12a
0508
(K12a
0508
)
A knot diagram
1
Linearized knot diagam
3 7 8 9 11 2 6 4 1 12 5 10
Solving Sequence
2,7
3 1 6 8 4 9 5 10 12 11
c
2
c
1
c
6
c
7
c
3
c
8
c
4
c
9
c
12
c
11
c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
64
− u
63
+ ··· − 2u − 1i
* 1 irreducible components of dim
C
= 0, with total 64 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
64
− u
63
+ · · · − 2u − 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
4
a
6
=
u
u
a
8
=
−u
3
−u
3
+ u
a
4
=
−u
8
+ u
6
− u
4
+ 1
−u
8
+ 2u
6
− 2u
4
+ 2u
2
a
9
=
u
13
− 2u
11
+ 3u
9
− 2u
7
− u
u
13
− 3u
11
+ 5u
9
− 6u
7
+ 4u
5
− 3u
3
+ u
a
5
=
u
18
− 3u
16
+ 6u
14
− 7u
12
+ 5u
10
− 3u
8
− u
2
+ 1
u
18
− 4u
16
+ 9u
14
− 14u
12
+ 15u
10
− 14u
8
+ 10u
6
− 6u
4
+ 3u
2
a
10
=
u
19
− 4u
17
+ 10u
15
− 16u
13
+ 19u
11
− 18u
9
+ 14u
7
− 10u
5
+ 5u
3
− 2u
u
21
− 3u
19
+ ··· − 3u
3
+ u
a
12
=
−u
36
+ 7u
34
+ ··· + u
2
+ 1
−u
38
+ 6u
36
+ ··· + 6u
4
− u
2
a
11
=
u
53
− 10u
51
+ ··· + 8u
3
− 3u
u
55
− 9u
53
+ ··· − 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
62
+ 44u
60
+ ··· − 24u − 14
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
64
+ 23u
63
+ ··· + 20u
2
+ 1
c
2
, c
6
u
64
− u
63
+ ··· − 2u − 1
c
3
, c
4
, c
8
u
64
+ u
63
+ ··· + 10u
2
− 25
c
5
, c
11
u
64
+ u
63
+ ··· − 2u − 1
c
9
, c
10
, c
12
u
64
+ 17u
63
+ ··· − 20u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
64
+ 37y
63
+ ··· + 40y + 1
c
2
, c
6
y
64
− 23y
63
+ ··· + 20y
2
+ 1
c
3
, c
4
, c
8
y
64
− 59y
63
+ ··· − 500y + 625
c
5
, c
11
y
64
+ 17y
63
+ ··· − 20y
2
+ 1
c
9
, c
10
, c
12
y
64
+ 61y
63
+ ··· − 40y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.734158 + 0.668101I
1.43109 − 1.27456I −5.99922 + 4.38526I
u = −0.734158 − 0.668101I
1.43109 + 1.27456I −5.99922 − 4.38526I
u = 0.577379 + 0.799136I
2.16758 + 9.15664I −5.96705 − 5.25155I
u = 0.577379 − 0.799136I
2.16758 − 9.15664I −5.96705 + 5.25155I
u = −0.583021 + 0.792165I
2.76820 − 3.03900I −4.88415 + 0.40129I
u = −0.583021 − 0.792165I
2.76820 + 3.03900I −4.88415 − 0.40129I
u = 0.548606 + 0.780128I
−4.67791 + 4.46553I −11.24364 − 4.14724I
u = 0.548606 − 0.780128I
−4.67791 − 4.46553I −11.24364 + 4.14724I
u = −0.880732 + 0.573012I
−1.06214 + 2.24840I −14.4188 − 2.9053I
u = −0.880732 − 0.573012I
−1.06214 − 2.24840I −14.4188 + 2.9053I
u = 0.908597 + 0.273784I
2.57323 − 5.53278I −10.06397 + 6.98986I
u = 0.908597 − 0.273784I
2.57323 + 5.53278I −10.06397 − 6.98986I
u = 0.814938 + 0.667976I
2.49514 − 2.06773I 0. + 3.71877I
u = 0.814938 − 0.667976I
2.49514 + 2.06773I 0. − 3.71877I
u = −0.887915 + 0.310649I
2.77545 − 0.35472I −9.35777 − 1.48918I
u = −0.887915 − 0.310649I
2.77545 + 0.35472I −9.35777 + 1.48918I
u = −0.551226 + 0.748282I
−1.69458 − 1.28164I −5.29966 + 0.30789I
u = −0.551226 − 0.748282I
−1.69458 + 1.28164I −5.29966 − 0.30789I
u = −0.770088 + 0.744037I
8.45315 − 3.75871I 0. + 2.83567I
u = −0.770088 − 0.744037I
8.45315 + 3.75871I 0. − 2.83567I
u = 0.781202 + 0.741537I
8.62407 − 2.42872I 0
u = 0.781202 − 0.741537I
8.62407 + 2.42872I 0
u = 0.512976 + 0.755043I
−4.91649 − 1.59221I −11.90786 + 3.54404I
u = 0.512976 − 0.755043I
−4.91649 + 1.59221I −11.90786 − 3.54404I
u = 0.886320 + 0.663470I
2.27623 − 3.08946I 0
u = 0.886320 − 0.663470I
2.27623 + 3.08946I 0
u = 1.11340
−7.26377 −11.6170
u = 1.117230 + 0.039150I
−3.19336 − 1.95587I −8.00000 + 0.I
u = 1.117230 − 0.039150I
−3.19336 + 1.95587I −8.00000 + 0.I
u = 0.869584 + 0.106333I
−3.24427 − 2.18605I −17.0727 + 6.0888I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.869584 − 0.106333I
−3.24427 + 2.18605I −17.0727 − 6.0888I
u = −1.125770 + 0.039610I
−3.84567 + 8.00446I 0
u = −1.125770 − 0.039610I
−3.84567 − 8.00446I 0
u = −1.128630 + 0.012921I
−10.47200 + 3.12169I 0
u = −1.128630 − 0.012921I
−10.47200 − 3.12169I 0
u = 0.461327 + 0.727213I
1.44734 − 6.27170I −6.61668 + 5.37317I
u = 0.461327 − 0.727213I
1.44734 + 6.27170I −6.61668 − 5.37317I
u = −0.936943 + 0.658530I
0.82334 + 6.42603I 0
u = −0.936943 − 0.658530I
0.82334 − 6.42603I 0
u = −0.465307 + 0.705263I
2.01780 + 0.29717I −5.58354 − 0.33676I
u = −0.465307 − 0.705263I
2.01780 − 0.29717I −5.58354 + 0.33676I
u = 0.925319 + 0.713366I
8.18783 − 3.10258I 0
u = 0.925319 − 0.713366I
8.18783 + 3.10258I 0
u = −0.933860 + 0.711939I
7.95805 + 9.29310I 0
u = −0.933860 − 0.711939I
7.95805 − 9.29310I 0
u = −1.038150 + 0.620576I
0.43349 + 4.75751I 0
u = −1.038150 − 0.620576I
0.43349 − 4.75751I 0
u = 1.047470 + 0.619918I
−0.204610 + 1.166380I 0
u = 1.047470 − 0.619918I
−0.204610 − 1.166380I 0
u = −1.041800 + 0.654112I
−3.12169 + 6.62412I 0
u = −1.041800 − 0.654112I
−3.12169 − 6.62412I 0
u = 1.051460 + 0.642911I
−6.47434 − 3.70666I 0
u = 1.051460 − 0.642911I
−6.47434 + 3.70666I 0
u = 1.052550 + 0.660953I
−6.16225 − 9.90913I 0
u = 1.052550 − 0.660953I
−6.16225 + 9.90913I 0
u = −1.046730 + 0.676138I
1.38825 + 8.57764I 0
u = −1.046730 − 0.676138I
1.38825 − 8.57764I 0
u = 1.050950 + 0.676533I
0.7567 − 14.7133I 0
u = 1.050950 − 0.676533I
0.7567 + 14.7133I 0
u = −0.684394
−1.02359 −9.41090
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.017794 + 0.518720I
5.24287 + 3.00974I −2.07163 − 2.91236I
u = −0.017794 − 0.518720I
5.24287 − 3.00974I −2.07163 + 2.91236I
u = −0.178281 + 0.318594I
−0.382136 + 0.981633I −6.53805 − 6.72584I
u = −0.178281 − 0.318594I
−0.382136 − 0.981633I −6.53805 + 6.72584I
7
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
u
64
+ 23u
63
+ ··· + 20u
2
+ 1
c
2
, c
6
u
64
− u
63
+ ··· − 2u − 1
c
3
, c
4
, c
8
u
64
+ u
63
+ ··· + 10u
2
− 25
c
5
, c
11
u
64
+ u
63
+ ··· − 2u − 1
c
9
, c
10
, c
12
u
64
+ 17u
63
+ ··· − 20u
2
+ 1
8
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
64
+ 37y
63
+ ··· + 40y + 1
c
2
, c
6
y
64
− 23y
63
+ ··· + 20y
2
+ 1
c
3
, c
4
, c
8
y
64
− 59y
63
+ ··· − 500y + 625
c
5
, c
11
y
64
+ 17y
63
+ ··· − 20y
2
+ 1
c
9
, c
10
, c
12
y
64
+ 61y
63
+ ··· − 40y + 1
9
