12a
0724
(K12a
0724
)
A knot diagram
1
Linearized knot diagam
3 8 9 10 11 12 1 2 4 7 6 5
Solving Sequence
2,9
8 3 4 10 5 1 7 11 12 6
c
8
c
2
c
3
c
9
c
4
c
1
c
7
c
10
c
12
c
6
c
5
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
53
− u
52
+ ··· − u − 1i
* 1 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
= hu
53
− u
52
+ · · · − u − 1i
(i) Arc colorings
a
2
=
0
u
a
9
=
1
0
a
8
=
1
u
2
a
3
=
u
u
3
+ u
a
4
=
−u
3
u
3
+ u
a
10
=
−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
7
=
−u
6
− u
4
+ 1
−u
8
− 2u
6
− 2u
4
a
11
=
−u
20
− 5u
18
− 11u
16
− 10u
14
+ 2u
12
+ 13u
10
+ 9u
8
− 2u
6
− 5u
4
− u
2
+ 1
−u
22
− 6u
20
− 17u
18
− 26u
16
− 20u
14
+ 13u
10
+ 10u
8
+ 3u
6
+ 2u
4
+ u
2
a
12
=
u
23
+ 6u
21
+ ··· + 6u
5
+ 2u
3
−u
23
− 7u
21
+ ··· − 3u
5
+ u
a
6
=
u
51
+ 14u
49
+ ··· − u
3
− 2u
u
52
− u
51
+ ··· + 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
52
− 4u
51
+ ··· − 8u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
53
+ 31u
52
+ ··· + 3u − 1
c
2
, c
8
u
53
− u
52
+ ··· − u − 1
c
3
, c
4
, c
7
c
9
u
53
+ u
52
+ ··· + 17u − 5
c
5
, c
6
, c
11
u
53
− u
52
+ ··· − 3u − 1
c
10
, c
12
u
53
+ 3u
52
+ ··· − 15u − 3
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
53
− 17y
52
+ ··· + 35y −1
c
2
, c
8
y
53
+ 31y
52
+ ··· + 3y −1
c
3
, c
4
, c
7
c
9
y
53
− 65y
52
+ ··· + 899y −25
c
5
, c
6
, c
11
y
53
− 45y
52
+ ··· + 3y −1
c
10
, c
12
y
53
+ 23y
52
+ ··· + 39y −9
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.128755 + 1.019230I
−4.90434 − 2.79376I −17.4626 + 3.5787I
u = −0.128755 − 1.019230I
−4.90434 + 2.79376I −17.4626 − 3.5787I
u = −0.911842
−15.0851 −15.7790
u = 0.452670 + 0.785816I
−0.74858 + 5.61733I −7.84958 − 7.79371I
u = 0.452670 − 0.785816I
−0.74858 − 5.61733I −7.84958 + 7.79371I
u = 0.904077 + 0.035348I
−10.76820 − 8.15829I −12.92275 + 4.70754I
u = 0.904077 − 0.035348I
−10.76820 + 8.15829I −12.92275 − 4.70754I
u = −0.894263 + 0.030410I
−5.73332 + 4.23818I −8.49578 − 3.55105I
u = −0.894263 − 0.030410I
−5.73332 − 4.23818I −8.49578 + 3.55105I
u = −0.435840 + 1.018520I
−3.18288 − 2.63277I −11.73208 + 4.34581I
u = −0.435840 − 1.018520I
−3.18288 + 2.63277I −11.73208 − 4.34581I
u = 0.270662 + 1.080700I
−1.60759 + 0.28866I −11.63603 + 0.60703I
u = 0.270662 − 1.080700I
−1.60759 − 0.28866I −11.63603 − 0.60703I
u = 0.885161 + 0.012504I
−7.93829 − 0.20732I −11.52979 − 0.97702I
u = 0.885161 − 0.012504I
−7.93829 + 0.20732I −11.52979 + 0.97702I
u = −0.366265 + 1.058630I
−3.39452 − 3.18652I −15.1026 + 5.9997I
u = −0.366265 − 1.058630I
−3.39452 + 3.18652I −15.1026 − 5.9997I
u = −0.438529 + 0.736447I
3.30946 − 1.89439I −2.29810 + 4.47995I
u = −0.438529 − 0.736447I
3.30946 + 1.89439I −2.29810 − 4.47995I
u = 0.459270 + 1.056820I
−0.23945 + 6.31415I −8.00000 − 7.97292I
u = 0.459270 − 1.056820I
−0.23945 − 6.31415I −8.00000 + 7.97292I
u = −0.271764 + 1.127110I
−6.33973 + 3.10223I −16.8942 − 2.0365I
u = −0.271764 − 1.127110I
−6.33973 − 3.10223I −16.8942 + 2.0365I
u = −0.471846 + 1.075850I
−4.85947 − 10.15110I −13.4245 + 9.4204I
u = −0.471846 − 1.075850I
−4.85947 + 10.15110I −13.4245 − 9.4204I
u = 0.147653 + 0.799603I
−0.621747 + 0.938340I −10.16569 − 6.80292I
u = 0.147653 − 0.799603I
−0.621747 − 0.938340I −10.16569 + 6.80292I
u = 0.443085 + 0.672743I
−0.43986 − 1.77287I −6.55805 − 0.17659I
u = 0.443085 − 0.672743I
−0.43986 + 1.77287I −6.55805 + 0.17659I
u = 0.386940 + 1.130380I
−9.24218 + 3.74726I −18.6066 + 0.I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.386940 − 1.130380I
−9.24218 − 3.74726I −18.6066 + 0.I
u = −0.626369 + 0.230684I
−2.49578 + 5.92364I −10.03468 − 5.69400I
u = −0.626369 − 0.230684I
−2.49578 − 5.92364I −10.03468 + 5.69400I
u = 0.461426 + 1.259080I
−11.80210 + 4.57211I 0
u = 0.461426 − 1.259080I
−11.80210 − 4.57211I 0
u = 0.474991 + 1.254660I
−11.70120 + 5.05761I 0
u = 0.474991 − 1.254660I
−11.70120 − 5.05761I 0
u = −0.452029 + 1.266230I
−9.69650 − 0.52013I 0
u = −0.452029 − 1.266230I
−9.69650 + 0.52013I 0
u = −0.485137 + 1.256070I
−9.45189 − 9.16998I 0
u = −0.485137 − 1.256070I
−9.45189 + 9.16998I 0
u = 0.450332 + 1.273070I
−14.7839 − 3.3745I 0
u = 0.450332 − 1.273070I
−14.7839 + 3.3745I 0
u = 0.489645 + 1.259800I
−14.4913 + 13.1415I 0
u = 0.489645 − 1.259800I
−14.4913 − 13.1415I 0
u = 0.647347
−6.05533 −14.6560
u = −0.472487 + 1.271320I
−18.9745 − 4.9215I 0
u = −0.472487 − 1.271320I
−18.9745 + 4.9215I 0
u = 0.573524 + 0.250742I
1.98282 − 2.24507I −4.39655 + 3.93832I
u = 0.573524 − 0.250742I
1.98282 + 2.24507I −4.39655 − 3.93832I
u = −0.505614 + 0.326260I
−1.29579 − 1.22009I −7.56412 + 0.50390I
u = −0.505614 − 0.326260I
−1.29579 + 1.22009I −7.56412 − 0.50390I
u = −0.436579
−0.780224 −12.8270
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
53
+ 31u
52
+ ··· + 3u − 1
c
2
, c
8
u
53
− u
52
+ ··· − u − 1
c
3
, c
4
, c
7
c
9
u
53
+ u
52
+ ··· + 17u − 5
c
5
, c
6
, c
11
u
53
− u
52
+ ··· − 3u − 1
c
10
, c
12
u
53
+ 3u
52
+ ··· − 15u − 3
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
53
− 17y
52
+ ··· + 35y −1
c
2
, c
8
y
53
+ 31y
52
+ ··· + 3y −1
c
3
, c
4
, c
7
c
9
y
53
− 65y
52
+ ··· + 899y −25
c
5
, c
6
, c
11
y
53
− 45y
52
+ ··· + 3y −1
c
10
, c
12
y
53
+ 23y
52
+ ··· + 39y −9
8
