12a
0454
(K12a
0454
)
A knot diagram
1
Linearized knot diagam
3 6 10 11 7 2 1 12 5 4 9 8
Solving Sequence
5,11
4 10 3 9 12 8 1 7 6 2
c
4
c
10
c
3
c
9
c
11
c
8
c
12
c
7
c
5
c
2
c
1
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
51
− u
50
+ ··· + 2u + 1i
* 1 irreducible components of dim
C
= 0, with total 51 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
51
− u
50
+ · · · + 2u + 1i
(i) Arc colorings
a
5
=
1
0
a
11
=
0
u
a
4
=
1
−u
2
a
10
=
u
−u
3
+ u
a
3
=
−u
2
+ 1
u
4
− 2u
2
a
9
=
−u
3
+ 2u
−u
3
+ u
a
12
=
u
7
− 4u
5
+ 4u
3
u
7
− 3u
5
+ 2u
3
+ u
a
8
=
−u
11
+ 6u
9
− 12u
7
+ 8u
5
− u
3
+ 2u
−u
11
+ 5u
9
− 8u
7
+ 3u
5
+ u
3
+ u
a
1
=
u
15
− 8u
13
+ 24u
11
− 32u
9
+ 18u
7
− 8u
5
+ 8u
3
u
15
− 7u
13
+ 18u
11
− 19u
9
+ 6u
7
− 2u
5
+ 4u
3
+ u
a
7
=
−u
19
+ 10u
17
− 40u
15
+ 80u
13
− 83u
11
+ 50u
9
− 36u
7
+ 24u
5
− u
3
+ 2u
−u
19
+ 9u
17
− 32u
15
+ 55u
13
− 45u
11
+ 19u
9
− 16u
7
+ 10u
5
+ 3u
3
+ u
a
6
=
u
38
− 19u
36
+ ··· + 2u
2
+ 1
u
38
− 18u
36
+ ··· + 6u
4
+ u
2
a
2
=
−u
21
+ 10u
19
+ ··· + 6u
3
− u
u
23
− 11u
21
+ ··· + 6u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
49
− 96u
47
+ ··· − 24u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
51
+ 19u
50
+ ··· − 6u − 1
c
2
, c
6
u
51
− u
50
+ ··· + 3u
2
+ 1
c
3
, c
4
, c
10
u
51
− u
50
+ ··· + 2u + 1
c
7
, c
8
, c
11
c
12
u
51
+ 5u
50
+ ··· + 60u + 7
c
9
u
51
+ 3u
50
+ ··· − 2294u − 851
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
51
+ 27y
50
+ ··· − 46y −1
c
2
, c
6
y
51
+ 19y
50
+ ··· − 6y −1
c
3
, c
4
, c
10
y
51
− 49y
50
+ ··· − 6y −1
c
7
, c
8
, c
11
c
12
y
51
+ 63y
50
+ ··· − 1230y −49
c
9
y
51
− 29y
50
+ ··· + 13032066y −724201
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.497159 + 0.687033I
−12.69280 + 2.28721I −7.99809 − 2.91340I
u = −0.497159 − 0.687033I
−12.69280 − 2.28721I −7.99809 + 2.91340I
u = −0.514014 + 0.669274I
−8.53754 − 4.85587I −4.54813 + 1.85081I
u = −0.514014 − 0.669274I
−8.53754 + 4.85587I −4.54813 − 1.85081I
u = −0.476017 + 0.696319I
−8.40118 + 9.40330I −4.15224 − 7.59831I
u = −0.476017 − 0.696319I
−8.40118 − 9.40330I −4.15224 + 7.59831I
u = 0.476316 + 0.685506I
−6.77395 − 3.90642I −1.99681 + 3.09904I
u = 0.476316 − 0.685506I
−6.77395 + 3.90642I −1.99681 − 3.09904I
u = 0.502151 + 0.665862I
−6.86920 − 0.58725I −2.24865 + 2.79780I
u = 0.502151 − 0.665862I
−6.86920 + 0.58725I −2.24865 − 2.79780I
u = −1.230860 + 0.105090I
−0.223422 − 0.095504I 0
u = −1.230860 − 0.105090I
−0.223422 + 0.095504I 0
u = 1.252200 + 0.136529I
−0.53663 − 5.16209I 0
u = 1.252200 − 0.136529I
−0.53663 + 5.16209I 0
u = −1.30322
−3.07453 0
u = 0.288268 + 0.603646I
0.37640 − 6.82727I −0.33113 + 9.54781I
u = 0.288268 − 0.603646I
0.37640 + 6.82727I −0.33113 − 9.54781I
u = 0.379930 + 0.522272I
−3.67336 − 1.67891I −7.63362 + 4.68207I
u = 0.379930 − 0.522272I
−3.67336 + 1.67891I −7.63362 − 4.68207I
u = −0.255571 + 0.575911I
1.26731 + 1.64575I 2.07350 − 4.49112I
u = −0.255571 − 0.575911I
1.26731 − 1.64575I 2.07350 + 4.49112I
u = 1.378680 + 0.061133I
−5.19429 − 2.37699I 0
u = 1.378680 − 0.061133I
−5.19429 + 2.37699I 0
u = 0.479281 + 0.376899I
−0.51645 + 3.54279I −3.97592 − 2.38269I
u = 0.479281 − 0.376899I
−0.51645 − 3.54279I −3.97592 + 2.38269I
u = 1.384830 + 0.203376I
−3.94332 − 4.47861I 0
u = 1.384830 − 0.203376I
−3.94332 + 4.47861I 0
u = 1.395460 + 0.136782I
−5.14352 − 2.78940I 0
u = 1.395460 − 0.136782I
−5.14352 + 2.78940I 0
u = −1.396860 + 0.217411I
−4.98503 + 9.81269I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.396860 − 0.217411I
−4.98503 − 9.81269I 0
u = −0.023873 + 0.568833I
3.28231 + 2.57953I 6.80307 − 3.85321I
u = −0.023873 − 0.568833I
3.28231 − 2.57953I 6.80307 + 3.85321I
u = −1.42782 + 0.18011I
−9.44989 + 4.23941I 0
u = −1.42782 − 0.18011I
−9.44989 − 4.23941I 0
u = −1.43604 + 0.12595I
−6.55813 − 1.72068I 0
u = −1.43604 − 0.12595I
−6.55813 + 1.72068I 0
u = −0.429960 + 0.261562I
0.244440 + 1.222020I −3.00114 − 3.76024I
u = −0.429960 − 0.261562I
0.244440 − 1.222020I −3.00114 + 3.76024I
u = −1.49311 + 0.24266I
−13.1596 + 7.2936I 0
u = −1.49311 − 0.24266I
−13.1596 − 7.2936I 0
u = 1.49510 + 0.24697I
−14.7939 − 12.8461I 0
u = 1.49510 − 0.24697I
−14.7939 + 12.8461I 0
u = −1.49838 + 0.22938I
−13.36680 + 3.84685I 0
u = −1.49838 − 0.22938I
−13.36680 − 3.84685I 0
u = 1.50158 + 0.23862I
−19.1879 − 5.6622I 0
u = 1.50158 − 0.23862I
−19.1879 + 5.6622I 0
u = 1.50376 + 0.22747I
−15.1019 + 1.5937I 0
u = 1.50376 − 0.22747I
−15.1019 − 1.5937I 0
u = −0.206294 + 0.390192I
0.029465 + 0.870630I 0.74153 − 7.77097I
u = −0.206294 − 0.390192I
0.029465 − 0.870630I 0.74153 + 7.77097I
6
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u
51
+ 19u
50
+ ··· − 6u − 1
c
2
, c
6
u
51
− u
50
+ ··· + 3u
2
+ 1
c
3
, c
4
, c
10
u
51
− u
50
+ ··· + 2u + 1
c
7
, c
8
, c
11
c
12
u
51
+ 5u
50
+ ··· + 60u + 7
c
9
u
51
+ 3u
50
+ ··· − 2294u − 851
7
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
51
+ 27y
50
+ ··· − 46y −1
c
2
, c
6
y
51
+ 19y
50
+ ··· − 6y −1
c
3
, c
4
, c
10
y
51
− 49y
50
+ ··· − 6y −1
c
7
, c
8
, c
11
c
12
y
51
+ 63y
50
+ ··· − 1230y −49
c
9
y
51
− 29y
50
+ ··· + 13032066y −724201
8
