12a
0721
(K12a
0721
)
A knot diagram
1
Linearized knot diagam
3 8 9 10 11 12 2 1 7 6 5 4
Solving Sequence
6,10
11 5 12 7 4 1 9 3 8 2
c
10
c
5
c
11
c
6
c
4
c
12
c
9
c
3
c
8
c
2
c
1
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
85
+ u
84
+ ··· + u + 1i
* 1 irreducible components of dim
C
= 0, with total 85 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
85
+ u
84
+ · · · + u + 1i
(i) Arc colorings
a
6
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
5
=
−u
u
3
+ u
a
12
=
u
2
+ 1
−u
4
− 2u
2
a
7
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
4
=
−u
3
− 2u
u
3
+ u
a
1
=
−u
10
− 5u
8
− 8u
6
− 3u
4
+ 3u
2
+ 1
u
10
+ 4u
8
+ 5u
6
− 3u
2
a
9
=
−u
12
− 5u
10
− 9u
8
− 6u
6
+ u
2
+ 1
u
14
+ 6u
12
+ 13u
10
+ 10u
8
− 2u
6
− 4u
4
+ u
2
a
3
=
u
29
+ 12u
27
+ ··· − 6u
3
− 3u
−u
31
− 13u
29
+ ··· + 24u
7
+ u
a
8
=
−u
34
− 15u
32
+ ··· − 3u
2
+ 1
u
34
+ 14u
32
+ ··· + 8u
4
+ u
2
a
2
=
u
70
+ 29u
68
+ ··· + 9u
4
+ 1
−u
72
− 30u
70
+ ··· − 6u
4
− 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
84
+ 4u
83
+ ··· − 8u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
85
+ 41u
84
+ ··· + 3u + 1
c
2
, c
7
u
85
+ u
84
+ ··· − u − 1
c
3
u
85
− u
84
+ ··· − 3u − 1
c
4
, c
6
u
85
+ u
84
+ ··· − 3u − 2
c
5
, c
10
, c
11
u
85
− u
84
+ ··· + u − 1
c
8
u
85
+ 3u
84
+ ··· − 703u − 192
c
9
, c
12
u
85
+ 7u
84
+ ··· + 433u + 37
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
85
+ 7y
84
+ ··· − 21y −1
c
2
, c
7
y
85
− 41y
84
+ ··· + 3y −1
c
3
y
85
− y
84
+ ··· + 67y −1
c
4
, c
6
y
85
− 45y
84
+ ··· + 9y −4
c
5
, c
10
, c
11
y
85
+ 71y
84
+ ··· + 3y −1
c
8
y
85
+ 27y
84
+ ··· + 92545y −36864
c
9
, c
12
y
85
+ 59y
84
+ ··· − 33697y −1369
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.248900 + 1.008010I
−5.94190 + 0.14890I 0
u = −0.248900 − 1.008010I
−5.94190 − 0.14890I 0
u = −0.297171 + 1.034020I
−4.01353 + 7.94113I 0
u = −0.297171 − 1.034020I
−4.01353 − 7.94113I 0
u = 0.280740 + 1.047720I
−1.61529 − 3.02389I 0
u = 0.280740 − 1.047720I
−1.61529 + 3.02389I 0
u = 0.270441 + 1.112840I
−0.089929 − 1.275270I 0
u = 0.270441 − 1.112840I
−0.089929 + 1.275270I 0
u = −0.278034 + 1.156140I
−0.94658 − 3.20959I 0
u = −0.278034 − 1.156140I
−0.94658 + 3.20959I 0
u = 0.209551 + 0.776333I
−6.17352 + 0.11601I −1.51814 − 0.75097I
u = 0.209551 − 0.776333I
−6.17352 − 0.11601I −1.51814 + 0.75097I
u = −0.781173 + 0.165218I
−1.36801 − 11.97790I 5.82993 + 9.11891I
u = −0.781173 − 0.165218I
−1.36801 + 11.97790I 5.82993 − 9.11891I
u = 0.775804 + 0.160398I
1.06779 + 6.99922I 9.02457 − 5.56636I
u = 0.775804 − 0.160398I
1.06779 − 6.99922I 9.02457 + 5.56636I
u = −0.766023 + 0.170399I
−3.38713 − 4.04703I 2.85028 + 3.33634I
u = −0.766023 − 0.170399I
−3.38713 + 4.04703I 2.85028 − 3.33634I
u = −0.778318 + 0.031728I
4.10355 − 5.77938I 10.92900 + 6.00912I
u = −0.778318 − 0.031728I
4.10355 + 5.77938I 10.92900 − 6.00912I
u = 0.764543 + 0.137328I
2.81551 + 5.13000I 10.71873 − 6.57822I
u = 0.764543 − 0.137328I
2.81551 − 5.13000I 10.71873 + 6.57822I
u = 0.772607 + 0.016397I
5.85261 + 1.00201I 14.4008 − 0.7177I
u = 0.772607 − 0.016397I
5.85261 − 1.00201I 14.4008 + 0.7177I
u = 0.264885 + 0.713873I
−4.48668 + 7.92179I 1.27242 − 7.67068I
u = 0.264885 − 0.713873I
−4.48668 − 7.92179I 1.27242 + 7.67068I
u = −0.750929 + 0.115587I
2.17145 − 0.56664I 9.65396 + 0.21560I
u = −0.750929 − 0.115587I
2.17145 + 0.56664I 9.65396 − 0.21560I
u = 0.719433 + 0.187623I
−4.06922 + 3.45026I 2.04119 − 4.45417I
u = 0.719433 − 0.187623I
−4.06922 − 3.45026I 2.04119 + 4.45417I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.228274 + 0.698343I
−2.02866 − 3.11378I 4.29634 + 4.05625I
u = −0.228274 − 0.698343I
−2.02866 + 3.11378I 4.29634 − 4.05625I
u = −0.326512 + 1.230870I
0.41498 + 1.78773I 0
u = −0.326512 − 1.230870I
0.41498 − 1.78773I 0
u = −0.184745 + 1.263020I
−3.07305 − 2.43438I 0
u = −0.184745 − 1.263020I
−3.07305 + 2.43438I 0
u = 0.688445 + 0.200059I
−2.66519 − 4.42958I 4.16803 + 2.23694I
u = 0.688445 − 0.200059I
−2.66519 + 4.42958I 4.16803 − 2.23694I
u = −0.691627 + 0.177718I
−0.102256 − 0.240928I 7.49528 + 1.37110I
u = −0.691627 − 0.177718I
−0.102256 + 0.240928I 7.49528 − 1.37110I
u = 0.324883 + 1.246210I
2.05753 + 2.96009I 0
u = 0.324883 − 1.246210I
2.05753 − 2.96009I 0
u = 0.116828 + 1.284750I
−6.28256 − 0.70523I 0
u = 0.116828 − 1.284750I
−6.28256 + 0.70523I 0
u = −0.699942
1.55328 6.87670
u = 0.329649 + 1.271860I
1.85314 + 4.97870I 0
u = 0.329649 − 1.271860I
1.85314 − 4.97870I 0
u = −0.290553 + 1.282560I
−2.46212 − 3.58115I 0
u = −0.290553 − 1.282560I
−2.46212 + 3.58115I 0
u = 0.182008 + 1.306080I
−5.59206 + 6.55252I 0
u = 0.182008 − 1.306080I
−5.59206 − 6.55252I 0
u = −0.334981 + 1.281900I
0.01669 − 9.79487I 0
u = −0.334981 − 1.281900I
0.01669 + 9.79487I 0
u = −0.317650 + 1.339370I
−2.41072 − 4.43457I 0
u = −0.317650 − 1.339370I
−2.41072 + 4.43457I 0
u = −0.013172 + 1.384380I
−6.34751 − 2.22482I 0
u = −0.013172 − 1.384380I
−6.34751 + 2.22482I 0
u = 0.324897 + 1.349170I
−1.86872 + 9.07355I 0
u = 0.324897 − 1.349170I
−1.86872 − 9.07355I 0
u = −0.293142 + 1.358620I
−4.94324 − 3.84382I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.293142 − 1.358620I
−4.94324 + 3.84382I 0
u = 0.288385 + 1.364890I
−7.59509 − 0.86185I 0
u = 0.288385 − 1.364890I
−7.59509 + 0.86185I 0
u = 0.301046 + 1.365670I
−8.97185 + 7.16378I 0
u = 0.301046 − 1.365670I
−8.97185 − 7.16378I 0
u = 0.328741 + 1.361380I
−3.73450 + 10.99570I 0
u = 0.328741 − 1.361380I
−3.73450 − 10.99570I 0
u = −0.323051 + 1.364830I
−8.23465 − 7.99145I 0
u = −0.323051 − 1.364830I
−8.23465 + 7.99145I 0
u = −0.330797 + 1.364300I
−6.1963 − 15.9996I 0
u = −0.330797 − 1.364300I
−6.1963 + 15.9996I 0
u = −0.023413 + 1.410440I
−8.46842 − 3.63559I 0
u = −0.023413 − 1.410440I
−8.46842 + 3.63559I 0
u = 0.02698 + 1.41604I
−11.02750 + 8.52728I 0
u = 0.02698 − 1.41604I
−11.02750 − 8.52728I 0
u = 0.01432 + 1.41634I
−12.76900 + 0.46813I 0
u = 0.01432 − 1.41634I
−12.76900 − 0.46813I 0
u = −0.159545 + 0.546638I
−0.43421 − 1.85967I 4.91547 + 5.62191I
u = −0.159545 − 0.546638I
−0.43421 + 1.85967I 4.91547 − 5.62191I
u = 0.408138 + 0.257025I
−0.93232 + 4.40586I 5.40236 − 8.36399I
u = 0.408138 − 0.257025I
−0.93232 − 4.40586I 5.40236 + 8.36399I
u = 0.266823 + 0.380763I
−1.43413 − 1.94308I 2.89552 − 0.96140I
u = 0.266823 − 0.380763I
−1.43413 + 1.94308I 2.89552 + 0.96140I
u = −0.391169 + 0.128683I
0.923187 − 0.319535I 11.07004 + 2.67367I
u = −0.391169 − 0.128683I
0.923187 + 0.319535I 11.07004 − 2.67367I
7
II. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
85
+ 41u
84
+ ··· + 3u + 1
c
2
, c
7
u
85
+ u
84
+ ··· − u − 1
c
3
u
85
− u
84
+ ··· − 3u − 1
c
4
, c
6
u
85
+ u
84
+ ··· − 3u − 2
c
5
, c
10
, c
11
u
85
− u
84
+ ··· + u − 1
c
8
u
85
+ 3u
84
+ ··· − 703u − 192
c
9
, c
12
u
85
+ 7u
84
+ ··· + 433u + 37
8
III. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
85
+ 7y
84
+ ··· − 21y −1
c
2
, c
7
y
85
− 41y
84
+ ··· + 3y −1
c
3
y
85
− y
84
+ ··· + 67y −1
c
4
, c
6
y
85
− 45y
84
+ ··· + 9y −4
c
5
, c
10
, c
11
y
85
+ 71y
84
+ ··· + 3y −1
c
8
y
85
+ 27y
84
+ ··· + 92545y −36864
c
9
, c
12
y
85
+ 59y
84
+ ··· − 33697y −1369
9
