12a
0533
(K12a
0533
)
A knot diagram
1
Linearized knot diagam
3 7 8 10 11 2 1 4 12 5 6 9
Solving Sequence
5,10
11 6 12 4 9 1 8 3 7 2
c
10
c
5
c
11
c
4
c
9
c
12
c
8
c
3
c
7
c
2
c
1
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
68
+ u
67
+ ··· + 3u
2
− 1i
* 1 irreducible components of dim
C
= 0, with total 68 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
68
+ u
67
+ · · · + 3u
2
− 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
u
−u
3
+ u
a
12
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
−u
u
a
9
=
−u
6
+ 3u
4
− 2u
2
+ 1
u
8
− 4u
6
+ 4u
4
a
1
=
−u
10
+ 5u
8
− 8u
6
+ 5u
4
− 3u
2
+ 1
u
12
− 6u
10
+ 12u
8
− 8u
6
+ u
4
− 2u
2
a
8
=
−u
10
+ 5u
8
− 8u
6
+ 5u
4
− 3u
2
+ 1
u
10
− 4u
8
+ 3u
6
+ 2u
4
+ u
2
a
3
=
u
19
− 10u
17
+ 40u
15
− 82u
13
+ 95u
11
− 72u
9
+ 44u
7
− 18u
5
+ 5u
3
− 2u
−u
19
+ 9u
17
− 30u
15
+ 43u
13
− 21u
11
+ u
9
− 6u
7
− u
3
+ u
a
7
=
u
32
− 17u
30
+ ··· − 6u
2
+ 1
−u
34
+ 18u
32
+ ··· + 8u
4
+ u
2
a
2
=
−u
50
+ 27u
48
+ ··· − 5u
2
+ 1
u
50
− 26u
48
+ ··· − 8u
4
− u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
65
− 144u
63
+ ··· − 20u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
68
+ 31u
67
+ ··· + 6u + 1
c
2
, c
6
u
68
− u
67
+ ··· + 2u − 1
c
3
, c
8
u
68
+ u
67
+ ··· + 92u − 13
c
4
, c
5
, c
10
c
11
u
68
− u
67
+ ··· + 3u
2
− 1
c
7
u
68
− 3u
67
+ ··· − 20u + 1
c
9
, c
12
u
68
+ 13u
67
+ ··· − 180u − 23
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
68
+ 13y
67
+ ··· − 6y + 1
c
2
, c
6
y
68
− 31y
67
+ ··· − 6y + 1
c
3
, c
8
y
68
− 47y
67
+ ··· − 9686y + 169
c
4
, c
5
, c
10
c
11
y
68
− 75y
67
+ ··· − 6y + 1
c
7
y
68
+ 5y
67
+ ··· − 110y + 1
c
9
, c
12
y
68
+ 33y
67
+ ··· + 7114y + 529
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.835401 + 0.076969I
3.98870 − 6.35660I 11.96357 + 5.97351I
u = −0.835401 − 0.076969I
3.98870 + 6.35660I 11.96357 − 5.97351I
u = 0.831050 + 0.041295I
5.76628 + 1.28729I 15.0699 − 0.8084I
u = 0.831050 − 0.041295I
5.76628 − 1.28729I 15.0699 + 0.8084I
u = 0.612556 + 0.561591I
0.04182 + 11.86150I 6.37979 − 10.46070I
u = 0.612556 − 0.561591I
0.04182 − 11.86150I 6.37979 + 10.46070I
u = −0.612136 + 0.551160I
2.09389 − 6.71465I 9.52044 + 6.37548I
u = −0.612136 − 0.551160I
2.09389 + 6.71465I 9.52044 − 6.37548I
u = −0.616362 + 0.518645I
2.80869 − 4.10090I 10.70764 + 6.36642I
u = −0.616362 − 0.518645I
2.80869 + 4.10090I 10.70764 − 6.36642I
u = 0.585980 + 0.550245I
−2.78920 + 4.53214I 2.74588 − 5.66472I
u = 0.585980 − 0.550245I
−2.78920 − 4.53214I 2.74588 + 5.66472I
u = 0.621345 + 0.497725I
1.38303 − 0.91928I 8.53731 − 1.06257I
u = 0.621345 − 0.497725I
1.38303 + 0.91928I 8.53731 + 1.06257I
u = −0.507777 + 0.569607I
−5.47539 − 5.67160I 0.47362 + 7.79451I
u = −0.507777 − 0.569607I
−5.47539 + 5.67160I 0.47362 − 7.79451I
u = −0.464449 + 0.572539I
−5.60242 + 1.75390I −0.214027 − 0.465353I
u = −0.464449 − 0.572539I
−5.60242 − 1.75390I −0.214027 + 0.465353I
u = 0.486921 + 0.541642I
−2.59155 + 1.87214I 4.19602 − 4.04466I
u = 0.486921 − 0.541642I
−2.59155 − 1.87214I 4.19602 + 4.04466I
u = −0.694797
0.907173 9.94330
u = 0.332805 + 0.598999I
−0.77445 − 7.89771I 4.14994 + 4.52279I
u = 0.332805 − 0.598999I
−0.77445 + 7.89771I 4.14994 − 4.52279I
u = 0.366988 + 0.568928I
−3.42846 − 0.68233I 0.547475 − 0.922163I
u = 0.366988 − 0.568928I
−3.42846 + 0.68233I 0.547475 + 0.922163I
u = −0.325745 + 0.583572I
1.26237 + 2.82639I 7.27704 − 0.30320I
u = −0.325745 − 0.583572I
1.26237 − 2.82639I 7.27704 + 0.30320I
u = 0.556194 + 0.317442I
−0.21535 + 3.63596I 8.68786 − 8.76844I
u = 0.556194 − 0.317442I
−0.21535 − 3.63596I 8.68786 + 8.76844I
u = −0.285763 + 0.538664I
1.87263 + 0.44543I 8.05642 + 0.19579I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.285763 − 0.538664I
1.87263 − 0.44543I 8.05642 − 0.19579I
u = 0.244521 + 0.522979I
0.33376 + 4.43660I 5.03394 − 5.63421I
u = 0.244521 − 0.522979I
0.33376 − 4.43660I 5.03394 + 5.63421I
u = −1.43293 + 0.08268I
4.69332 + 5.60280I 0
u = −1.43293 − 0.08268I
4.69332 − 5.60280I 0
u = 1.45312 + 0.06829I
6.74595 − 0.72677I 0
u = 1.45312 − 0.06829I
6.74595 + 0.72677I 0
u = −0.524220 + 0.080652I
0.800382 − 0.035159I 12.87268 + 1.03494I
u = −0.524220 − 0.080652I
0.800382 + 0.035159I 12.87268 − 1.03494I
u = −1.46881 + 0.10794I
2.45112 − 1.55036I 0
u = −1.46881 − 0.10794I
2.45112 + 1.55036I 0
u = 1.50453 + 0.15035I
0.861268 + 0.774157I 0
u = 1.50453 − 0.15035I
0.861268 − 0.774157I 0
u = −1.52221 + 0.14559I
4.06744 − 4.28308I 0
u = −1.52221 − 0.14559I
4.06744 + 4.28308I 0
u = 1.52815 + 0.06626I
7.60675 + 0.77642I 0
u = 1.52815 − 0.06626I
7.60675 − 0.77642I 0
u = 1.52367 + 0.16029I
1.24045 + 8.26818I 0
u = 1.52367 − 0.16029I
1.24045 − 8.26818I 0
u = −1.54603 + 0.09488I
6.83939 − 5.14255I 0
u = −1.54603 − 0.09488I
6.83939 + 5.14255I 0
u = −1.55970 + 0.16290I
4.38474 − 7.12790I 0
u = −1.55970 − 0.16290I
4.38474 + 7.12790I 0
u = 1.56912 + 0.16482I
9.39965 + 9.33939I 0
u = 1.56912 − 0.16482I
9.39965 − 9.33939I 0
u = −1.56899 + 0.16875I
7.3423 − 14.5405I 0
u = −1.56899 − 0.16875I
7.3423 + 14.5405I 0
u = 1.57101 + 0.15333I
10.15460 + 6.56396I 0
u = 1.57101 − 0.15333I
10.15460 − 6.56396I 0
u = −1.57221 + 0.14666I
8.75910 − 1.44505I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.57221 − 0.14666I
8.75910 + 1.44505I 0
u = 1.58883
8.73667 0
u = −1.60657 + 0.00745I
14.01720 − 1.44111I 0
u = −1.60657 − 0.00745I
14.01720 + 1.44111I 0
u = 1.60735 + 0.01384I
12.25660 + 6.64311I 0
u = 1.60735 − 0.01384I
12.25660 − 6.64311I 0
u = 0.106973 + 0.377298I
−1.48571 − 1.30445I 0.644293 + 0.844711I
u = 0.106973 − 0.377298I
−1.48571 + 1.30445I 0.644293 − 0.844711I
7
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
68
+ 31u
67
+ ··· + 6u + 1
c
2
, c
6
u
68
− u
67
+ ··· + 2u − 1
c
3
, c
8
u
68
+ u
67
+ ··· + 92u − 13
c
4
, c
5
, c
10
c
11
u
68
− u
67
+ ··· + 3u
2
− 1
c
7
u
68
− 3u
67
+ ··· − 20u + 1
c
9
, c
12
u
68
+ 13u
67
+ ··· − 180u − 23
8
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
68
+ 13y
67
+ ··· − 6y + 1
c
2
, c
6
y
68
− 31y
67
+ ··· − 6y + 1
c
3
, c
8
y
68
− 47y
67
+ ··· − 9686y + 169
c
4
, c
5
, c
10
c
11
y
68
− 75y
67
+ ··· − 6y + 1
c
7
y
68
+ 5y
67
+ ··· − 110y + 1
c
9
, c
12
y
68
+ 33y
67
+ ··· + 7114y + 529
9
