12n
0238
(K12n
0238
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 10 12 11 4 5 6 7 9
Solving Sequence
5,9
10 6
3,11
2 1 4 8 7 12
c
9
c
5
c
10
c
2
c
1
c
4
c
8
c
7
c
12
c
3
, c
6
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−404722938u
20
− 1078466313u
19
+ ··· + 5499202867b − 58406387,
4406297082u
20
+ 8871000551u
19
+ ··· + 5499202867a + 15450577476, u
21
+ 2u
20
+ ··· + 3u + 1i
I
u
2
= hb, −u
5
+ u
4
+ 3u
3
− 2u
2
+ a − 2u − 1, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 27 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
= h−4.05 × 10
8
u
20
− 1.08 × 10
9
u
19
+ · · · + 5.50 × 10
9
b − 5.84 × 10
7
, 4.41 ×
10
9
u
20
+8.87×10
9
u
19
+· · ·+5.50×10
9
a+1.55×10
10
, u
21
+2u
20
+· · ·+3u+1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
6
=
u
−u
3
+ u
a
3
=
−0.801261u
20
− 1.61314u
19
+ ··· + 8.20107u − 2.80960
0.0735967u
20
+ 0.196113u
19
+ ··· + 1.83312u + 0.0106209
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
−0.801261u
20
− 1.61314u
19
+ ··· + 8.20107u − 2.80960
u
a
1
=
0.265288u
20
+ 0.496064u
19
+ ··· + 3.71973u + 0.0807825
−0.0710076u
20
− 0.0622947u
19
+ ··· − 1.04835u − 0.0356490
a
4
=
−0.874858u
20
− 1.80926u
19
+ ··· + 6.36794u − 2.82022
0.0735967u
20
+ 0.196113u
19
+ ··· + 1.83312u + 0.0106209
a
8
=
0.0356490u
20
+ 0.000290302u
19
+ ··· − 0.113612u − 0.941407
0.0345127u
20
− 0.0516588u
19
+ ··· + 0.715083u + 0.265288
a
7
=
0.0129414u
20
+ 0.219376u
19
+ ··· − 0.396043u − 1.03717
0.329095u
20
+ 0.210059u
19
+ ··· + 1.76622u + 0.634234
a
12
=
0.336296u
20
+ 0.558359u
19
+ ··· + 4.76808u + 0.116432
−0.0710076u
20
− 0.0622947u
19
+ ··· − 1.04835u − 0.0356490
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
19796050041
5499202867
u
20
−
34864711542
5499202867
u
19
+ ··· +
105657040340
5499202867
u −
58195456402
5499202867
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
21
− u
20
+ ··· − 10u + 1
c
2
, c
4
u
21
− 7u
20
+ ··· − 4u + 1
c
3
, c
8
u
21
− u
20
+ ··· + 64u + 64
c
5
, c
9
, c
10
u
21
+ 2u
20
+ ··· + 3u + 1
c
6
, c
7
, c
11
u
21
− 2u
20
+ ··· + u + 1
c
12
u
21
− 8u
20
+ ··· + 15665u − 2537
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
21
+ 53y
20
+ ··· + 14y −1
c
2
, c
4
y
21
+ y
20
+ ··· − 10y −1
c
3
, c
8
y
21
+ 39y
20
+ ··· + 71680y
2
− 4096
c
5
, c
9
, c
10
y
21
− 32y
20
+ ··· + 29y −1
c
6
, c
7
, c
11
y
21
+ 16y
20
+ ··· + 29y −1
c
12
y
21
− 116y
20
+ ··· + 451761953y −6436369
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.533638 + 0.732854I
a = −0.021350 + 0.475268I
b = −0.156517 + 0.629640I
−3.26941 − 2.37868I 2.23871 + 4.16638I
u = −0.533638 − 0.732854I
a = −0.021350 − 0.475268I
b = −0.156517 − 0.629640I
−3.26941 + 2.37868I 2.23871 − 4.16638I
u = 1.270070 + 0.160273I
a = −0.214865 + 0.848945I
b = 0.58338 + 1.42040I
4.18543 − 2.07978I 5.06109 + 1.69933I
u = 1.270070 − 0.160273I
a = −0.214865 − 0.848945I
b = 0.58338 − 1.42040I
4.18543 + 2.07978I 5.06109 − 1.69933I
u = 0.578863 + 0.221418I
a = 0.276951 + 0.590407I
b = 0.506741 + 0.461302I
1.037710 + 0.275110I 9.16776 − 1.72750I
u = 0.578863 − 0.221418I
a = 0.276951 − 0.590407I
b = 0.506741 − 0.461302I
1.037710 − 0.275110I 9.16776 + 1.72750I
u = −0.602512 + 0.112972I
a = −0.615716 − 1.104380I
b = −0.968514 − 0.190025I
−1.76723 − 2.46823I 2.72362 + 4.48751I
u = −0.602512 − 0.112972I
a = −0.615716 + 1.104380I
b = −0.968514 + 0.190025I
−1.76723 + 2.46823I 2.72362 − 4.48751I
u = −1.378270 + 0.285291I
a = 0.265526 + 0.752482I
b = −0.30372 + 1.44468I
7.49754 − 2.42009I 8.20075 + 2.52746I
u = −1.378270 − 0.285291I
a = 0.265526 − 0.752482I
b = −0.30372 − 1.44468I
7.49754 + 2.42009I 8.20075 − 2.52746I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.43377 + 0.44455I
a = −0.268209 + 0.659406I
b = 0.094833 + 1.327870I
3.10341 + 6.65319I 3.92005 − 5.62951I
u = 1.43377 − 0.44455I
a = −0.268209 − 0.659406I
b = 0.094833 − 1.327870I
3.10341 − 6.65319I 3.92005 + 5.62951I
u = 0.192099 + 0.306787I
a = 1.15825 − 2.94256I
b = 0.472659 + 0.611670I
−4.24122 + 1.01092I 0.113832 + 1.103661I
u = 0.192099 − 0.306787I
a = 1.15825 + 2.94256I
b = 0.472659 − 0.611670I
−4.24122 − 1.01092I 0.113832 − 1.103661I
u = −0.208424
a = −4.36014
b = −0.397831
−1.31628 −11.0260
u = −1.83563 + 0.08011I
a = 0.602185 + 0.714479I
b = 0.39971 + 2.30587I
15.7454 + 0.6574I 4.16491 − 0.88898I
u = −1.83563 − 0.08011I
a = 0.602185 − 0.714479I
b = 0.39971 − 2.30587I
15.7454 − 0.6574I 4.16491 + 0.88898I
u = 1.86695 + 0.09495I
a = −0.612105 + 0.689039I
b = −0.50531 + 2.27337I
19.6581 + 4.5242I 7.10782 − 2.01921I
u = 1.86695 − 0.09495I
a = −0.612105 − 0.689039I
b = −0.50531 − 2.27337I
19.6581 − 4.5242I 7.10782 + 2.01921I
u = −1.88749 + 0.12006I
a = 0.609399 + 0.663862I
b = 0.57565 + 2.19938I
15.4586 − 9.6359I 3.81453 + 4.73258I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.88749 − 0.12006I
a = 0.609399 − 0.663862I
b = 0.57565 − 2.19938I
15.4586 + 9.6359I 3.81453 − 4.73258I
7
II.
I
u
2
= hb, −u
5
+ u
4
+ 3u
3
− 2u
2
+ a − 2u − 1, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
(i) Arc colorings
a
5
=
0
u
a
9
=
1
0
a
10
=
1
−u
2
a
6
=
u
−u
3
+ u
a
3
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
0
a
11
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
−u
a
1
=
0
−u
a
4
=
u
5
− u
4
− 3u
3
+ 2u
2
+ 2u + 1
0
a
8
=
1
0
a
7
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
a
12
=
u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
5
− 8u
3
+ 12u + 5
8
(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
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
6
, c
7
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
9
, c
10
, c
12
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
11
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
9
(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
9
, c
10
c
12
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
c
6
, c
7
, c
11
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.493180 + 0.575288I
a = −0.858925 − 1.001920I
b = 0
−4.60518 − 1.97241I −3.77811 + 4.83849I
u = −0.493180 − 0.575288I
a = −0.858925 + 1.001920I
b = 0
−4.60518 + 1.97241I −3.77811 − 4.83849I
u = 0.483672
a = 2.06752
b = 0
−0.906083 9.92530
u = 1.52087 + 0.16310I
a = 0.650045 − 0.069710I
b = 0
2.05064 + 4.59213I 3.28527 − 2.79936I
u = 1.52087 − 0.16310I
a = 0.650045 + 0.069710I
b = 0
2.05064 − 4.59213I 3.28527 + 2.79936I
u = −1.53904
a = −0.649754
b = 0
6.01515 7.06030
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
21
− u
20
+ ··· − 10u + 1)
c
2
((u − 1)
6
)(u
21
− 7u
20
+ ··· − 4u + 1)
c
3
, c
8
u
6
(u
21
− u
20
+ ··· + 64u + 64)
c
4
((u + 1)
6
)(u
21
− 7u
20
+ ··· − 4u + 1)
c
5
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
21
+ 2u
20
+ ··· + 3u + 1)
c
6
, c
7
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
21
− 2u
20
+ ··· + u + 1)
c
9
, c
10
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
21
+ 2u
20
+ ··· + 3u + 1)
c
11
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
21
− 2u
20
+ ··· + u + 1)
c
12
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
21
− 8u
20
+ ··· + 15665u − 2537)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
21
+ 53y
20
+ ··· + 14y −1)
c
2
, c
4
((y −1)
6
)(y
21
+ y
20
+ ··· − 10y −1)
c
3
, c
8
y
6
(y
21
+ 39y
20
+ ··· + 71680y
2
− 4096)
c
5
, c
9
, c
10
(y
6
− 7y
5
+ ··· − 5y + 1)(y
21
− 32y
20
+ ··· + 29y −1)
c
6
, c
7
, c
11
(y
6
+ 5y
5
+ ··· − 5y + 1)(y
21
+ 16y
20
+ ··· + 29y −1)
c
12
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
21
− 116y
20
+ ··· + 451761953y −6436369)
13
