12a
0540
(K12a
0540
)
A knot diagram
1
Linearized knot diagam
3 7 8 10 11 12 2 1 6 5 4 9
Solving Sequence
2,7
3 8 4 1 9 12 6 10 11 5
c
2
c
7
c
3
c
1
c
8
c
12
c
6
c
9
c
11
c
5
c
4
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
82
− u
81
+ ··· + 3u − 1i
* 1 irreducible components of dim
C
= 0, with total 82 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
82
− u
81
+ · · · + 3u − 1i
(i) Arc colorings
a
2
=
1
0
a
7
=
0
u
a
3
=
1
−u
2
a
8
=
u
u
a
4
=
u
4
+ u
2
+ 1
u
4
a
1
=
u
2
+ 1
−u
4
a
9
=
−u
7
− 2u
5
− 2u
3
u
9
+ u
7
+ u
5
+ u
a
12
=
u
12
+ 3u
10
+ 5u
8
+ 4u
6
+ 2u
4
+ u
2
+ 1
−u
14
− 2u
12
− 3u
10
− 2u
8
− 2u
6
− 2u
4
− u
2
a
6
=
u
25
+ 6u
23
+ ··· + 2u
3
+ u
−u
27
− 5u
25
+ ··· − u
3
+ u
a
10
=
−u
43
− 10u
41
+ ··· − 8u
5
− 3u
3
u
45
+ 9u
43
+ ··· − u
3
+ u
a
11
=
−u
22
− 5u
20
+ ··· − 3u
4
+ 1
−u
22
− 4u
20
+ ··· − 2u
4
− u
2
a
5
=
−u
71
− 16u
69
+ ··· + 2u
3
+ 2u
−u
71
− 15u
69
+ ··· − 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
80
− 4u
79
+ ··· + 16u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
82
+ 37u
81
+ ··· − 3u + 1
c
2
, c
7
u
82
+ u
81
+ ··· − 3u − 1
c
3
u
82
− u
81
+ ··· + 13u − 2
c
4
, c
5
, c
10
u
82
− u
81
+ ··· − 3u − 1
c
6
u
82
+ u
81
+ ··· − 1107u − 521
c
8
, c
12
u
82
+ 5u
81
+ ··· − 208u − 16
c
9
, c
11
u
82
+ 3u
81
+ ··· + 503u + 88
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
82
+ 17y
81
+ ··· − 31y + 1
c
2
, c
7
y
82
+ 37y
81
+ ··· − 3y + 1
c
3
y
82
− 3y
81
+ ··· − 281y + 4
c
4
, c
5
, c
10
y
82
− 67y
81
+ ··· − 3y + 1
c
6
y
82
+ 17y
81
+ ··· + 5469401y + 271441
c
8
, c
12
y
82
+ 65y
81
+ ··· − 5920y + 256
c
9
, c
11
y
82
+ 57y
81
+ ··· − 88625y + 7744
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.133466 + 1.000640I
−1.27367 − 1.16757I 0
u = 0.133466 − 1.000640I
−1.27367 + 1.16757I 0
u = −0.532862 + 0.830572I
−0.02430 − 6.14834I 0
u = −0.532862 − 0.830572I
−0.02430 + 6.14834I 0
u = 0.524056 + 0.795819I
4.00847 + 2.15498I 0. − 3.78589I
u = 0.524056 − 0.795819I
4.00847 − 2.15498I 0. + 3.78589I
u = −0.284834 + 0.897551I
−0.67768 − 1.28884I −8.00000 + 4.92802I
u = −0.284834 − 0.897551I
−0.67768 + 1.28884I −8.00000 − 4.92802I
u = −0.170246 + 1.062460I
−6.83445 + 2.33727I 0
u = −0.170246 − 1.062460I
−6.83445 − 2.33727I 0
u = −0.749314 + 0.538316I
4.42162 − 7.61184I −4.45987 + 5.77896I
u = −0.749314 − 0.538316I
4.42162 + 7.61184I −4.45987 − 5.77896I
u = 0.752586 + 0.529461I
8.86798 + 3.35426I 0. − 3.38737I
u = 0.752586 − 0.529461I
8.86798 − 3.35426I 0. + 3.38737I
u = −0.525332 + 0.751819I
0.19898 + 1.80626I −5.15609 + 0.I
u = −0.525332 − 0.751819I
0.19898 − 1.80626I −5.15609 + 0.I
u = −0.755993 + 0.517707I
5.56056 + 0.94896I −3.15680 + 0.I
u = −0.755993 − 0.517707I
5.56056 − 0.94896I −3.15680 + 0.I
u = 0.287923 + 1.044900I
−4.82508 + 3.82270I 0
u = 0.287923 − 1.044900I
−4.82508 − 3.82270I 0
u = 0.081808 + 1.082030I
−0.0248733 − 0.0847749I 0
u = 0.081808 − 1.082030I
−0.0248733 + 0.0847749I 0
u = −0.098958 + 1.094980I
3.18345 + 4.26960I 0
u = −0.098958 − 1.094980I
3.18345 − 4.26960I 0
u = 0.781133 + 0.445371I
5.16257 − 1.98343I −3.74120 + 0.I
u = 0.781133 − 0.445371I
5.16257 + 1.98343I −3.74120 + 0.I
u = −0.785449 + 0.435883I
8.35181 + 6.28657I −0.78678 − 3.84368I
u = −0.785449 − 0.435883I
8.35181 − 6.28657I −0.78678 + 3.84368I
u = 0.787695 + 0.428827I
3.81832 − 10.53440I −5.33981 + 6.01257I
u = 0.787695 − 0.428827I
3.81832 + 10.53440I −5.33981 − 6.01257I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.109809 + 1.102990I
−1.33118 − 8.43930I 0
u = 0.109809 − 1.102990I
−1.33118 + 8.43930I 0
u = −0.359135 + 1.056470I
−1.102440 − 0.747841I 0
u = −0.359135 − 1.056470I
−1.102440 + 0.747841I 0
u = −0.724858 + 0.480472I
3.67849 − 0.28703I −3.38751 + 3.80148I
u = −0.724858 − 0.480472I
3.67849 + 0.28703I −3.38751 − 3.80148I
u = 0.743794 + 0.448446I
3.49632 − 2.84575I −4.21837 + 4.55562I
u = 0.743794 − 0.448446I
3.49632 + 2.84575I −4.21837 − 4.55562I
u = 0.430453 + 1.053190I
−3.25125 + 3.38062I 0
u = 0.430453 − 1.053190I
−3.25125 − 3.38062I 0
u = 0.365552 + 1.084130I
−5.57182 − 2.81011I 0
u = 0.365552 − 1.084130I
−5.57182 + 2.81011I 0
u = −0.744053 + 0.414739I
−2.14948 + 4.40174I −9.50950 − 3.74202I
u = −0.744053 − 0.414739I
−2.14948 − 4.40174I −9.50950 + 3.74202I
u = 0.679349 + 0.510566I
−1.62052 + 2.05298I −8.39395 − 3.31884I
u = 0.679349 − 0.510566I
−1.62052 − 2.05298I −8.39395 + 3.31884I
u = 0.495189 + 1.047400I
−3.38134 + 2.83643I 0
u = 0.495189 − 1.047400I
−3.38134 − 2.83643I 0
u = −0.423498 + 1.091700I
−9.08228 − 3.64788I 0
u = −0.423498 − 1.091700I
−9.08228 + 3.64788I 0
u = −0.475657 + 1.085560I
−0.31884 − 6.30832I 0
u = −0.475657 − 1.085560I
−0.31884 + 6.30832I 0
u = 0.563996 + 1.045740I
−3.20812 + 2.76694I 0
u = 0.563996 − 1.045740I
−3.20812 − 2.76694I 0
u = 0.470135 + 1.098680I
−4.87213 + 10.11490I 0
u = 0.470135 − 1.098680I
−4.87213 − 10.11490I 0
u = −0.620596 + 1.036150I
2.94040 + 2.41061I 0
u = −0.620596 − 1.036150I
2.94040 − 2.41061I 0
u = 0.620020 + 1.042550I
7.34120 + 1.85374I 0
u = 0.620020 − 1.042550I
7.34120 − 1.85374I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.592434 + 1.063710I
1.94954 − 4.75248I 0
u = −0.592434 − 1.063710I
1.94954 + 4.75248I 0
u = −0.618735 + 1.050250I
3.97605 − 6.16137I 0
u = −0.618735 − 1.050250I
3.97605 + 6.16137I 0
u = 0.595132 + 1.081620I
1.62331 + 7.94105I 0
u = 0.595132 − 1.081620I
1.62331 − 7.94105I 0
u = −0.587329 + 1.094110I
−4.15024 − 9.46471I 0
u = −0.587329 − 1.094110I
−4.15024 + 9.46471I 0
u = 0.609197 + 1.092830I
3.23683 + 7.22327I 0
u = 0.609197 − 1.092830I
3.23683 − 7.22327I 0
u = −0.608118 + 1.098090I
6.38214 − 11.53260I 0
u = −0.608118 − 1.098090I
6.38214 + 11.53260I 0
u = 0.606813 + 1.101570I
1.8176 + 15.7804I 0
u = 0.606813 − 1.101570I
1.8176 − 15.7804I 0
u = 0.567507 + 0.269437I
−1.29725 + 1.34677I −7.56967 − 0.28082I
u = 0.567507 − 0.269437I
−1.29725 − 1.34677I −7.56967 + 0.28082I
u = 0.609013 + 0.145464I
−2.26753 − 6.00853I −9.43299 + 5.77985I
u = 0.609013 − 0.145464I
−2.26753 + 6.00853I −9.43299 − 5.77985I
u = −0.581311 + 0.180322I
2.13139 + 2.21650I −4.07689 − 3.88447I
u = −0.581311 − 0.180322I
2.13139 − 2.21650I −4.07689 + 3.88447I
u = −0.572870
−6.20998 −14.2450
u = 0.421035
−0.786796 −12.6530
7
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
82
+ 37u
81
+ ··· − 3u + 1
c
2
, c
7
u
82
+ u
81
+ ··· − 3u − 1
c
3
u
82
− u
81
+ ··· + 13u − 2
c
4
, c
5
, c
10
u
82
− u
81
+ ··· − 3u − 1
c
6
u
82
+ u
81
+ ··· − 1107u − 521
c
8
, c
12
u
82
+ 5u
81
+ ··· − 208u − 16
c
9
, c
11
u
82
+ 3u
81
+ ··· + 503u + 88
8
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
82
+ 17y
81
+ ··· − 31y + 1
c
2
, c
7
y
82
+ 37y
81
+ ··· − 3y + 1
c
3
y
82
− 3y
81
+ ··· − 281y + 4
c
4
, c
5
, c
10
y
82
− 67y
81
+ ··· − 3y + 1
c
6
y
82
+ 17y
81
+ ··· + 5469401y + 271441
c
8
, c
12
y
82
+ 65y
81
+ ··· − 5920y + 256
c
9
, c
11
y
82
+ 57y
81
+ ··· − 88625y + 7744
9
