12a
0306
(K12a
0306
)
A knot diagram
1
Linearized knot diagam
3 6 8 10 7 2 1 12 11 4 9 5
Solving Sequence
4,11
10 5 9 12 1 8 3 2 7 6
c
10
c
4
c
9
c
11
c
12
c
8
c
3
c
1
c
7
c
6
c
2
, c
5
Ideals for irreducible components
2
of X
par
I
u
1
= hu
66
− u
65
+ ··· + 2u + 1i
I
u
2
= hu
7
+ u
5
+ 2u
3
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 73 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
66
− u
65
+ · · · + 2u + 1i
(i) Arc colorings
a
4
=
0
u
a
11
=
1
0
a
10
=
1
−u
2
a
5
=
−u
u
3
+ u
a
9
=
u
2
+ 1
−u
2
a
12
=
u
4
+ u
2
+ 1
−u
4
a
1
=
u
8
+ u
6
+ 3u
4
+ 2u
2
+ 1
−u
10
− 2u
8
− 3u
6
− 4u
4
− u
2
a
8
=
u
6
+ u
4
+ 2u
2
+ 1
−u
6
− u
2
a
3
=
−u
13
− 2u
11
− 5u
9
− 6u
7
− 6u
5
− 4u
3
− u
u
13
+ u
11
+ 3u
9
+ 2u
7
+ 2u
5
+ u
3
+ u
a
2
=
u
36
+ 5u
34
+ ··· + u
2
+ 1
−u
36
− 4u
34
+ ··· − 12u
8
− u
4
a
7
=
−u
24
− 3u
22
+ ··· + 2u
2
+ 1
u
26
+ 4u
24
+ ··· + 3u
6
− u
2
a
6
=
−u
47
− 6u
45
+ ··· − 4u
3
− 2u
u
49
+ 7u
47
+ ··· + 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
64
− 4u
63
+ ··· + 8u
2
+ 2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
66
+ 21u
65
+ ··· + 4u + 1
c
2
, c
6
u
66
− u
65
+ ··· − 2u + 1
c
3
, c
12
u
66
+ 6u
65
+ ··· + 1260u + 392
c
4
, c
10
u
66
− u
65
+ ··· + 2u + 1
c
7
u
66
+ 5u
65
+ ··· − 4u + 37
c
8
, c
9
, c
11
u
66
− 17u
65
+ ··· − 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
66
+ 49y
65
+ ··· + 76y + 1
c
2
, c
6
y
66
+ 21y
65
+ ··· + 4y + 1
c
3
, c
12
y
66
− 42y
65
+ ··· + 3127376y + 153664
c
4
, c
10
y
66
+ 17y
65
+ ··· + 4y + 1
c
7
y
66
− 7y
65
+ ··· − 10820y + 1369
c
8
, c
9
, c
11
y
66
+ 65y
65
+ ··· + 4y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.276142 + 0.973147I
4.50590 − 2.85550I 12.05805 + 4.41993I
u = 0.276142 − 0.973147I
4.50590 + 2.85550I 12.05805 − 4.41993I
u = −0.311087 + 0.967783I
1.20301 + 5.52787I 4.00000 − 8.08112I
u = −0.311087 − 0.967783I
1.20301 − 5.52787I 4.00000 + 8.08112I
u = −0.225553 + 0.996987I
7.62104 − 4.67869I 10.89032 + 2.00709I
u = −0.225553 − 0.996987I
7.62104 + 4.67869I 10.89032 − 2.00709I
u = 0.236153 + 0.997386I
8.32399 − 1.19963I 12.16015 + 3.25993I
u = 0.236153 − 0.997386I
8.32399 + 1.19963I 12.16015 − 3.25993I
u = 0.304523 + 1.001500I
7.92106 − 4.82325I 11.07978 + 4.36296I
u = 0.304523 − 1.001500I
7.92106 + 4.82325I 11.07978 − 4.36296I
u = −0.312943 + 1.002150I
7.10672 + 10.70210I 9.44810 − 9.41252I
u = −0.312943 − 1.002150I
7.10672 − 10.70210I 9.44810 + 9.41252I
u = 0.456777 + 0.785683I
1.23819 − 6.54922I 4.17677 + 9.67765I
u = 0.456777 − 0.785683I
1.23819 + 6.54922I 4.17677 − 9.67765I
u = −0.397793 + 0.800537I
1.82221 + 1.25981I 6.09781 − 4.41931I
u = −0.397793 − 0.800537I
1.82221 − 1.25981I 6.09781 + 4.41931I
u = 0.784027 + 0.781220I
1.10586 − 5.81643I 0
u = 0.784027 − 0.781220I
1.10586 + 5.81643I 0
u = −0.023490 + 0.865695I
3.89741 + 2.71211I 12.15997 − 3.32953I
u = −0.023490 − 0.865695I
3.89741 − 2.71211I 12.15997 + 3.32953I
u = −0.836827 + 0.807352I
−2.60329 − 1.13595I 0
u = −0.836827 − 0.807352I
−2.60329 + 1.13595I 0
u = −0.858606 + 0.795301I
0.37310 − 3.30299I 0
u = −0.858606 − 0.795301I
0.37310 + 3.30299I 0
u = 0.863434 + 0.797395I
−0.53580 + 9.14737I 0
u = 0.863434 − 0.797395I
−0.53580 − 9.14737I 0
u = 0.827585 + 0.837312I
−4.93871 − 2.14028I 0
u = 0.827585 − 0.837312I
−4.93871 + 2.14028I 0
u = 0.854921 + 0.812716I
−6.26873 + 3.63660I 0
u = 0.854921 − 0.812716I
−6.26873 − 3.63660I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.827211 + 0.875814I
−5.15023 − 2.33627I 0
u = 0.827211 − 0.875814I
−5.15023 + 2.33627I 0
u = −0.845186 + 0.880793I
−6.18098 − 2.57775I 0
u = −0.845186 − 0.880793I
−6.18098 + 2.57775I 0
u = 0.459757 + 0.623778I
−3.05863 − 1.75367I −4.05356 + 5.04373I
u = 0.459757 − 0.623778I
−3.05863 + 1.75367I −4.05356 − 5.04373I
u = 0.815789 + 0.918639I
−5.01899 − 3.79645I 0
u = 0.815789 − 0.918639I
−5.01899 + 3.79645I 0
u = −0.761181 + 0.966217I
2.20192 + 5.87663I 0
u = −0.761181 − 0.966217I
2.20192 − 5.87663I 0
u = −0.837522 + 0.902945I
−9.94428 + 3.11601I 0
u = −0.837522 − 0.902945I
−9.94428 − 3.11601I 0
u = 0.791445 + 0.947693I
−4.59364 − 3.91730I 0
u = 0.791445 − 0.947693I
−4.59364 + 3.91730I 0
u = −0.830031 + 0.923795I
−6.04701 + 8.80946I 0
u = −0.830031 − 0.923795I
−6.04701 − 8.80946I 0
u = −0.787054 + 0.967827I
−2.10788 + 7.20390I 0
u = −0.787054 − 0.967827I
−2.10788 − 7.20390I 0
u = 0.798774 + 0.972679I
−5.77098 − 9.79547I 0
u = 0.798774 − 0.972679I
−5.77098 + 9.79547I 0
u = −0.792645 + 0.983270I
0.95662 + 9.45046I 0
u = −0.792645 − 0.983270I
0.95662 − 9.45046I 0
u = 0.795929 + 0.984558I
0.0466 − 15.3200I 0
u = 0.795929 − 0.984558I
0.0466 + 15.3200I 0
u = −0.223454 + 0.622794I
0.329896 + 0.949161I 5.93547 − 7.10571I
u = −0.223454 − 0.622794I
0.329896 − 0.949161I 5.93547 + 7.10571I
u = 0.487684 + 0.434861I
0.22976 + 2.99529I 0.08554 − 2.50316I
u = 0.487684 − 0.434861I
0.22976 − 2.99529I 0.08554 + 2.50316I
u = −0.643478 + 0.086667I
4.24877 − 7.37536I 3.63430 + 5.60307I
u = −0.643478 − 0.086667I
4.24877 + 7.37536I 3.63430 − 5.60307I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.638810 + 0.068850I
5.02009 + 1.56359I 5.22655 − 0.46859I
u = 0.638810 − 0.068850I
5.02009 − 1.56359I 5.22655 + 0.46859I
u = −0.565065 + 0.112376I
−1.37963 − 2.39285I −2.15770 + 3.99262I
u = −0.565065 − 0.112376I
−1.37963 + 2.39285I −2.15770 − 3.99262I
u = −0.467047 + 0.336383I
0.51184 + 1.97273I 0.43404 − 3.18104I
u = −0.467047 − 0.336383I
0.51184 − 1.97273I 0.43404 + 3.18104I
7
II. I
u
2
= hu
7
+ u
5
+ 2u
3
+ u − 1i
(i) Arc colorings
a
4
=
0
u
a
11
=
1
0
a
10
=
1
−u
2
a
5
=
−u
u
3
+ u
a
9
=
u
2
+ 1
−u
2
a
12
=
u
4
+ u
2
+ 1
−u
4
a
1
=
u
4
+ u
2
+ u + 1
−u
4
− u
3
− u
a
8
=
u
6
+ u
4
+ 2u
2
+ 1
−u
6
− u
2
a
3
=
−u
6
− u
4
− 2u
2
− 1
u
6
+ u
2
+ u
a
2
=
u
4
+ u
2
+ 1
−u
6
− 2u
4
− u
2
a
7
=
u
3
−u
5
− u
3
+ u
a
6
=
−u
5
− u
−2u
3
+ 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
7
+ 2u
6
+ 5u
5
+ 6u
4
+ 6u
3
+ 4u
2
+ u − 1
c
2
, c
4
, c
6
c
10
u
7
+ u
5
+ 2u
3
+ u − 1
c
3
, c
12
(u − 1)
7
c
7
u
7
− 3u
5
− 2u
4
+ 8u
3
+ 2u
2
− u − 3
c
8
, c
9
, c
11
u
7
− 2u
6
+ 5u
5
− 6u
4
+ 6u
3
− 4u
2
+ u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
8
c
9
, c
11
y
7
+ 6y
6
+ 13y
5
+ 10y
4
+ 2y
3
+ 8y
2
+ 9y −1
c
2
, c
4
, c
6
c
10
y
7
+ 2y
6
+ 5y
5
+ 6y
4
+ 6y
3
+ 4y
2
+ y −1
c
3
, c
12
(y −1)
7
c
7
y
7
− 6y
6
+ 25y
5
− 54y
4
+ 78y
3
− 32y
2
+ 13y −9
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.237779 + 0.943997I
1.64493 6.00000
u = −0.237779 − 0.943997I
1.64493 6.00000
u = −0.799839 + 0.781167I
1.64493 6.00000
u = −0.799839 − 0.781167I
1.64493 6.00000
u = 0.755347 + 0.961681I
1.64493 6.00000
u = 0.755347 − 0.961681I
1.64493 6.00000
u = 0.564540
1.64493 6.00000
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
7
+ 2u
6
+ ··· + u − 1)(u
66
+ 21u
65
+ ··· + 4u + 1)
c
2
, c
6
(u
7
+ u
5
+ 2u
3
+ u − 1)(u
66
− u
65
+ ··· − 2u + 1)
c
3
, c
12
((u − 1)
7
)(u
66
+ 6u
65
+ ··· + 1260u + 392)
c
4
, c
10
(u
7
+ u
5
+ 2u
3
+ u − 1)(u
66
− u
65
+ ··· + 2u + 1)
c
7
(u
7
− 3u
5
− 2u
4
+ 8u
3
+ 2u
2
− u − 3)(u
66
+ 5u
65
+ ··· − 4u + 37)
c
8
, c
9
, c
11
(u
7
− 2u
6
+ ··· + u + 1)(u
66
− 17u
65
+ ··· − 4u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
7
+ 6y
6
+ 13y
5
+ 10y
4
+ 2y
3
+ 8y
2
+ 9y −1)
· (y
66
+ 49y
65
+ ··· + 76y + 1)
c
2
, c
6
(y
7
+ 2y
6
+ ··· + y −1)(y
66
+ 21y
65
+ ··· + 4y + 1)
c
3
, c
12
((y −1)
7
)(y
66
− 42y
65
+ ··· + 3127376y + 153664)
c
4
, c
10
(y
7
+ 2y
6
+ ··· + y −1)(y
66
+ 17y
65
+ ··· + 4y + 1)
c
7
(y
7
− 6y
6
+ 25y
5
− 54y
4
+ 78y
3
− 32y
2
+ 13y −9)
· (y
66
− 7y
65
+ ··· − 10820y + 1369)
c
8
, c
9
, c
11
(y
7
+ 6y
6
+ 13y
5
+ 10y
4
+ 2y
3
+ 8y
2
+ 9y −1)
· (y
66
+ 65y
65
+ ··· + 4y + 1)
13
