4.61664I
u = 0.158391 − 0.848063I
0.73193 + 2.06229I 4.26089 − 4.61664I
u = 0.032760 + 0.848874I
2.77969 − 2.00175I 7.49779 + 4.64090I
u = 0.032760 − 0.848874I
2.77969 + 2.00175I 7.49779 − 4.64090I
u = 0.406358 + 0.693825I
0.093723 − 1.399000I 1.38184 + 4.66196I
u = 0.406358 − 0.693825I
0.093723 + 1.399000I 1.38184 − 4.66196I
u = −0.548612 + 0.514408I
−1.05882 − 2.06467I −4.27290 + 3.60449I
u = −0.548612 − 0.514408I
−1.05882 + 2.06467I −4.27290 − 3.60449I
u = 0.894461 + 0.903176I
−8.78655 + 1.96573I 0
u = 0.894461 − 0.903176I
−8.78655 − 1.96573I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.884957 + 0.914000I
−7.60137 + 2.47945I 0
u = −0.884957 − 0.914000I
−7.60137 − 2.47945I 0
u = −0.876087 + 0.934697I
−7.53561 + 4.03332I 0
u = −0.876087 − 0.934697I
−7.53561 − 4.03332I 0
u = 0.874986 + 0.947470I
−8.64564 − 8.50368I 0
u = 0.874986 − 0.947470I
−8.64564 + 8.50368I 0
u = 0.920221 + 0.904244I
−12.94750 + 1.94957I 0
u = 0.920221 − 0.904244I
−12.94750 − 1.94957I 0
u = −0.924130 + 0.901108I
−16.2298 − 6.8466I 0
u = −0.924130 − 0.901108I
−16.2298 + 6.8466I 0
u = 0.899259 + 0.932836I
−12.49120 − 3.31619I 0
u = 0.899259 − 0.932836I
−12.49120 + 3.31619I 0
u = −0.923024 + 0.910030I
−17.0363 + 2.2847I 0
u = −0.923024 − 0.910030I
−17.0363 − 2.2847I 0
u = 0.889204 + 0.964577I
−12.7512 − 8.6150I 0
u = 0.889204 − 0.964577I
−12.7512 + 8.6150I 0
u = −0.888934 + 0.968984I
−16.0087 + 13.5233I 0
u = −0.888934 − 0.968984I
−16.0087 − 13.5233I 0
u = −0.894950 + 0.963625I
−16.8614 + 4.4090I 0
u = −0.894950 − 0.963625I
−16.8614 − 4.4090I 0
u = −0.569536 + 0.038892I
−5.25644 + 4.29003I −8.82901 − 3.69791I
u = −0.569536 − 0.038892I
−5.25644 − 4.29003I −8.82901 + 3.69791I
u = 0.522568
−1.89518 −5.69250
u = 0.368693 + 0.284269I
−0.336797 − 1.233170I −4.20998 + 5.17134I
u = 0.368693 − 0.284269I
−0.336797 + 1.233170I −4.20998 − 5.17134I
6
II. I
u
2
= hu
2
+ u + 1i
(i) Arc colorings
a
4
=
0
u
a
11
=
1
0
a
10
=
1
u + 1
a
5
=
−u
u + 1
a
9
=
−u
u + 1
a
12
=
0
−u
a
8
=
−u
u
a
1
=
−1
−u + 1
a
6
=
0
−u
a
7
=
1
u − 1
a
3
=
u
−u − 1
a
2
=
u
−u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −12u − 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
u
2
+ u + 1
c
3
, c
4
, c
5
c
8
, c
9
, c
10
c
11
, c
12
u
2
− u + 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y
2
+ y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
6.08965I 0. − 10.39230I
u = −0.500000 − 0.866025I
− 6.08965I 0. + 10.39230I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
+ u + 1)(u
55
+ 30u
54
+ ··· − 2u − 1)
c
2
, c
6
(u
2
+ u + 1)(u
55
− 2u
54
+ ··· − 4u + 1)
c
3
, c
5
(u
2
− u + 1)(u
55
+ 2u
54
+ ··· + 20u + 1)
c
4
, c
10
(u
2
− u + 1)(u
55
+ 5u
53
+ ··· + 2u + 1)
c
7
(u
2
+ u + 1)(u
55
− 10u
54
+ ··· − 10716u + 797)
c
8
, c
9
, c
11
c
12
(u
2
− u + 1)(u
55
− 10u
54
+ ··· − 2u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
2
+ y + 1)(y
55
− 10y
54
+ ··· − 10y −1)
c
2
, c
6
(y
2
+ y + 1)(y
55
+ 30y
54
+ ··· − 2y −1)
c
3
, c
5
(y
2
+ y + 1)(y
55
− 50y
54
+ ··· + 94y −1)
c
4
, c
10
(y
2
+ y + 1)(y
55
+ 10y
54
+ ··· − 2y −1)
c
7
(y
2
+ y + 1)(y
55
− 30y
54
+ ··· + 4.13381 × 10
7
y −635209)
c
8
, c
9
, c
11
c
12
(y
2
+ y + 1)(y
55
+ 70y
54
+ ··· − 26y −1)
12
