12n
0237
(K12n
0237
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 10 11 12 3 1 6 8 7
Solving Sequence
8,11 4,12
3 9 7 1 2 6 10 5
c
11
c
3
c
8
c
7
c
12
c
1
c
6
c
10
c
5
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
41
+ u
40
+ ··· + b + 2u, u
41
+ u
40
+ ··· + a − 4u, u
42
+ 2u
41
+ ··· + u + 1i
I
u
2
= h−u
4
+ u
3
− 2u
2
+ b + u, a, u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 48 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
41
+ u
40
+ · · · +b + 2u, u
41
+ u
40
+ · · · +a − 4u, u
42
+ 2u
41
+ · · · +u + 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
4
=
−u
41
− u
40
+ ··· + 2u
2
+ 4u
−u
41
− u
40
+ ··· − 8u
2
− 2u
a
12
=
1
u
2
a
3
=
−u
41
− u
40
+ ··· + 2u
2
+ 4u
−u
41
− 2u
40
+ ··· − 2u − 1
a
9
=
u
12
+ 5u
10
+ 9u
8
+ 4u
6
− 6u
4
− 5u
2
+ 1
u
14
+ 6u
12
+ 13u
10
+ 10u
8
− 4u
6
− 8u
4
− u
2
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
u
9
+ 4u
7
+ 5u
5
− 3u
−u
37
− u
36
+ ··· + 8u
2
+ 2u
a
6
=
u
3
+ 2u
u
3
+ u
a
10
=
−u
6
− 3u
4
− 2u
2
+ 1
−u
6
− 2u
4
− u
2
a
5
=
−u
9
− 4u
7
− 5u
5
+ 3u
−u
9
− 3u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
41
− 8u
40
+ ··· + 15u − 9
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
42
+ 13u
41
+ ··· + 28u + 1
c
2
, c
4
u
42
− 7u
41
+ ··· − 8u + 1
c
3
, c
8
u
42
+ u
41
+ ··· + 192u + 64
c
5
, c
6
, c
10
u
42
− 2u
41
+ ··· − 55u + 17
c
7
, c
11
, c
12
u
42
+ 2u
41
+ ··· + u + 1
c
9
u
42
+ 2u
41
+ ··· + 5u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
42
+ 39y
41
+ ··· − 212y + 1
c
2
, c
4
y
42
− 13y
41
+ ··· − 28y + 1
c
3
, c
8
y
42
− 39y
41
+ ··· − 90112y + 4096
c
5
, c
6
, c
10
y
42
− 38y
41
+ ··· − 4011y + 289
c
7
, c
11
, c
12
y
42
+ 34y
41
+ ··· − 19y + 1
c
9
y
42
+ 46y
41
+ ··· − 19y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.884997
a = −0.685840
b = −0.554744
−8.59934 −6.83080
u = −0.860411 + 0.091340I
a = 1.60817 + 0.83486I
b = 0.281208 + 1.248210I
−1.28738 + 8.63776I −11.30852 − 5.50570I
u = −0.860411 − 0.091340I
a = 1.60817 − 0.83486I
b = 0.281208 − 1.248210I
−1.28738 − 8.63776I −11.30852 + 5.50570I
u = −0.819362 + 0.102352I
a = −1.76027 − 0.48960I
b = −0.358516 − 0.802687I
0.15821 + 2.18866I −9.63326 − 1.25581I
u = −0.819362 − 0.102352I
a = −1.76027 + 0.48960I
b = −0.358516 + 0.802687I
0.15821 − 2.18866I −9.63326 + 1.25581I
u = −0.818580
a = 1.24094
b = −0.564691
−7.20526 −12.4480
u = 0.813444 + 0.036008I
a = −0.240335 − 1.364030I
b = −0.236106 − 1.145350I
−5.56172 − 2.39562I −13.13795 + 3.14651I
u = 0.813444 − 0.036008I
a = −0.240335 + 1.364030I
b = −0.236106 + 1.145350I
−5.56172 + 2.39562I −13.13795 − 3.14651I
u = −0.360418 + 1.144850I
a = 0.461015 + 1.052370I
b = 0.396025 + 0.874829I
3.34850 + 2.08589I −6.37713 − 2.84313I
u = −0.360418 − 1.144850I
a = 0.461015 − 1.052370I
b = 0.396025 − 0.874829I
3.34850 − 2.08589I −6.37713 + 2.84313I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.039780 + 1.228480I
a = −0.408011 + 0.443145I
b = 0.69289 + 2.43964I
1.48060 − 0.96666I −6.26750 − 1.40491I
u = 0.039780 − 1.228480I
a = −0.408011 − 0.443145I
b = 0.69289 − 2.43964I
1.48060 + 0.96666I −6.26750 + 1.40491I
u = −0.193377 + 1.231330I
a = 0.262816 + 0.483914I
b = 0.802069 + 0.546478I
2.73948 + 2.47038I −3.06420 − 3.92417I
u = −0.193377 − 1.231330I
a = 0.262816 − 0.483914I
b = 0.802069 − 0.546478I
2.73948 − 2.47038I −3.06420 + 3.92417I
u = −0.415470 + 1.179020I
a = −0.775903 − 0.927476I
b = −0.095078 − 0.588213I
2.05152 − 4.06193I −8.19749 + 0.I
u = −0.415470 − 1.179020I
a = −0.775903 + 0.927476I
b = −0.095078 + 0.588213I
2.05152 + 4.06193I −8.19749 + 0.I
u = 0.357741 + 1.238750I
a = −0.744213 − 0.408723I
b = 0.418778 − 1.322100I
−1.85166 − 1.82090I −9.63780 + 0.I
u = 0.357741 − 1.238750I
a = −0.744213 + 0.408723I
b = 0.418778 + 1.322100I
−1.85166 + 1.82090I −9.63780 + 0.I
u = −0.079247 + 1.295310I
a = 0.637223 − 0.258877I
b = 1.01766 − 1.34640I
4.02758 + 2.15699I 0
u = −0.079247 − 1.295310I
a = 0.637223 + 0.258877I
b = 1.01766 + 1.34640I
4.02758 − 2.15699I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.365234 + 1.269190I
a = −0.284658 − 0.718402I
b = −0.34989 − 2.35592I
−3.26541 + 4.25757I 0
u = −0.365234 − 1.269190I
a = −0.284658 + 0.718402I
b = −0.34989 + 2.35592I
−3.26541 − 4.25757I 0
u = 0.517463 + 0.417209I
a = 1.47848 − 1.77950I
b = 0.187293 − 1.152780I
4.73868 − 4.97358I −7.62188 + 6.30595I
u = 0.517463 − 0.417209I
a = 1.47848 + 1.77950I
b = 0.187293 + 1.152780I
4.73868 + 4.97358I −7.62188 − 6.30595I
u = 0.443990 + 0.494284I
a = −1.67760 + 1.53589I
b = −0.210344 + 0.860490I
5.02884 + 1.46246I −6.53686 + 0.89586I
u = 0.443990 − 0.494284I
a = −1.67760 − 1.53589I
b = −0.210344 − 0.860490I
5.02884 − 1.46246I −6.53686 − 0.89586I
u = 0.418536 + 1.277910I
a = 0.134196 − 0.433061I
b = 0.636951 − 0.103857I
−4.63050 − 4.66445I 0
u = 0.418536 − 1.277910I
a = 0.134196 + 0.433061I
b = 0.636951 + 0.103857I
−4.63050 + 4.66445I 0
u = 0.361948 + 1.295330I
a = 0.843712 + 0.183840I
b = 0.34096 + 1.49506I
−1.40855 − 6.62685I 0
u = 0.361948 − 1.295330I
a = 0.843712 − 0.183840I
b = 0.34096 − 1.49506I
−1.40855 + 6.62685I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.361210 + 1.333990I
a = 0.133728 + 1.145250I
b = 1.68733 + 2.67654I
4.66389 + 6.44264I 0
u = −0.361210 − 1.333990I
a = 0.133728 − 1.145250I
b = 1.68733 − 2.67654I
4.66389 − 6.44264I 0
u = 0.108456 + 1.380140I
a = 0.156921 − 1.186400I
b = 1.07791 − 4.11722I
10.88470 − 0.27094I 0
u = 0.108456 − 1.380140I
a = 0.156921 + 1.186400I
b = 1.07791 + 4.11722I
10.88470 + 0.27094I 0
u = 0.146369 + 1.377200I
a = 0.141022 + 1.192190I
b = −0.84503 + 4.22749I
10.39390 − 7.18776I 0
u = 0.146369 − 1.377200I
a = 0.141022 − 1.192190I
b = −0.84503 − 4.22749I
10.39390 + 7.18776I 0
u = −0.385158 + 1.335160I
a = 0.110632 − 1.155620I
b = −1.67195 − 2.97617I
3.18564 + 13.11100I 0
u = −0.385158 − 1.335160I
a = 0.110632 + 1.155620I
b = −1.67195 + 2.97617I
3.18564 − 13.11100I 0
u = −0.492418
a = −0.979682
b = −0.240678
−0.983238 −9.79060
u = −0.283222 + 0.253637I
a = −1.15749 + 1.43241I
b = −0.099702 + 0.442777I
−0.614323 + 0.919516I −9.08610 − 7.37537I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.283222 − 0.253637I
a = −1.15749 − 1.43241I
b = −0.099702 − 0.442777I
−0.614323 − 0.919516I −9.08610 + 7.37537I
u = 0.256764
a = 1.58568
b = −0.984824
−2.02811 1.75710
9
II. I
u
2
= h−u
4
+ u
3
− 2u
2
+ b + u, a, u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1i
(i) Arc colorings
a
8
=
0
u
a
11
=
1
0
a
4
=
0
u
4
− u
3
+ 2u
2
− u
a
12
=
1
u
2
a
3
=
0
u
4
− u
3
+ 2u
2
− u
a
9
=
0
u
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
4
+ 2u
2
a
2
=
u
2
+ 1
2u
4
− u
3
+ 4u
2
− u
a
6
=
u
3
+ 2u
u
3
+ u
a
10
=
−u
5
− 2u
3
− u
−u
5
+ u
4
− 2u
3
+ u
2
− u − 1
a
5
=
−u
2
− 1
−u
4
− 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
4
+ 6u
3
− 11u
2
+ 6u − 17
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
8
u
6
c
4
(u + 1)
6
c
5
, c
6
, c
9
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
7
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
10
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
11
, c
12
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
8
y
6
c
5
, c
6
, c
9
c
10
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
7
, c
11
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.873214
a = 0
b = 0.567375
−9.30502 −19.0600
u = −0.138835 + 1.234450I
a = 0
b = −1.35607 + 0.92119I
1.31531 + 1.97241I −8.22189 − 4.83849I
u = −0.138835 − 1.234450I
a = 0
b = −1.35607 − 0.92119I
1.31531 − 1.97241I −8.22189 + 4.83849I
u = 0.408802 + 1.276380I
a = 0
b = −0.354716 − 0.801205I
−5.34051 − 4.59213I −15.2853 + 2.7994I
u = 0.408802 − 1.276380I
a = 0
b = −0.354716 + 0.801205I
−5.34051 + 4.59213I −15.2853 − 2.7994I
u = −0.413150
a = 0
b = 0.854195
−2.38379 −21.9250
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
42
+ 13u
41
+ ··· + 28u + 1)
c
2
((u − 1)
6
)(u
42
− 7u
41
+ ··· − 8u + 1)
c
3
, c
8
u
6
(u
42
+ u
41
+ ··· + 192u + 64)
c
4
((u + 1)
6
)(u
42
− 7u
41
+ ··· − 8u + 1)
c
5
, c
6
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
42
− 2u
41
+ ··· − 55u + 17)
c
7
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
42
+ 2u
41
+ ··· + u + 1)
c
9
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
42
+ 2u
41
+ ··· + 5u + 1)
c
10
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
42
− 2u
41
+ ··· − 55u + 17)
c
11
, c
12
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
42
+ 2u
41
+ ··· + u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
42
+ 39y
41
+ ··· − 212y + 1)
c
2
, c
4
((y −1)
6
)(y
42
− 13y
41
+ ··· − 28y + 1)
c
3
, c
8
y
6
(y
42
− 39y
41
+ ··· − 90112y + 4096)
c
5
, c
6
, c
10
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
42
− 38y
41
+ ··· − 4011y + 289)
c
7
, c
11
, c
12
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
42
+ 34y
41
+ ··· − 19y + 1)
c
9
(y
6
− 7y
5
+ ··· − 5y + 1)(y
42
+ 46y
41
+ ··· − 19y + 1)
15
