12a
0759
(K12a
0759
)
A knot diagram
1
Linearized knot diagam
3 8 10 11 12 9 2 7 1 4 5 6
Solving Sequence
2,7
8 3 9 1 10 4 6 12 5 11
c
7
c
2
c
8
c
1
c
9
c
3
c
6
c
12
c
5
c
11
c
4
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
30
− u
29
+ ··· − u − 1i
* 1 irreducible components of dim
C
= 0, with total 30 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
30
− u
29
+ · · · − u − 1i
(i) Arc colorings
a
2
=
0
u
a
7
=
1
0
a
8
=
1
u
2
a
3
=
−u
−u
3
+ u
a
9
=
−u
2
+ 1
u
2
a
1
=
u
3
u
5
− u
3
+ u
a
10
=
u
10
− u
8
+ 2u
6
− u
4
− u
2
+ 1
u
12
− 2u
10
+ 4u
8
− 4u
6
+ 3u
4
a
4
=
u
19
− 2u
17
+ 6u
15
− 8u
13
+ 9u
11
− 6u
9
+ 4u
5
− 3u
3
u
21
− 3u
19
+ 9u
17
− 16u
15
+ 24u
13
− 25u
11
+ 21u
9
− 10u
7
+ 3u
5
− u
3
+ u
a
6
=
u
4
− u
2
+ 1
−u
4
a
12
=
−u
13
+ 2u
11
− 5u
9
+ 6u
7
− 6u
5
+ 4u
3
− u
u
13
− u
11
+ 3u
9
− 2u
7
+ 2u
5
− u
3
+ u
a
5
=
−u
22
+ 3u
20
+ ··· − 2u
2
+ 1
u
22
− 2u
20
+ ··· − 4u
4
+ u
2
a
11
=
−u
28
+ 3u
26
+ ··· − u
2
+ 1
−u
29
+ u
28
+ ··· − u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
29
+ 16u
27
− 4u
26
− 60u
25
+ 12u
24
+ 148u
23
− 44u
22
−
304u
21
+ 88u
20
+ 508u
19
− 160u
18
− 692u
17
+ 212u
16
+ 796u
15
− 224u
14
− 736u
13
+
168u
12
+ 568u
11
− 80u
10
− 344u
9
+ 180u
7
+ 24u
6
− 84u
5
+ 32u
3
− 8u
2
− 12u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
8
u
30
+ 7u
29
+ ··· + 5u + 1
c
2
, c
7
u
30
+ u
29
+ ··· + u − 1
c
3
, c
4
, c
5
c
10
, c
11
, c
12
u
30
+ u
29
+ ··· − u − 1
c
9
u
30
− 7u
29
+ ··· + 521u − 295
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
y
30
+ 33y
29
+ ··· + 39y + 1
c
2
, c
7
y
30
− 7y
29
+ ··· − 5y + 1
c
3
, c
4
, c
5
c
10
, c
11
, c
12
y
30
− 43y
29
+ ··· − 5y + 1
c
9
y
30
− 23y
29
+ ··· − 1109241y + 87025
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.920005 + 0.430378I
4.66034 − 4.78463I 7.25047 + 6.81855I
u = 0.920005 − 0.430378I
4.66034 + 4.78463I 7.25047 − 6.81855I
u = 0.979433
12.9125 3.79610
u = −0.968807 + 0.456896I
15.5155 + 5.5117I 7.63592 − 5.51087I
u = −0.968807 − 0.456896I
15.5155 − 5.5117I 7.63592 + 5.51087I
u = −0.850370 + 0.353923I
−0.53732 + 3.17807I 2.96134 − 9.77982I
u = −0.850370 − 0.353923I
−0.53732 − 3.17807I 2.96134 + 9.77982I
u = −0.895297
2.40603 3.06770
u = 0.782312 + 0.234321I
−1.26656 − 0.80671I −2.39136 + 0.48620I
u = 0.782312 − 0.234321I
−1.26656 + 0.80671I −2.39136 − 0.48620I
u = 0.873425 + 0.850032I
6.78596 − 0.15290I 9.21207 − 2.20813I
u = 0.873425 − 0.850032I
6.78596 + 0.15290I 9.21207 + 2.20813I
u = −0.353083 + 0.696158I
17.4896 − 1.3046I 12.03831 + 0.06444I
u = −0.353083 − 0.696158I
17.4896 + 1.3046I 12.03831 − 0.06444I
u = −0.902387 + 0.826249I
4.85853 + 3.08395I 4.14772 − 2.46951I
u = −0.902387 − 0.826249I
4.85853 − 3.08395I 4.14772 + 2.46951I
u = −0.858968 + 0.882764I
12.99360 − 1.81516I 11.64969 + 0.86495I
u = −0.858968 − 0.882764I
12.99360 + 1.81516I 11.64969 − 0.86495I
u = 0.853261 + 0.904188I
−15.0818 + 2.8449I 11.96540 − 0.16863I
u = 0.853261 − 0.904188I
−15.0818 − 2.8449I 11.96540 + 0.16863I
u = 0.935818 + 0.828568I
6.59163 − 6.09371I 8.55797 + 7.37822I
u = 0.935818 − 0.828568I
6.59163 + 6.09371I 8.55797 − 7.37822I
u = −0.962584 + 0.839703I
12.6661 + 8.1956I 11.00485 − 5.80701I
u = −0.962584 − 0.839703I
12.6661 − 8.1956I 11.00485 + 5.80701I
u = 0.355798 + 0.609144I
6.41234 + 0.93846I 12.13091 − 0.39281I
u = 0.355798 − 0.609144I
6.41234 − 0.93846I 12.13091 + 0.39281I
u = 0.978792 + 0.847377I
−15.4823 − 9.3158I 11.28938 + 4.91125I
u = 0.978792 − 0.847377I
−15.4823 + 9.3158I 11.28938 − 4.91125I
u = −0.345278 + 0.364509I
0.887471 − 0.222734I 11.11542 + 1.64999I
u = −0.345278 − 0.364509I
0.887471 + 0.222734I 11.11542 − 1.64999I
5
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
8
u
30
+ 7u
29
+ ··· + 5u + 1
c
2
, c
7
u
30
+ u
29
+ ··· + u − 1
c
3
, c
4
, c
5
c
10
, c
11
, c
12
u
30
+ u
29
+ ··· − u − 1
c
9
u
30
− 7u
29
+ ··· + 521u − 295
6
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
, c
8
y
30
+ 33y
29
+ ··· + 39y + 1
c
2
, c
7
y
30
− 7y
29
+ ··· − 5y + 1
c
3
, c
4
, c
5
c
10
, c
11
, c
12
y
30
− 43y
29
+ ··· − 5y + 1
c
9
y
30
− 23y
29
+ ··· − 1109241y + 87025
7
