12a
0519
(K12a
0519
)
A knot diagram
1
Linearized knot diagam
3 7 8 9 11 12 2 4 1 5 6 10
Solving Sequence
2,8
7 3 4 9 5 1 10 11 12 6
c
7
c
2
c
3
c
8
c
4
c
1
c
9
c
10
c
12
c
6
c
5
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
55
+ u
54
+ ··· − 2u − 1i
* 1 irreducible components of dim
C
= 0, with total 55 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
+ u
54
+ · · · − 2u − 1i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
9
=
−u
6
− u
4
+ 1
u
6
+ 2u
4
+ u
2
a
5
=
u
9
+ 2u
7
+ u
5
− 2u
3
− u
−u
9
− 3u
7
− 3u
5
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
10
=
−u
14
− 3u
12
− 4u
10
− u
8
+ 1
−u
16
− 4u
14
− 8u
12
− 8u
10
− 4u
8
+ 2u
6
+ 4u
4
+ 2u
2
a
11
=
u
34
+ 9u
32
+ ··· − u
2
+ 1
−u
34
− 10u
32
+ ··· + 6u
4
+ 3u
2
a
12
=
u
25
+ 6u
23
+ ··· + 2u
3
+ u
u
27
+ 7u
25
+ ··· + 3u
3
+ u
a
6
=
−u
50
− 13u
48
+ ··· − u
2
+ 1
−u
52
− 14u
50
+ ··· − 18u
6
− 5u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
53
+ 4u
52
+ ··· − 4u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
55
+ 31u
54
+ ··· + 4u − 1
c
2
, c
7
u
55
+ u
54
+ ··· − 2u − 1
c
3
, c
4
, c
8
u
55
− u
54
+ ··· + u − 2
c
5
, c
6
, c
10
c
11
u
55
+ u
54
+ ··· − 2u − 1
c
9
, c
12
u
55
− 11u
54
+ ··· + 8u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
55
− 13y
54
+ ··· + 68y −1
c
2
, c
7
y
55
+ 31y
54
+ ··· + 4y −1
c
3
, c
4
, c
8
y
55
− 57y
54
+ ··· + 293y −4
c
5
, c
6
, c
10
c
11
y
55
− 61y
54
+ ··· + 4y −1
c
9
, c
12
y
55
+ 23y
54
+ ··· − 36y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.102308 + 0.993927I
−1.51781 + 1.35357I −14.4750 − 4.2304I
u = −0.102308 − 0.993927I
−1.51781 − 1.35357I −14.4750 + 4.2304I
u = −0.485253 + 0.877654I
−4.60903 − 0.50394I −9.84299 + 3.20622I
u = −0.485253 − 0.877654I
−4.60903 + 0.50394I −9.84299 − 3.20622I
u = 0.479095 + 0.928930I
1.53724 + 2.73421I −6.40413 − 2.99488I
u = 0.479095 − 0.928930I
1.53724 − 2.73421I −6.40413 + 2.99488I
u = −0.328532 + 0.998033I
−3.02866 − 2.73775I −17.4070 + 6.3920I
u = −0.328532 − 0.998033I
−3.02866 + 2.73775I −17.4070 − 6.3920I
u = 0.116890 + 1.065220I
−8.61843 − 3.30249I −17.5441 + 2.1403I
u = 0.116890 − 1.065220I
−8.61843 + 3.30249I −17.5441 − 2.1403I
u = −0.491393 + 0.960867I
1.09428 − 6.52024I −8.28646 + 9.91432I
u = −0.491393 − 0.960867I
1.09428 + 6.52024I −8.28646 − 9.91432I
u = 0.503891 + 0.983618I
−5.90815 + 9.04807I −12.0898 − 8.5316I
u = 0.503891 − 0.983618I
−5.90815 − 9.04807I −12.0898 + 8.5316I
u = 0.243423 + 0.854516I
−0.623812 + 1.209860I −7.64614 − 4.90268I
u = 0.243423 − 0.854516I
−0.623812 − 1.209860I −7.64614 + 4.90268I
u = −0.870822
−14.9278 −15.7210
u = 0.333283 + 1.082310I
−10.49970 + 3.34960I −17.9466 − 4.0527I
u = 0.333283 − 1.082310I
−10.49970 − 3.34960I −17.9466 + 4.0527I
u = −0.861168 + 0.072729I
−10.59330 + 8.29227I −12.98061 − 4.57740I
u = −0.861168 − 0.072729I
−10.59330 − 8.29227I −12.98061 + 4.57740I
u = 0.845961 + 0.068541I
−3.20212 − 5.70812I −9.97360 + 5.97545I
u = 0.845961 − 0.068541I
−3.20212 + 5.70812I −9.97360 − 5.97545I
u = 0.840103
−6.53703 −14.4920
u = −0.825031 + 0.057929I
−2.24307 + 1.86074I −7.46523 + 0.06131I
u = −0.825031 − 0.057929I
−2.24307 − 1.86074I −7.46523 − 0.06131I
u = −0.514519 + 0.590131I
−3.81103 − 3.61233I −7.78381 + 3.89227I
u = −0.514519 − 0.590131I
−3.81103 + 3.61233I −7.78381 − 3.89227I
u = 0.503991 + 0.523196I
2.66446 + 1.32817I −3.49967 − 4.01662I
u = 0.503991 − 0.523196I
2.66446 − 1.32817I −3.49967 + 4.01662I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.726197
−7.37532 −11.5790
u = 0.572655 + 0.425482I
−4.35800 − 4.75776I −8.70806 + 3.36680I
u = 0.572655 − 0.425482I
−4.35800 + 4.75776I −8.70806 − 3.36680I
u = 0.463308 + 1.201860I
−10.77920 + 4.40954I 0
u = 0.463308 − 1.201860I
−10.77920 − 4.40954I 0
u = −0.530920 + 0.462145I
2.47339 + 2.35970I −4.46729 − 4.22291I
u = −0.530920 − 0.462145I
2.47339 − 2.35970I −4.46729 + 4.22291I
u = −0.431213 + 1.227700I
−6.07588 − 2.53923I 0
u = −0.431213 − 1.227700I
−6.07588 + 2.53923I 0
u = 0.423975 + 1.239910I
−7.15358 − 1.27519I 0
u = 0.423975 − 1.239910I
−7.15358 + 1.27519I 0
u = −0.485017 + 1.219700I
−5.68867 − 6.60293I 0
u = −0.485017 − 1.219700I
−5.68867 + 6.60293I 0
u = 0.459979 + 1.232640I
−10.22260 + 4.63419I 0
u = 0.459979 − 1.232640I
−10.22260 − 4.63419I 0
u = −0.421595 + 1.249750I
−14.6184 + 3.8161I 0
u = −0.421595 − 1.249750I
−14.6184 − 3.8161I 0
u = 0.492780 + 1.225990I
−6.65767 + 10.54660I 0
u = 0.492780 − 1.225990I
−6.65767 − 10.54660I 0
u = −0.497583 + 1.231530I
−14.0678 − 13.1961I 0
u = −0.497583 − 1.231530I
−14.0678 + 13.1961I 0
u = −0.464095 + 1.248480I
−18.7025 − 4.7501I 0
u = −0.464095 − 1.248480I
−18.7025 + 4.7501I 0
u = 0.653534
−7.36372 −12.0180
u = −0.350221
−0.718328 −13.6840
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
55
+ 31u
54
+ ··· + 4u − 1
c
2
, c
7
u
55
+ u
54
+ ··· − 2u − 1
c
3
, c
4
, c
8
u
55
− u
54
+ ··· + u − 2
c
5
, c
6
, c
10
c
11
u
55
+ u
54
+ ··· − 2u − 1
c
9
, c
12
u
55
− 11u
54
+ ··· + 8u − 1
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
55
− 13y
54
+ ··· + 68y −1
c
2
, c
7
y
55
+ 31y
54
+ ··· + 4y −1
c
3
, c
4
, c
8
y
55
− 57y
54
+ ··· + 293y −4
c
5
, c
6
, c
10
c
11
y
55
− 61y
54
+ ··· + 4y −1
c
9
, c
12
y
55
+ 23y
54
+ ··· − 36y −1
8
