12a
0128
(K12a
0128
)
A knot diagram
1
Linearized knot diagam
3 5 8 6 2 11 4 1 12 7 10 9
Solving Sequence
6,11 2,7
5 3 1 4 8 10 12 9
c
6
c
5
c
2
c
1
c
4
c
7
c
10
c
11
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
56
+ 2u
55
+ ··· + 2b − 4u, u
54
− 2u
53
+ ··· + 2a − 1, u
58
− 3u
57
+ ··· − 3u + 1i
I
u
2
= hu
2
a + u
2
+ b, −u
2
a − u
3
+ a
2
+ au + u
2
+ a − u, u
4
− u
3
+ u
2
+ 1i
* 2 irreducible components of dim
C
= 0, with total 66 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−u
56
+2u
55
+· · ·+2b −4u, u
54
−2u
53
+· · ·+2a −1, u
58
−3u
57
+· · ·−3u +1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
−
1
2
u
54
+ u
53
+ ··· −
1
2
u +
1
2
1
2
u
56
− u
55
+ ··· −
5
2
u
2
+ 2u
a
7
=
1
−u
2
a
5
=
−u
57
+
3
2
u
56
+ ··· +
13
2
u −
1
2
−u
57
+
5
2
u
56
+ ··· +
17
2
u
2
− u
a
3
=
2u
56
− 2u
55
+ ··· +
15
2
u −
1
2
−2u
57
+
7
2
u
56
+ ··· +
17
2
u
2
− u
a
1
=
−u
7
− 2u
3
u
9
+ u
7
+ 3u
5
+ 2u
3
+ u
a
4
=
−2u
57
+ 4u
56
+ ··· +
11
2
u −
1
2
−u
57
+
5
2
u
56
+ ··· +
17
2
u
2
− u
a
8
=
u
9
+ 3u
5
+ u
−u
11
− u
9
− 4u
7
− 3u
5
− 3u
3
− u
a
10
=
−u
u
3
+ u
a
12
=
−u
3
u
5
+ u
3
+ u
a
9
=
−u
5
− u
u
7
+ u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
15
2
u
57
+ 12u
56
+ ··· −
51
2
u −
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
58
+ 17u
57
+ ··· + 49u + 1
c
2
, c
5
u
58
+ 5u
57
+ ··· + u + 1
c
3
, c
7
u
58
− u
57
+ ··· + 384u + 256
c
6
, c
10
u
58
− 3u
57
+ ··· − 3u + 1
c
8
, c
9
, c
11
c
12
u
58
+ 11u
57
+ ··· + 21u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
58
+ 53y
57
+ ··· − 451y + 1
c
2
, c
5
y
58
+ 17y
57
+ ··· + 49y + 1
c
3
, c
7
y
58
+ 45y
57
+ ··· + 475136y + 65536
c
6
, c
10
y
58
+ 11y
57
+ ··· + 21y + 1
c
8
, c
9
, c
11
c
12
y
58
+ 75y
57
+ ··· + 45y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.236171 + 0.966829I
a = −0.773305 + 0.432884I
b = −0.777739 + 0.842309I
2.58788 − 0.06234I −2.29623 − 1.17460I
u = 0.236171 − 0.966829I
a = −0.773305 − 0.432884I
b = −0.777739 − 0.842309I
2.58788 + 0.06234I −2.29623 + 1.17460I
u = −0.789321 + 0.603582I
a = 1.49361 − 0.38105I
b = −0.866625 + 0.810195I
8.82733 − 0.53338I 4.24509 + 1.90410I
u = −0.789321 − 0.603582I
a = 1.49361 + 0.38105I
b = −0.866625 − 0.810195I
8.82733 + 0.53338I 4.24509 − 1.90410I
u = −0.557363 + 0.839839I
a = −1.49324 − 1.38719I
b = 0.257230 − 1.060260I
0.27509 − 5.53347I −2.68605 + 8.82067I
u = −0.557363 − 0.839839I
a = −1.49324 + 1.38719I
b = 0.257230 + 1.060260I
0.27509 + 5.53347I −2.68605 − 8.82067I
u = 0.191407 + 0.973334I
a = 0.61484 − 1.91027I
b = −0.761623 − 0.924314I
2.33477 + 5.75175I −3.31343 − 6.58381I
u = 0.191407 − 0.973334I
a = 0.61484 + 1.91027I
b = −0.761623 + 0.924314I
2.33477 − 5.75175I −3.31343 + 6.58381I
u = −0.650775 + 0.773636I
a = −0.562703 − 0.501035I
b = 0.694164 − 0.069962I
3.85863 − 2.42965I 4.20800 + 3.88872I
u = −0.650775 − 0.773636I
a = −0.562703 + 0.501035I
b = 0.694164 + 0.069962I
3.85863 + 2.42965I 4.20800 − 3.88872I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.621083 + 0.799921I
a = −2.68797 + 0.17282I
b = 0.714179 + 0.895252I
3.02078 + 5.11389I 0. − 6.57141I
u = 0.621083 − 0.799921I
a = −2.68797 − 0.17282I
b = 0.714179 − 0.895252I
3.02078 − 5.11389I 0. + 6.57141I
u = 0.641113 + 0.736885I
a = −1.05326 + 1.50993I
b = 0.727655 − 0.829091I
3.22659 − 0.38005I 1.076438 − 0.381418I
u = 0.641113 − 0.736885I
a = −1.05326 − 1.50993I
b = 0.727655 + 0.829091I
3.22659 + 0.38005I 1.076438 + 0.381418I
u = −0.784621 + 0.562056I
a = 1.34703 + 0.49250I
b = −0.805827 − 0.978477I
8.30451 + 5.68111I 3.40199 − 3.19190I
u = −0.784621 − 0.562056I
a = 1.34703 − 0.49250I
b = −0.805827 + 0.978477I
8.30451 − 5.68111I 3.40199 + 3.19190I
u = 0.419617 + 0.831418I
a = 1.92901 − 0.95806I
b = −0.048744 − 0.737006I
−1.20572 + 2.00199I −7.02325 − 3.53170I
u = 0.419617 − 0.831418I
a = 1.92901 + 0.95806I
b = −0.048744 + 0.737006I
−1.20572 − 2.00199I −7.02325 + 3.53170I
u = −0.584881 + 0.658903I
a = 0.148792 + 0.449303I
b = 0.354250 + 1.034950I
0.86366 + 1.17340I 0.879783 − 1.104139I
u = −0.584881 − 0.658903I
a = 0.148792 − 0.449303I
b = 0.354250 − 1.034950I
0.86366 − 1.17340I 0.879783 + 1.104139I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.598169 + 0.963503I
a = 2.16232 + 1.27766I
b = −0.779295 + 0.996450I
6.97066 − 10.76520I 0
u = −0.598169 − 0.963503I
a = 2.16232 − 1.27766I
b = −0.779295 − 0.996450I
6.97066 + 10.76520I 0
u = −0.627970 + 0.950567I
a = −0.095056 + 1.088790I
b = −0.853835 − 0.767019I
7.67631 − 4.67661I 0
u = −0.627970 − 0.950567I
a = −0.095056 − 1.088790I
b = −0.853835 + 0.767019I
7.67631 + 4.67661I 0
u = 0.071034 + 0.820875I
a = 0.48145 + 2.63173I
b = 0.134528 + 0.904687I
−2.92758 + 1.95595I −11.89625 − 4.81261I
u = 0.071034 − 0.820875I
a = 0.48145 − 2.63173I
b = 0.134528 − 0.904687I
−2.92758 − 1.95595I −11.89625 + 4.81261I
u = −0.864927 + 0.895250I
a = 0.697824 + 0.131404I
b = −0.302360 − 0.659717I
6.57199 − 1.97448I 0
u = −0.864927 − 0.895250I
a = 0.697824 − 0.131404I
b = −0.302360 + 0.659717I
6.57199 + 1.97448I 0
u = −0.851879 + 0.930952I
a = 1.53417 + 0.29283I
b = −0.284323 + 0.694959I
6.46068 − 4.39375I 0
u = −0.851879 − 0.930952I
a = 1.53417 − 0.29283I
b = −0.284323 − 0.694959I
6.46068 + 4.39375I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.910198 + 0.920175I
a = −0.1069100 − 0.0175219I
b = 0.318618 − 1.173080I
9.62869 − 0.79499I 0
u = 0.910198 − 0.920175I
a = −0.1069100 + 0.0175219I
b = 0.318618 + 1.173080I
9.62869 + 0.79499I 0
u = 0.325035 + 0.621241I
a = 0.691818 + 0.030726I
b = −0.045053 + 0.234286I
−0.201748 + 1.112070I −3.30409 − 5.91204I
u = 0.325035 − 0.621241I
a = 0.691818 − 0.030726I
b = −0.045053 − 0.234286I
−0.201748 − 1.112070I −3.30409 + 5.91204I
u = 0.942009 + 0.898668I
a = 1.31780 − 0.56148I
b = −0.811125 + 1.045670I
17.4170 − 7.1616I 0
u = 0.942009 − 0.898668I
a = 1.31780 + 0.56148I
b = −0.811125 − 1.045670I
17.4170 + 7.1616I 0
u = 0.895499 + 0.949477I
a = −1.43215 + 0.42653I
b = 0.303547 + 1.175220I
9.53363 + 7.45134I 0
u = 0.895499 − 0.949477I
a = −1.43215 − 0.42653I
b = 0.303547 − 1.175220I
9.53363 − 7.45134I 0
u = −0.914772 + 0.933791I
a = −1.32356 − 1.26003I
b = 0.823322 + 0.892050I
12.70460 − 0.29180I 0
u = −0.914772 − 0.933791I
a = −1.32356 + 1.26003I
b = 0.823322 − 0.892050I
12.70460 + 0.29180I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.690394 + 0.030120I
a = 1.39889 + 0.41674I
b = −0.809868 − 0.894099I
5.73169 + 3.03027I 4.19424 − 2.78424I
u = 0.690394 − 0.030120I
a = 1.39889 − 0.41674I
b = −0.809868 + 0.894099I
5.73169 − 3.03027I 4.19424 + 2.78424I
u = 0.942436 + 0.909290I
a = 1.57800 + 0.37798I
b = −0.942649 − 0.753265I
18.3412 − 0.7119I 0
u = 0.942436 − 0.909290I
a = 1.57800 − 0.37798I
b = −0.942649 + 0.753265I
18.3412 + 0.7119I 0
u = −0.908706 + 0.945270I
a = −2.36231 + 0.27916I
b = 0.818870 − 0.903943I
12.6670 − 6.4226I 0
u = −0.908706 − 0.945270I
a = −2.36231 − 0.27916I
b = 0.818870 + 0.903943I
12.6670 + 6.4226I 0
u = 0.914486 + 0.940989I
a = −1.045780 + 0.497688I
b = 0.892652 + 0.010539I
13.59950 + 3.36619I 0
u = 0.914486 − 0.940989I
a = −1.045780 − 0.497688I
b = 0.892652 − 0.010539I
13.59950 − 3.36619I 0
u = 0.895550 + 0.983231I
a = 2.40578 − 0.45339I
b = −0.803885 − 1.049880I
17.1389 + 13.9172I 0
u = 0.895550 − 0.983231I
a = 2.40578 + 0.45339I
b = −0.803885 + 1.049880I
17.1389 − 13.9172I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.903709 + 0.978699I
a = 0.66497 − 1.36041I
b = −0.941425 + 0.741601I
18.1123 + 7.4971I 0
u = 0.903709 − 0.978699I
a = 0.66497 + 1.36041I
b = −0.941425 − 0.741601I
18.1123 − 7.4971I 0
u = −0.138946 + 0.644931I
a = −2.02349 − 2.26505I
b = 0.559591 − 0.926866I
−0.64003 − 2.84760I −8.08127 + 0.14345I
u = −0.138946 − 0.644931I
a = −2.02349 + 2.26505I
b = 0.559591 + 0.926866I
−0.64003 + 2.84760I −8.08127 − 0.14345I
u = 0.344616 + 0.363333I
a = 0.547401 − 0.207447I
b = 0.251942 + 0.577672I
−0.068529 + 1.208380I −0.22566 − 4.75539I
u = 0.344616 − 0.363333I
a = 0.547401 + 0.207447I
b = 0.251942 − 0.577672I
−0.068529 − 1.208380I −0.22566 + 4.75539I
u = −0.172028 + 0.378457I
a = 0.94604 − 1.19928I
b = 0.483828 + 0.745563I
0.00258 + 1.44884I −2.25105 − 5.51745I
u = −0.172028 − 0.378457I
a = 0.94604 + 1.19928I
b = 0.483828 − 0.745563I
0.00258 − 1.44884I −2.25105 + 5.51745I
10
II. I
u
2
= hu
2
a + u
2
+ b, −u
2
a − u
3
+ a
2
+ au + u
2
+ a − u, u
4
− u
3
+ u
2
+ 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
a
−u
2
a − u
2
a
7
=
1
−u
2
a
5
=
u
2
a + a + u + 1
−u
2
a − u
2
− 1
a
3
=
−u
2
+ a + u
−u
2
a − u
2
− 1
a
1
=
−1
0
a
4
=
−u
2
+ a + u
−u
2
a − u
2
− 1
a
8
=
1
−u
2
a
10
=
−u
u
3
+ u
a
12
=
−u
3
u
3
− u
2
− 1
a
9
=
u
2
+ 1
−u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
3
a + 4u
2
a + 2au + u
2
− a + 6u − 2
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
3
, c
7
u
8
c
6
(u
4
− u
3
+ u
2
+ 1)
2
c
8
, c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)
2
c
10
(u
4
+ u
3
+ u
2
+ 1)
2
c
11
, c
12
(u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
(y
2
+ y + 1)
4
c
3
, c
7
y
8
c
6
, c
10
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
c
8
, c
9
, c
11
c
12
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.351808 + 0.720342I
a = 0.541116 + 0.214920I
b = 0.500000 + 0.866025I
−0.211005 + 0.614778I −5.86133 + 2.84273I
u = −0.351808 + 0.720342I
a = −1.58443 − 1.44211I
b = 0.500000 − 0.866025I
−0.21101 − 3.44499I −1.10064 + 8.92228I
u = −0.351808 − 0.720342I
a = 0.541116 − 0.214920I
b = 0.500000 − 0.866025I
−0.211005 − 0.614778I −5.86133 − 2.84273I
u = −0.351808 − 0.720342I
a = −1.58443 + 1.44211I
b = 0.500000 + 0.866025I
−0.21101 + 3.44499I −1.10064 − 8.92228I
u = 0.851808 + 0.911292I
a = −0.423047 + 0.283088I
b = 0.500000 − 0.866025I
6.79074 + 1.13408I 0.90087 + 2.75771I
u = 0.851808 + 0.911292I
a = −1.53364 + 0.35811I
b = 0.500000 + 0.866025I
6.79074 + 5.19385I 1.56110 − 7.61722I
u = 0.851808 − 0.911292I
a = −0.423047 − 0.283088I
b = 0.500000 + 0.866025I
6.79074 − 1.13408I 0.90087 − 2.75771I
u = 0.851808 − 0.911292I
a = −1.53364 − 0.35811I
b = 0.500000 − 0.866025I
6.79074 − 5.19385I 1.56110 + 7.61722I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
((u
2
− u + 1)
4
)(u
58
+ 17u
57
+ ··· + 49u + 1)
c
2
((u
2
+ u + 1)
4
)(u
58
+ 5u
57
+ ··· + u + 1)
c
3
, c
7
u
8
(u
58
− u
57
+ ··· + 384u + 256)
c
5
((u
2
− u + 1)
4
)(u
58
+ 5u
57
+ ··· + u + 1)
c
6
((u
4
− u
3
+ u
2
+ 1)
2
)(u
58
− 3u
57
+ ··· − 3u + 1)
c
8
, c
9
((u
4
− u
3
+ 3u
2
− 2u + 1)
2
)(u
58
+ 11u
57
+ ··· + 21u + 1)
c
10
((u
4
+ u
3
+ u
2
+ 1)
2
)(u
58
− 3u
57
+ ··· − 3u + 1)
c
11
, c
12
((u
4
+ u
3
+ 3u
2
+ 2u + 1)
2
)(u
58
+ 11u
57
+ ··· + 21u + 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
((y
2
+ y + 1)
4
)(y
58
+ 53y
57
+ ··· − 451y + 1)
c
2
, c
5
((y
2
+ y + 1)
4
)(y
58
+ 17y
57
+ ··· + 49y + 1)
c
3
, c
7
y
8
(y
58
+ 45y
57
+ ··· + 475136y + 65536)
c
6
, c
10
((y
4
+ y
3
+ 3y
2
+ 2y + 1)
2
)(y
58
+ 11y
57
+ ··· + 21y + 1)
c
8
, c
9
, c
11
c
12
((y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
2
)(y
58
+ 75y
57
+ ··· + 45y + 1)
16
