12a
0506
(K12a
0506
)
A knot diagram
1
Linearized knot diagam
3 7 8 9 10 1 2 12 11 5 6 4
Solving Sequence
5,10
6 11 12 9 4 1 7 8 3 2
c
5
c
10
c
11
c
9
c
4
c
12
c
6
c
8
c
3
c
2
c
1
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
90
+ 24u
88
+ ··· + u + 1i
I
u
2
= hu
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 92 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
90
+ 24u
88
+ · · · + u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
11
=
−u
u
a
12
=
u
3
u
5
+ u
3
+ u
a
9
=
−u
3
u
3
+ u
a
4
=
−u
6
− u
4
+ 1
u
6
+ 2u
4
+ u
2
a
1
=
−u
17
− 4u
15
− 7u
13
− 4u
11
+ 3u
9
+ 6u
7
+ 2u
5
− u
u
17
+ 5u
15
+ 11u
13
+ 12u
11
+ 5u
9
− 2u
7
− 2u
5
+ u
a
7
=
−u
36
− 9u
34
+ ··· − u
2
+ 1
u
36
+ 10u
34
+ ··· + u
4
+ 2u
2
a
8
=
−u
11
− 2u
9
− 2u
7
− u
3
−u
13
− 3u
11
− 5u
9
− 4u
7
− 2u
5
+ u
3
+ u
a
3
=
u
30
+ 7u
28
+ ··· − 2u
12
+ 1
u
32
+ 8u
30
+ ··· + 12u
8
+ 4u
6
a
2
=
−u
79
− 20u
77
+ ··· + 4u
5
− 2u
−u
81
− 21u
79
+ ··· − 2u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
89
− 96u
87
+ ··· − 8u − 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
90
+ 48u
89
+ ··· + u + 1
c
2
, c
7
u
90
+ 24u
88
+ ··· − u + 1
c
3
, c
6
u
90
− 36u
88
+ ··· + 35u + 1
c
4
, c
11
u
90
− 36u
88
+ ··· − 35u + 1
c
5
, c
10
u
90
+ 24u
88
+ ··· + u + 1
c
8
u
90
+ 12u
89
+ ··· + 355u + 29
c
9
u
90
− 48u
89
+ ··· − u + 1
c
12
u
90
− 12u
89
+ ··· − 355u + 29
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
9
y
90
− 12y
89
+ ··· + 5y + 1
c
2
, c
5
, c
7
c
10
y
90
+ 48y
89
+ ··· + y + 1
c
3
, c
4
, c
6
c
11
y
90
− 72y
89
+ ··· − 767y + 1
c
8
, c
12
y
90
+ 8y
89
+ ··· + 16481y + 841
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.556433 + 0.835324I
−7.18763 + 2.08490I 0
u = −0.556433 − 0.835324I
−7.18763 − 2.08490I 0
u = −0.024996 + 0.994939I
3.43520 + 2.09623I 8.58300 + 0.I
u = −0.024996 − 0.994939I
3.43520 − 2.09623I 8.58300 + 0.I
u = −0.123552 + 0.999729I
1.18678 + 2.46011I 0
u = −0.123552 − 0.999729I
1.18678 − 2.46011I 0
u = 0.548923 + 0.847737I
−3.24727 − 6.11532I 0
u = 0.548923 − 0.847737I
−3.24727 + 6.11532I 0
u = −0.557501 + 0.853846I
−6.39706 + 10.87620I 0
u = −0.557501 − 0.853846I
−6.39706 − 10.87620I 0
u = 0.183246 + 1.004240I
−2.41430 + 1.30442I 0
u = 0.183246 − 1.004240I
−2.41430 − 1.30442I 0
u = −0.451800 + 0.860540I
0.69661 + 2.01708I 0
u = −0.451800 − 0.860540I
0.69661 − 2.01708I 0
u = 0.128657 + 1.032900I
−1.77995 − 7.01061I 0
u = 0.128657 − 1.032900I
−1.77995 + 7.01061I 0
u = 0.510603 + 0.767949I
−3.43520 − 2.09623I −8.58300 + 4.19142I
u = 0.510603 − 0.767949I
−3.43520 + 2.09623I −8.58300 − 4.19142I
u = −0.565146 + 0.686956I
−7.60942 + 2.39548I −8.03590 − 3.45801I
u = −0.565146 − 0.686956I
−7.60942 − 2.39548I −8.03590 + 3.45801I
u = −0.570856 + 0.661159I
−6.94292 − 6.37762I −6.79931 + 3.36470I
u = −0.570856 − 0.661159I
−6.94292 + 6.37762I −6.79931 − 3.36470I
u = 0.556299 + 0.668324I
−3.75583 + 1.68060I −3.85601 − 0.15656I
u = 0.556299 − 0.668324I
−3.75583 − 1.68060I −3.85601 + 0.15656I
u = −0.362656 + 0.775470I
0.22941 + 1.46786I 2.13380 − 4.56718I
u = −0.362656 − 0.775470I
0.22941 − 1.46786I 2.13380 + 4.56718I
u = 0.804947 + 0.154678I
−3.05581 + 11.52890I −3.10731 − 7.64918I
u = 0.804947 − 0.154678I
−3.05581 − 11.52890I −3.10731 + 7.64918I
u = −0.798246 + 0.151160I
− 6.69000I 0. + 4.60895I
u = −0.798246 − 0.151160I
6.69000I 0. − 4.60895I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.791540 + 0.160688I
−4.06433 + 2.78100I −4.82687 − 1.51453I
u = 0.791540 − 0.160688I
−4.06433 − 2.78100I −4.82687 + 1.51453I
u = −0.793161 + 0.120606I
3.24727 − 6.11532I 2.14726 + 6.93728I
u = −0.793161 − 0.120606I
3.24727 + 6.11532I 2.14726 − 6.93728I
u = 0.482440 + 1.099170I
−3.44119 + 0.81536I 0
u = 0.482440 − 1.099170I
−3.44119 − 0.81536I 0
u = 0.784724 + 0.103510I
3.75583 + 1.68060I 3.85601 − 0.15656I
u = 0.784724 − 0.103510I
3.75583 − 1.68060I 3.85601 + 0.15656I
u = −0.479940 + 1.114160I
−0.05773 + 3.67296I 0
u = −0.479940 − 1.114160I
−0.05773 − 3.67296I 0
u = 0.493654 + 1.117110I
−3.67430 − 7.96529I 0
u = 0.493654 − 1.117110I
−3.67430 + 7.96529I 0
u = −0.776515 + 0.033180I
0.05773 + 3.67296I −0.34241 − 3.66459I
u = −0.776515 − 0.033180I
0.05773 − 3.67296I −0.34241 + 3.66459I
u = 0.469286 + 0.612610I
−0.69661 + 2.01708I −2.53308 − 3.61771I
u = 0.469286 − 0.612610I
−0.69661 − 2.01708I −2.53308 + 3.61771I
u = −0.406847 + 1.161990I
2.41430 + 1.30442I 0
u = −0.406847 − 1.161990I
2.41430 − 1.30442I 0
u = 0.758452 + 0.064597I
2.47004 + 0.60077I 3.44076 − 0.47239I
u = 0.758452 − 0.064597I
2.47004 − 0.60077I 3.44076 + 0.47239I
u = 0.365897 + 1.197990I
− 1.05575I 0
u = 0.365897 − 1.197990I
1.05575I 0
u = −0.371458 + 1.204960I
4.06433 − 2.78100I 0
u = −0.371458 − 1.204960I
4.06433 + 2.78100I 0
u = 0.367726 + 1.209060I
1.05381 + 7.61321I 0
u = 0.367726 − 1.209060I
1.05381 − 7.61321I 0
u = 0.422336 + 1.195770I
6.11748 − 3.54885I 0
u = 0.422336 − 1.195770I
6.11748 + 3.54885I 0
u = −0.391217 + 1.206550I
7.18763 − 2.08490I 0
u = −0.391217 − 1.206550I
7.18763 + 2.08490I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.493698 + 1.168560I
1.77995 + 7.01061I 0
u = −0.493698 − 1.168560I
1.77995 − 7.01061I 0
u = 0.401574 + 1.204250I
7.60942 − 2.39548I 0
u = 0.401574 − 1.204250I
7.60942 + 2.39548I 0
u = −0.716511 + 0.137526I
−1.18678 − 2.46011I −5.77540 + 3.61513I
u = −0.716511 − 0.137526I
−1.18678 + 2.46011I −5.77540 − 3.61513I
u = −0.433989 + 1.202730I
3.67430 + 7.96529I 0
u = −0.433989 − 1.202730I
3.67430 − 7.96529I 0
u = 0.478981 + 1.188060I
5.71402 − 5.12960I 0
u = 0.478981 − 1.188060I
5.71402 + 5.12960I 0
u = −0.466527 + 1.195460I
3.44119 + 0.81536I 0
u = −0.466527 − 1.195460I
3.44119 − 0.81536I 0
u = 0.495551 + 1.192980I
6.94292 − 6.37762I 0
u = 0.495551 − 1.192980I
6.94292 + 6.37762I 0
u = 0.516466 + 1.184590I
−1.05381 − 7.61321I 0
u = 0.516466 − 1.184590I
−1.05381 + 7.61321I 0
u = −0.502991 + 1.193330I
6.39706 + 10.87620I 0
u = −0.502991 − 1.193330I
6.39706 − 10.87620I 0
u = −0.514902 + 1.189080I
3.05581 + 11.52890I 0
u = −0.514902 − 1.189080I
3.05581 − 11.52890I 0
u = 0.517655 + 1.190580I
− 16.3978I 0
u = 0.517655 − 1.190580I
16.3978I 0
u = 0.645018 + 0.270245I
−6.11748 + 3.54885I −7.14499 − 3.21882I
u = 0.645018 − 0.270245I
−6.11748 − 3.54885I −7.14499 + 3.21882I
u = 0.619550 + 0.309895I
−5.71402 − 5.12960I −6.45992 + 3.90146I
u = 0.619550 − 0.309895I
−5.71402 + 5.12960I −6.45992 − 3.90146I
u = −0.607875 + 0.278085I
−2.47004 + 0.60077I −3.44076 − 0.47239I
u = −0.607875 − 0.278085I
−2.47004 − 0.60077I −3.44076 + 0.47239I
u = −0.376707 + 0.399911I
−0.22941 + 1.46786I −2.13380 − 4.56718I
u = −0.376707 − 0.399911I
−0.22941 − 1.46786I −2.13380 + 4.56718I
7
II. I
u
2
= hu
2
− u + 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u − 1
a
11
=
−u
u
a
12
=
−1
0
a
9
=
1
u − 1
a
4
=
u
−u
a
1
=
u − 2
−u + 1
a
7
=
−u
u
a
8
=
−u + 2
u − 1
a
3
=
−1
−u + 1
a
2
=
−2
−u + 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 12u − 6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
7
, c
11
, c
12
u
2
+ u + 1
c
3
, c
5
, c
6
c
8
, c
9
, c
10
u
2
− u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y
2
+ y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
− 6.08965I 0. + 10.39230I
u = 0.500000 − 0.866025I
6.08965I 0. − 10.39230I
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
+ u + 1)(u
90
+ 48u
89
+ ··· + u + 1)
c
2
, c
7
(u
2
+ u + 1)(u
90
+ 24u
88
+ ··· − u + 1)
c
3
, c
6
(u
2
− u + 1)(u
90
− 36u
88
+ ··· + 35u + 1)
c
4
, c
11
(u
2
+ u + 1)(u
90
− 36u
88
+ ··· − 35u + 1)
c
5
, c
10
(u
2
− u + 1)(u
90
+ 24u
88
+ ··· + u + 1)
c
8
(u
2
− u + 1)(u
90
+ 12u
89
+ ··· + 355u + 29)
c
9
(u
2
− u + 1)(u
90
− 48u
89
+ ··· − u + 1)
c
12
(u
2
+ u + 1)(u
90
− 12u
89
+ ··· − 355u + 29)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
9
(y
2
+ y + 1)(y
90
− 12y
89
+ ··· + 5y + 1)
c
2
, c
5
, c
7
c
10
(y
2
+ y + 1)(y
90
+ 48y
89
+ ··· + y + 1)
c
3
, c
4
, c
6
c
11
(y
2
+ y + 1)(y
90
− 72y
89
+ ··· − 767y + 1)
c
8
, c
12
(y
2
+ y + 1)(y
90
+ 8y
89
+ ··· + 16481y + 841)
13
