12a
0034
(K12a
0034
)
A knot diagram
1
Linearized knot diagam
3 5 6 10 2 11 12 1 4 9 7 8
Solving Sequence
7,12
8
1,3
2 9 11 6 4 5 10
c
7
c
12
c
1
c
8
c
11
c
6
c
3
c
5
c
10
c
2
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−11u
60
+ 15u
59
+ ··· + 2b + 7, −8u
60
+ 9u
59
+ ··· + 2a + 7, u
61
− 3u
60
+ ··· + u + 1i
I
u
2
= hau + b − a, a
2
+ au + a + u + 2, u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 65 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−11u
60
+15u
59
+· · ·+2b+7, −8u
60
+9u
59
+· · ·+2a+7, u
61
−3u
60
+· · ·+u+1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
u
2
a
1
=
−u
−u
3
+ u
a
3
=
4u
60
−
9
2
u
59
+ ··· − 6u −
7
2
11
2
u
60
−
15
2
u
59
+ ··· −
11
2
u −
7
2
a
2
=
−
1
2
u
59
+ u
58
+ ··· + 6u −
1
2
−
1
2
u
60
+
1
2
u
59
+ ··· +
5
2
u +
1
2
a
9
=
−u
2
+ 1
−u
4
+ 2u
2
a
11
=
u
u
a
6
=
−u
2
+ 1
−u
2
a
4
=
19
2
u
60
− 12u
59
+ ··· −
21
2
u − 6
27
2
u
60
− 18u
59
+ ··· −
25
2
u − 8
a
5
=
−
7
2
u
60
+ 5u
59
+ ··· +
5
2
u + 2
−
15
2
u
60
+ 10u
59
+ ··· +
15
2
u + 4
a
10
=
−u
7
+ 4u
5
− 4u
3
+ 2u
−u
9
+ 5u
7
− 7u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
5
2
u
60
+ 95u
58
+ ··· −
29
2
u − 5
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
61
+ 29u
60
+ ··· + 23u − 1
c
2
, c
5
u
61
+ 3u
60
+ ··· + 9u + 1
c
3
u
61
− 3u
60
+ ··· − 129u + 241
c
4
, c
9
u
61
+ u
60
+ ··· − 48u − 16
c
6
, c
7
, c
8
c
11
, c
12
u
61
+ 3u
60
+ ··· + u − 1
c
10
u
61
+ 25u
60
+ ··· + 2432u + 256
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
61
+ 9y
60
+ ··· + 707y −1
c
2
, c
5
y
61
+ 29y
60
+ ··· + 23y −1
c
3
y
61
− 11y
60
+ ··· + 2638239y −58081
c
4
, c
9
y
61
− 25y
60
+ ··· + 2432y −256
c
6
, c
7
, c
8
c
11
, c
12
y
61
− 79y
60
+ ··· + 19y −1
c
10
y
61
+ 15y
60
+ ··· + 2564096y −65536
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.941549 + 0.362904I
a = −0.926027 − 0.288989I
b = 0.684803 + 0.361766I
−1.57693 + 6.68347I 0
u = −0.941549 − 0.362904I
a = −0.926027 + 0.288989I
b = 0.684803 − 0.361766I
−1.57693 − 6.68347I 0
u = −0.962161 + 0.388897I
a = 1.091940 + 0.880454I
b = −0.891980 − 0.179165I
−3.85828 + 11.82190I 0
u = −0.962161 − 0.388897I
a = 1.091940 − 0.880454I
b = −0.891980 + 0.179165I
−3.85828 − 11.82190I 0
u = −0.991302 + 0.313503I
a = −0.066752 + 0.371102I
b = −0.789116 − 0.780549I
−6.39349 + 4.01306I 0
u = −0.991302 − 0.313503I
a = −0.066752 − 0.371102I
b = −0.789116 + 0.780549I
−6.39349 − 4.01306I 0
u = 0.907581 + 0.284646I
a = −1.43188 + 0.77881I
b = 1.194390 − 0.037845I
−1.92448 − 5.85744I 0
u = 0.907581 − 0.284646I
a = −1.43188 − 0.77881I
b = 1.194390 + 0.037845I
−1.92448 + 5.85744I 0
u = −1.06207
a = −0.867883
b = 0.339270
−5.58398 0
u = 0.916155 + 0.124387I
a = −0.395310 + 0.861467I
b = 0.790911 − 1.053260I
−3.56819 + 0.84716I −15.9906 + 0.I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.916155 − 0.124387I
a = −0.395310 − 0.861467I
b = 0.790911 + 1.053260I
−3.56819 − 0.84716I −15.9906 + 0.I
u = −0.858548 + 0.285771I
a = −0.240294 + 1.130700I
b = 0.426613 + 0.684373I
−0.14613 + 4.14771I −8.00000 − 7.53022I
u = −0.858548 − 0.285771I
a = −0.240294 − 1.130700I
b = 0.426613 − 0.684373I
−0.14613 − 4.14771I −8.00000 + 7.53022I
u = 0.833612 + 0.279640I
a = 1.133180 − 0.204262I
b = −0.769914 + 0.142240I
0.014847 − 1.230250I −8.00000 + 2.00002I
u = 0.833612 − 0.279640I
a = 1.133180 + 0.204262I
b = −0.769914 − 0.142240I
0.014847 + 1.230250I −8.00000 − 2.00002I
u = −1.138410 + 0.086480I
a = 0.760786 + 0.178531I
b = −0.780459 + 0.648714I
−8.82704 + 3.92280I 0
u = −1.138410 − 0.086480I
a = 0.760786 − 0.178531I
b = −0.780459 − 0.648714I
−8.82704 − 3.92280I 0
u = −0.819047 + 0.217033I
a = −0.57338 − 1.48189I
b = −0.544215 − 0.744173I
−1.09072 − 0.91618I −13.20392 − 2.90788I
u = −0.819047 − 0.217033I
a = −0.57338 + 1.48189I
b = −0.544215 + 0.744173I
−1.09072 + 0.91618I −13.20392 + 2.90788I
u = 0.651545 + 0.456443I
a = 0.43685 − 1.47583I
b = 0.607385 − 0.419645I
−2.01345 + 4.76879I −12.80458 − 3.25405I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.651545 − 0.456443I
a = 0.43685 + 1.47583I
b = 0.607385 + 0.419645I
−2.01345 − 4.76879I −12.80458 + 3.25405I
u = 0.672132 + 0.371659I
a = 0.204778 + 0.807835I
b = −0.503235 + 0.231907I
0.006977 + 0.191932I −9.14973 + 0.93355I
u = 0.672132 − 0.371659I
a = 0.204778 − 0.807835I
b = −0.503235 − 0.231907I
0.006977 − 0.191932I −9.14973 − 0.93355I
u = 0.486313 + 0.449825I
a = 1.064050 + 0.128423I
b = 0.750517 + 0.088709I
−3.44842 − 2.18500I −15.5614 + 4.6731I
u = 0.486313 − 0.449825I
a = 1.064050 − 0.128423I
b = 0.750517 − 0.088709I
−3.44842 + 2.18500I −15.5614 − 4.6731I
u = 0.142006 + 0.624563I
a = 0.949731 − 0.841604I
b = 1.063740 + 0.386697I
−0.47167 − 8.39775I −9.16009 + 8.05408I
u = 0.142006 − 0.624563I
a = 0.949731 + 0.841604I
b = 1.063740 − 0.386697I
−0.47167 + 8.39775I −9.16009 − 8.05408I
u = 0.119463 + 0.584881I
a = −0.402048 + 0.694739I
b = −0.574572 − 0.527063I
1.67223 − 3.47107I −5.43201 + 4.12123I
u = 0.119463 − 0.584881I
a = −0.402048 − 0.694739I
b = −0.574572 + 0.527063I
1.67223 + 3.47107I −5.43201 − 4.12123I
u = 0.222473 + 0.540240I
a = 0.01204 − 1.43662I
b = 0.230433 − 0.262534I
−2.65865 − 1.12584I −12.84225 + 3.10905I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.222473 − 0.540240I
a = 0.01204 + 1.43662I
b = 0.230433 + 0.262534I
−2.65865 + 1.12584I −12.84225 − 3.10905I
u = 0.011799 + 0.502580I
a = 0.736143 + 0.780204I
b = 0.443073 − 0.686495I
2.47447 − 1.45390I −2.95032 + 3.27697I
u = 0.011799 − 0.502580I
a = 0.736143 − 0.780204I
b = 0.443073 + 0.686495I
2.47447 + 1.45390I −2.95032 − 3.27697I
u = −0.071181 + 0.471890I
a = −1.39302 − 1.27147I
b = −0.971413 + 0.506703I
1.06320 + 3.26175I −5.16927 − 3.30010I
u = −0.071181 − 0.471890I
a = −1.39302 + 1.27147I
b = −0.971413 − 0.506703I
1.06320 − 3.26175I −5.16927 + 3.30010I
u = 0.441573
a = 0.291655
b = −0.318830
−0.703516 −13.9150
u = −1.57089 + 0.05487I
a = −0.108181 − 0.454338I
b = −0.73199 − 1.31069I
−9.35339 − 3.05356I 0
u = −1.57089 − 0.05487I
a = −0.108181 + 0.454338I
b = −0.73199 + 1.31069I
−9.35339 + 3.05356I 0
u = −1.62164 + 0.05572I
a = −0.661023 + 0.422089I
b = −1.01599 + 1.16132I
−7.88099 + 1.12428I 0
u = −1.62164 − 0.05572I
a = −0.661023 − 0.422089I
b = −1.01599 − 1.16132I
−7.88099 − 1.12428I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.67660 + 0.06327I
a = −2.46813 − 0.26761I
b = −4.28495 − 0.21565I
−8.84015 + 2.47805I 0
u = −1.67660 − 0.06327I
a = −2.46813 + 0.26761I
b = −4.28495 + 0.21565I
−8.84015 − 2.47805I 0
u = 1.67898 + 0.05115I
a = 0.276936 − 0.341290I
b = 0.80641 − 1.34802I
−9.96853 − 0.06844I 0
u = 1.67898 − 0.05115I
a = 0.276936 + 0.341290I
b = 0.80641 + 1.34802I
−9.96853 + 0.06844I 0
u = 1.68198 + 0.06781I
a = 0.485407 + 0.293294I
b = 0.76562 + 1.24151I
−9.10899 − 5.46550I 0
u = 1.68198 − 0.06781I
a = 0.485407 − 0.293294I
b = 0.76562 − 1.24151I
−9.10899 + 5.46550I 0
u = −1.69350 + 0.07169I
a = 3.58656 + 0.38893I
b = 6.19427 + 0.54411I
−11.11280 + 7.23397I 0
u = −1.69350 − 0.07169I
a = 3.58656 − 0.38893I
b = 6.19427 − 0.54411I
−11.11280 − 7.23397I 0
u = −1.69643 + 0.03492I
a = 2.03123 + 2.16248I
b = 3.41894 + 3.43232I
−12.85740 − 0.20377I 0
u = −1.69643 − 0.03492I
a = 2.03123 − 2.16248I
b = 3.41894 − 3.43232I
−12.85740 + 0.20377I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.69964 + 0.09553I
a = 2.24617 − 0.46709I
b = 4.02268 − 0.47212I
−10.86090 − 8.49023I 0
u = 1.69964 − 0.09553I
a = 2.24617 + 0.46709I
b = 4.02268 + 0.47212I
−10.86090 + 8.49023I 0
u = 1.70503 + 0.10421I
a = −2.95201 + 0.71525I
b = −5.24696 + 1.06747I
−13.2248 − 13.7857I 0
u = 1.70503 − 0.10421I
a = −2.95201 − 0.71525I
b = −5.24696 − 1.06747I
−13.2248 + 13.7857I 0
u = 1.71354 + 0.08181I
a = −1.37523 + 1.51380I
b = −2.28865 + 2.24136I
−15.9607 − 5.5984I 0
u = 1.71354 − 0.08181I
a = −1.37523 − 1.51380I
b = −2.28865 − 2.24136I
−15.9607 + 5.5984I 0
u = 1.72573
a = 2.16104
b = 3.90890
−15.5407 0
u = 1.73947 + 0.01560I
a = −2.65245 − 0.84909I
b = −4.61334 − 1.29091I
−19.1104 − 4.2965I 0
u = 1.73947 − 0.01560I
a = −2.65245 + 0.84909I
b = −4.61334 + 1.29091I
−19.1104 + 4.2965I 0
u = −0.193073 + 0.125473I
a = −0.16246 − 3.64433I
b = −0.357674 − 0.618866I
−0.31183 − 1.80289I −2.01842 + 2.61178I
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.193073 − 0.125473I
a = −0.16246 + 3.64433I
b = −0.357674 + 0.618866I
−0.31183 + 1.80289I −2.01842 − 2.61178I
11
II. I
u
2
= hau + b − a, a
2
+ au + a + u + 2, u
2
+ u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
−u + 1
a
1
=
−u
−u + 1
a
3
=
a
−au + a
a
2
=
a + 1
−au + a + 1
a
9
=
u
u
a
11
=
u
u
a
6
=
u
u − 1
a
4
=
au
au
a
5
=
au
au
a
10
=
u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3au − 2a − u − 16
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
2
c
2
(u
2
+ u + 1)
2
c
4
, c
9
, c
10
u
4
c
6
, c
7
, c
8
(u
2
+ u − 1)
2
c
11
, c
12
(u
2
− u − 1)
2
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
2
c
4
, c
9
, c
10
y
4
c
6
, c
7
, c
8
c
11
, c
12
(y
2
− 3y + 1)
2
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −0.80902 + 1.40126I
b = −0.309017 + 0.535233I
−0.98696 + 2.02988I −13.5000 − 5.4006I
u = 0.618034
a = −0.80902 − 1.40126I
b = −0.309017 − 0.535233I
−0.98696 − 2.02988I −13.5000 + 5.4006I
u = −1.61803
a = 0.309017 + 0.535233I
b = 0.80902 + 1.40126I
−8.88264 − 2.02988I −13.50000 + 1.52761I
u = −1.61803
a = 0.309017 − 0.535233I
b = 0.80902 − 1.40126I
−8.88264 + 2.02988I −13.50000 − 1.52761I
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
2
)(u
61
+ 29u
60
+ ··· + 23u − 1)
c
2
((u
2
+ u + 1)
2
)(u
61
+ 3u
60
+ ··· + 9u + 1)
c
3
((u
2
− u + 1)
2
)(u
61
− 3u
60
+ ··· − 129u + 241)
c
4
, c
9
u
4
(u
61
+ u
60
+ ··· − 48u − 16)
c
5
((u
2
− u + 1)
2
)(u
61
+ 3u
60
+ ··· + 9u + 1)
c
6
, c
7
, c
8
((u
2
+ u − 1)
2
)(u
61
+ 3u
60
+ ··· + u − 1)
c
10
u
4
(u
61
+ 25u
60
+ ··· + 2432u + 256)
c
11
, c
12
((u
2
− u − 1)
2
)(u
61
+ 3u
60
+ ··· + u − 1)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
2
)(y
61
+ 9y
60
+ ··· + 707y −1)
c
2
, c
5
((y
2
+ y + 1)
2
)(y
61
+ 29y
60
+ ··· + 23y −1)
c
3
((y
2
+ y + 1)
2
)(y
61
− 11y
60
+ ··· + 2638239y −58081)
c
4
, c
9
y
4
(y
61
− 25y
60
+ ··· + 2432y −256)
c
6
, c
7
, c
8
c
11
, c
12
((y
2
− 3y + 1)
2
)(y
61
− 79y
60
+ ··· + 19y −1)
c
10
y
4
(y
61
+ 15y
60
+ ··· + 2564096y −65536)
17
