12n
0048
(K12n
0048
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 12 11 4 6 8 7 10
Solving Sequence
6,12 2,7
5 3 1 11 8 4 9 10
c
6
c
5
c
2
c
1
c
11
c
7
c
4
c
8
c
10
c
3
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
31
− 2u
30
+ ··· + 2b + 2u, u
30
− 2u
29
+ ··· + 2a + 4, u
32
− 3u
31
+ ··· − 4u + 1i
I
u
2
= h−au + b − u, u
3
a − u
2
a − u
3
+ a
2
+ 3au − 2u, u
4
− u
3
+ 3u
2
− 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 40 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
31
−2u
30
+· · ·+2b+2u, u
30
−2u
29
+· · ·+2a+4, u
32
−3u
31
+· · ·−4u+1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
2
=
−
1
2
u
30
+ u
29
+ ··· − 2u − 2
−
1
2
u
31
+ u
30
+ ··· − 2u
2
− u
a
7
=
1
u
2
a
5
=
5
2
u
31
−
13
2
u
30
+ ··· + 15u − 4
1
2
u
31
− 2u
30
+ ··· + 3u − 2
a
3
=
u
31
−
7
2
u
30
+ ··· + 12u − 4
−
1
2
u
31
+ u
30
+ ··· − 4u
2
− 1
a
1
=
u
7
+ 4u
5
+ 4u
3
u
9
+ 5u
7
+ 7u
5
+ 2u
3
+ u
a
11
=
u
u
3
+ u
a
8
=
u
2
+ 1
u
4
+ 2u
2
a
4
=
3
2
u
31
−
9
2
u
30
+ ··· + 12u − 3
−
1
2
u
31
+ u
30
+ ··· − 4u
2
− 1
a
9
=
u
5
+ 2u
3
− u
−u
5
− 3u
3
− u
a
10
=
u
3
+ 2u
u
5
+ 3u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes =
5
2
u
31
− 6u
30
+ ··· − 4u +
11
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
32
+ 21u
31
+ ··· + 16u + 1
c
2
, c
5
u
32
+ 5u
31
+ ··· + 8u + 1
c
3
u
32
− 5u
31
+ ··· + 8u
2
+ 1
c
4
, c
8
u
32
− u
31
+ ··· + 128u + 256
c
6
, c
7
, c
10
c
11
u
32
− 3u
31
+ ··· − 4u + 1
c
9
u
32
− 3u
31
+ ··· − 10410u + 8329
c
12
u
32
+ 3u
31
+ ··· + 8u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
32
− 15y
31
+ ··· + 432y + 1
c
2
, c
5
y
32
+ 21y
31
+ ··· + 16y + 1
c
3
y
32
− 51y
31
+ ··· + 16y + 1
c
4
, c
8
y
32
+ 45y
31
+ ··· + 475136y + 65536
c
6
, c
7
, c
10
c
11
y
32
+ 35y
31
+ ··· + 16y + 1
c
9
y
32
+ 63y
31
+ ··· + 4526986928y + 69372241
c
12
y
32
+ 43y
31
+ ··· + 16y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.712603 + 0.575097I
a = 1.17357 + 2.13596I
b = −0.53596 + 1.34356I
−12.5938 − 8.0140I −3.11734 + 5.63402I
u = 0.712603 − 0.575097I
a = 1.17357 − 2.13596I
b = −0.53596 − 1.34356I
−12.5938 + 8.0140I −3.11734 − 5.63402I
u = 0.744220 + 0.468541I
a = −0.60836 − 1.69690I
b = −0.49608 − 1.36523I
−12.91530 + 3.15697I −3.83308 − 0.28463I
u = 0.744220 − 0.468541I
a = −0.60836 + 1.69690I
b = −0.49608 + 1.36523I
−12.91530 − 3.15697I −3.83308 + 0.28463I
u = 0.701586 + 0.517087I
a = −0.496748 + 0.685785I
b = −1.054030 − 0.033443I
−8.50888 − 2.34942I −0.92927 + 2.77248I
u = 0.701586 − 0.517087I
a = −0.496748 − 0.685785I
b = −1.054030 + 0.033443I
−8.50888 + 2.34942I −0.92927 − 2.77248I
u = −0.490239 + 0.668149I
a = 1.72439 − 1.21578I
b = −0.125255 − 1.087690I
−2.20368 + 2.67014I −2.97925 − 3.94706I
u = −0.490239 − 0.668149I
a = 1.72439 + 1.21578I
b = −0.125255 + 1.087690I
−2.20368 − 2.67014I −2.97925 + 3.94706I
u = −0.599531 + 0.288987I
a = 0.01061 + 2.20427I
b = 0.026786 + 1.136550I
−3.37323 + 1.08981I −5.51512 − 2.69237I
u = −0.599531 − 0.288987I
a = 0.01061 − 2.20427I
b = 0.026786 − 1.136550I
−3.37323 − 1.08981I −5.51512 + 2.69237I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.14693 + 1.41081I
a = −0.351474 + 1.272230I
b = 0.195534 + 1.213220I
2.00066 + 3.66255I 0. − 3.26134I
u = −0.14693 − 1.41081I
a = −0.351474 − 1.272230I
b = 0.195534 − 1.213220I
2.00066 − 3.66255I 0. + 3.26134I
u = −0.294946 + 0.485626I
a = 0.628330 − 0.123530I
b = −0.164482 + 0.198738I
0.031344 + 1.111830I 0.46087 − 6.46007I
u = −0.294946 − 0.485626I
a = 0.628330 + 0.123530I
b = −0.164482 − 0.198738I
0.031344 − 1.111830I 0.46087 + 6.46007I
u = 0.05664 + 1.45455I
a = −1.51625 − 0.70222I
b = 0.648137 − 1.003850I
5.53715 − 3.59224I 0. + 2.25541I
u = 0.05664 − 1.45455I
a = −1.51625 + 0.70222I
b = 0.648137 + 1.003850I
5.53715 + 3.59224I 0. − 2.25541I
u = −0.01210 + 1.48445I
a = −0.270899 − 0.532093I
b = 0.656453 + 0.503553I
6.89090 + 1.46785I 4.83746 − 2.83876I
u = −0.01210 − 1.48445I
a = −0.270899 + 0.532093I
b = 0.656453 − 0.503553I
6.89090 − 1.46785I 4.83746 + 2.83876I
u = 0.26257 + 1.48581I
a = −0.261839 − 0.635491I
b = −0.44075 − 1.37939I
−6.59902 − 0.50025I 0
u = 0.26257 − 1.48581I
a = −0.261839 + 0.635491I
b = −0.44075 + 1.37939I
−6.59902 + 0.50025I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.23235 + 1.52047I
a = 0.332026 + 0.658818I
b = −1.044910 − 0.104621I
−1.84666 − 5.75102I 0
u = 0.23235 − 1.52047I
a = 0.332026 − 0.658818I
b = −1.044910 + 0.104621I
−1.84666 + 5.75102I 0
u = −0.07381 + 1.55331I
a = 0.782474 − 0.225010I
b = −0.293198 + 0.473940I
7.02752 + 2.34797I 0
u = −0.07381 − 1.55331I
a = 0.782474 + 0.225010I
b = −0.293198 − 0.473940I
7.02752 − 2.34797I 0
u = 0.038702 + 0.442217I
a = 0.809540 − 1.072710I
b = 0.420247 + 0.724423I
0.54316 + 1.39338I 5.51393 − 4.82316I
u = 0.038702 − 0.442217I
a = 0.809540 + 1.072710I
b = 0.420247 − 0.724423I
0.54316 − 1.39338I 5.51393 + 4.82316I
u = 0.23771 + 1.55222I
a = 1.49649 + 1.06121I
b = −0.56783 + 1.31462I
−5.59324 − 11.51330I 0
u = 0.23771 − 1.55222I
a = 1.49649 − 1.06121I
b = −0.56783 − 1.31462I
−5.59324 + 11.51330I 0
u = −0.12413 + 1.59090I
a = 1.46814 − 0.31922I
b = −0.237070 − 1.031430I
5.45590 + 4.87027I 0
u = −0.12413 − 1.59090I
a = 1.46814 + 0.31922I
b = −0.237070 + 1.031430I
5.45590 − 4.87027I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.255308 + 0.267111I
a = −1.91999 − 2.90858I
b = 0.512407 − 0.945593I
−0.17176 − 2.59226I 1.49287 + 0.71136I
u = 0.255308 − 0.267111I
a = −1.91999 + 2.90858I
b = 0.512407 + 0.945593I
−0.17176 + 2.59226I 1.49287 − 0.71136I
8
II.
I
u
2
= h−au + b − u, u
3
a − u
2
a − u
3
+ a
2
+ 3au − 2u, u
4
− u
3
+ 3u
2
− 2u + 1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
2
=
a
au + u
a
7
=
1
u
2
a
5
=
u
3
− au − u
2
+ a + 2u
au + u − 1
a
3
=
u
3
− u
2
+ a + 3u − 1
au + u − 1
a
1
=
−1
0
a
11
=
u
u
3
+ u
a
8
=
u
2
+ 1
u
3
− u
2
+ 2u − 1
a
4
=
u
3
− au − u
2
+ a + 2u
au + u − 1
a
9
=
u
2
+ 1
u
3
− u
2
+ 2u − 1
a
10
=
u
3
+ 2u
u
3
− u
2
+ 2u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
a + 4u
3
− 4au − 3u
2
− a + 7u − 3
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
4
, c
8
u
8
c
6
, c
7
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
9
, c
12
(u
4
+ u
3
+ u
2
+ 1)
2
c
10
, c
11
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
4
c
4
, c
8
y
8
c
6
, c
7
, c
10
c
11
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
c
9
, c
12
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.395123 + 0.506844I
a = 0.541116 + 0.214920I
b = 0.500000 + 0.866025I
−0.211005 + 0.614778I −2.00436 + 1.31849I
u = 0.395123 + 0.506844I
a = −1.58443 − 1.44211I
b = 0.500000 − 0.866025I
−0.21101 − 3.44499I 0.99907 + 9.21934I
u = 0.395123 − 0.506844I
a = 0.541116 − 0.214920I
b = 0.500000 − 0.866025I
−0.211005 − 0.614778I −2.00436 − 1.31849I
u = 0.395123 − 0.506844I
a = −1.58443 + 1.44211I
b = 0.500000 + 0.866025I
−0.21101 + 3.44499I 0.99907 − 9.21934I
u = 0.10488 + 1.55249I
a = −0.423047 − 0.283088I
b = 0.500000 + 0.866025I
6.79074 − 1.13408I 1.85285 − 1.30164I
u = 0.10488 + 1.55249I
a = −1.53364 − 0.35811I
b = 0.500000 − 0.866025I
6.79074 − 5.19385I 5.65243 + 5.51994I
u = 0.10488 − 1.55249I
a = −0.423047 + 0.283088I
b = 0.500000 − 0.866025I
6.79074 + 1.13408I 1.85285 + 1.30164I
u = 0.10488 − 1.55249I
a = −1.53364 + 0.35811I
b = 0.500000 + 0.866025I
6.79074 + 5.19385I 5.65243 − 5.51994I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
4
)(u
32
+ 21u
31
+ ··· + 16u + 1)
c
2
((u
2
+ u + 1)
4
)(u
32
+ 5u
31
+ ··· + 8u + 1)
c
3
((u
2
− u + 1)
4
)(u
32
− 5u
31
+ ··· + 8u
2
+ 1)
c
4
, c
8
u
8
(u
32
− u
31
+ ··· + 128u + 256)
c
5
((u
2
− u + 1)
4
)(u
32
+ 5u
31
+ ··· + 8u + 1)
c
6
, c
7
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
32
− 3u
31
+ ··· − 4u + 1)
c
9
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
32
− 3u
31
+ ··· − 10410u + 8329)
c
10
, c
11
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
32
− 3u
31
+ ··· − 4u + 1)
c
12
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
32
+ 3u
31
+ ··· + 8u
2
+ 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
4
)(y
32
− 15y
31
+ ··· + 432y + 1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
32
+ 21y
31
+ ··· + 16y + 1)
c
3
((y
2
+ y + 1)
4
)(y
32
− 51y
31
+ ··· + 16y + 1)
c
4
, c
8
y
8
(y
32
+ 45y
31
+ ··· + 475136y + 65536)
c
6
, c
7
, c
10
c
11
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
32
+ 35y
31
+ ··· + 16y + 1)
c
9
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
· (y
32
+ 63y
31
+ ··· + 4526986928y + 69372241)
c
12
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
32
+ 43y
31
+ ··· + 16y + 1)
14
