12a
0378
(K12a
0378
)
A knot diagram
1
Linearized knot diagam
3 6 9 10 2 1 12 5 11 4 8 7
Solving Sequence
5,10
4 11 9 3 8 12 7 1 2 6
c
4
c
10
c
9
c
3
c
8
c
11
c
7
c
12
c
1
c
6
c
2
, c
5
Ideals for irreducible components
2
of X
par
I
u
1
= hu
63
− u
62
+ ··· + 2u − 1i
* 1 irreducible components of dim
C
= 0, with total 63 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
63
− u
62
+ · · · + 2u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
11
=
u
u
3
+ u
a
9
=
u
3
u
5
+ u
3
+ u
a
3
=
−u
6
− u
4
+ 1
−u
8
− 2u
6
− 2u
4
a
8
=
u
5
+ 2u
3
+ u
u
5
+ u
3
+ u
a
12
=
u
13
+ 4u
11
+ 7u
9
+ 6u
7
+ 2u
5
+ u
u
13
+ 3u
11
+ 5u
9
+ 4u
7
+ 2u
5
+ u
3
+ u
a
7
=
u
21
+ 6u
19
+ 17u
17
+ 28u
15
+ 28u
13
+ 16u
11
+ 5u
9
+ 2u
7
+ 3u
5
+ 2u
3
+ u
u
21
+ 5u
19
+ 13u
17
+ 20u
15
+ 20u
13
+ 13u
11
+ 7u
9
+ 4u
7
+ 3u
5
+ u
3
+ u
a
1
=
u
29
+ 8u
27
+ ··· + 4u
5
+ u
u
29
+ 7u
27
+ ··· + u
3
+ u
a
2
=
u
43
+ 10u
41
+ ··· + 6u
7
− u
3
u
45
+ 11u
43
+ ··· + u
3
+ u
a
6
=
u
37
+ 10u
35
+ ··· + 2u
3
+ u
u
37
+ 9u
35
+ ··· + u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
61
− 4u
60
+ ··· + 4u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
63
+ 37u
62
+ ··· + 16u
3
+ 1
c
2
, c
5
u
63
+ u
62
+ ··· + 2u + 1
c
3
u
63
− u
62
+ ··· − 386u + 317
c
4
, c
10
u
63
+ u
62
+ ··· + 2u + 1
c
6
, c
7
, c
11
c
12
u
63
+ 3u
62
+ ··· + 16u + 1
c
8
u
63
+ 5u
62
+ ··· + 360u + 31
c
9
u
63
+ 31u
62
+ ··· + 16u
3
− 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
63
− 21y
62
+ ··· − 64y
2
− 1
c
2
, c
5
y
63
− 37y
62
+ ··· + 16y
3
− 1
c
3
y
63
− 25y
62
+ ··· + 1252156y −100489
c
4
, c
10
y
63
+ 31y
62
+ ··· + 16y
3
− 1
c
6
, c
7
, c
11
c
12
y
63
+ 79y
62
+ ··· − 80y −1
c
8
y
63
− 13y
62
+ ··· + 27176y −961
c
9
y
63
+ 3y
62
+ ··· + 128y
2
− 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.621935 + 0.782730I
−9.96990 + 2.39006I −4.17438 + 0.I
u = −0.621935 − 0.782730I
−9.96990 − 2.39006I −4.17438 + 0.I
u = 0.438527 + 0.907377I
−1.96094 − 0.80799I −3.24469 + 2.24299I
u = 0.438527 − 0.907377I
−1.96094 + 0.80799I −3.24469 − 2.24299I
u = −0.630886 + 0.761128I
−9.90322 − 7.24194I −3.95069 + 6.24276I
u = −0.630886 − 0.761128I
−9.90322 + 7.24194I −3.95069 − 6.24276I
u = 0.618288 + 0.768075I
−6.12298 + 2.40025I −0.83833 − 3.23354I
u = 0.618288 − 0.768075I
−6.12298 − 2.40025I −0.83833 + 3.23354I
u = −0.473047 + 1.008350I
−0.44679 − 2.73565I 0
u = −0.473047 − 1.008350I
−0.44679 + 2.73565I 0
u = 0.309687 + 1.086540I
−4.07179 + 0.35561I 0
u = 0.309687 − 1.086540I
−4.07179 − 0.35561I 0
u = 0.281866 + 0.810813I
−2.03136 − 0.82581I −5.70235 + 1.89666I
u = 0.281866 − 0.810813I
−2.03136 + 0.82581I −5.70235 − 1.89666I
u = 0.563966 + 0.645928I
−1.23878 + 4.99873I −1.02686 − 8.32445I
u = 0.563966 − 0.645928I
−1.23878 − 4.99873I −1.02686 + 8.32445I
u = −0.274490 + 1.109110I
−6.89845 + 3.68623I 0
u = −0.274490 − 1.109110I
−6.89845 − 3.68623I 0
u = −0.512226 + 1.042810I
0.00316 − 3.27794I 0
u = −0.512226 − 1.042810I
0.00316 + 3.27794I 0
u = 0.795174 + 0.261887I
−12.4090 − 9.0599I −4.87131 + 5.08954I
u = 0.795174 − 0.261887I
−12.4090 + 9.0599I −4.87131 − 5.08954I
u = 0.792169 + 0.247301I
−12.62010 + 0.73676I −5.26880 − 0.86219I
u = 0.792169 − 0.247301I
−12.62010 − 0.73676I −5.26880 + 0.86219I
u = −0.788814 + 0.256117I
−8.63854 + 4.09024I −1.84772 − 2.15081I
u = −0.788814 − 0.256117I
−8.63854 − 4.09024I −1.84772 + 2.15081I
u = −0.335883 + 1.123950I
−7.59429 − 3.78030I 0
u = −0.335883 − 1.123950I
−7.59429 + 3.78030I 0
u = 0.444654 + 1.090310I
−4.15081 + 3.62964I 0
u = 0.444654 − 1.090310I
−4.15081 − 3.62964I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.530104 + 1.071360I
−0.56847 + 7.03450I 0
u = 0.530104 − 1.071360I
−0.56847 − 7.03450I 0
u = −0.286256 + 1.181430I
−13.08250 + 0.76656I 0
u = −0.286256 − 1.181430I
−13.08250 − 0.76656I 0
u = 0.280566 + 1.184730I
−16.9082 − 5.7443I 0
u = 0.280566 − 1.184730I
−16.9082 + 5.7443I 0
u = −0.720190 + 0.301321I
−2.72172 + 6.49499I −2.54742 − 7.00667I
u = −0.720190 − 0.301321I
−2.72172 − 6.49499I −2.54742 + 7.00667I
u = 0.291741 + 1.185570I
−17.0499 + 4.1182I 0
u = 0.291741 − 1.185570I
−17.0499 − 4.1182I 0
u = 0.531414 + 1.112570I
−2.57205 + 7.06874I 0
u = 0.531414 − 1.112570I
−2.57205 − 7.06874I 0
u = −0.505728 + 0.573150I
0.85780 − 1.28412I 3.73246 + 4.21223I
u = −0.505728 − 0.573150I
0.85780 + 1.28412I 3.73246 − 4.21223I
u = −0.514558 + 1.129920I
−6.38558 − 3.99438I 0
u = −0.514558 − 1.129920I
−6.38558 + 3.99438I 0
u = −0.544940 + 1.121930I
−5.10210 − 11.30890I 0
u = −0.544940 − 1.121930I
−5.10210 + 11.30890I 0
u = −0.572727 + 0.467677I
1.68698 − 1.08392I 5.60962 + 3.91399I
u = −0.572727 − 0.467677I
1.68698 + 1.08392I 5.60962 − 3.91399I
u = 0.620953 + 0.393540I
1.38897 − 2.48970I 3.99601 + 5.45980I
u = 0.620953 − 0.393540I
1.38897 + 2.48970I 3.99601 − 5.45980I
u = 0.673225 + 0.294907I
−0.22891 − 2.41591I 0.95752 + 3.52560I
u = 0.673225 − 0.294907I
−0.22891 + 2.41591I 0.95752 − 3.52560I
u = −0.685484 + 0.219433I
−3.79957 − 0.58028I −5.57705 + 0.79096I
u = −0.685484 − 0.219433I
−3.79957 + 0.58028I −5.57705 − 0.79096I
u = −0.549892 + 1.156170I
−11.2870 − 9.0785I 0
u = −0.549892 − 1.156170I
−11.2870 + 9.0785I 0
u = 0.547368 + 1.159540I
−15.3064 + 4.2467I 0
u = 0.547368 − 1.159540I
−15.3064 − 4.2467I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.553587 + 1.157060I
−15.0484 + 14.0814I 0
u = 0.553587 − 1.157060I
−15.0484 − 14.0814I 0
u = 0.487531
−1.48768 −6.31320
7
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
63
+ 37u
62
+ ··· + 16u
3
+ 1
c
2
, c
5
u
63
+ u
62
+ ··· + 2u + 1
c
3
u
63
− u
62
+ ··· − 386u + 317
c
4
, c
10
u
63
+ u
62
+ ··· + 2u + 1
c
6
, c
7
, c
11
c
12
u
63
+ 3u
62
+ ··· + 16u + 1
c
8
u
63
+ 5u
62
+ ··· + 360u + 31
c
9
u
63
+ 31u
62
+ ··· + 16u
3
− 1
8
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
63
− 21y
62
+ ··· − 64y
2
− 1
c
2
, c
5
y
63
− 37y
62
+ ··· + 16y
3
− 1
c
3
y
63
− 25y
62
+ ··· + 1252156y −100489
c
4
, c
10
y
63
+ 31y
62
+ ··· + 16y
3
− 1
c
6
, c
7
, c
11
c
12
y
63
+ 79y
62
+ ··· − 80y −1
c
8
y
63
− 13y
62
+ ··· + 27176y −961
c
9
y
63
+ 3y
62
+ ··· + 128y
2
− 1
9
