12a
0026
(K12a
0026
)
A knot diagram
1
Linearized knot diagam
3 5 6 9 2 11 12 10 4 1 7 8
Solving Sequence
7,12
8
1,3
2 11 6 4 5 10 9
c
7
c
12
c
1
c
11
c
6
c
3
c
5
c
10
c
9
c
2
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h15u
75
+ 20u
74
+ ··· + 2b − 12, −3u
75
+ 135u
73
+ ··· + 2a + 8, u
76
+ 3u
75
+ ··· + 2u − 1i
I
u
2
= hb + a, a
2
− a + 1, u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 80 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
= h15u
75
+ 20u
74
+ · · · + 2b − 12, −3u
75
+ 135u
73
+ · · · + 2a + 8, u
76
+
3u
75
+ · · · + 2u − 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
=
3
2
u
75
−
135
2
u
73
+ ··· +
11
2
u − 4
−
15
2
u
75
− 10u
74
+ ··· −
25
2
u + 6
a
2
=
−
1
2
u
75
− u
74
+ ··· +
3
2
u + 1
1
2
u
75
+ u
74
+ ··· −
3
2
u
2
+
1
2
u
a
11
=
−u
u
a
6
=
−u
2
+ 1
u
2
a
4
=
8u
75
+
17
2
u
74
+ ··· +
33
2
u −
19
2
−
39
2
u
75
−
55
2
u
74
+ ··· − 33u +
27
2
a
5
=
6u
75
+
21
2
u
74
+ ··· +
17
2
u −
1
2
−
9
2
u
75
−
15
2
u
74
+ ··· − 7u +
3
2
a
10
=
u
5
− 2u
3
− u
−u
7
+ 3u
5
− 2u
3
+ u
a
9
=
u
10
− 5u
8
+ 6u
6
+ u
4
− u
2
+ 1
−u
12
+ 6u
10
− 12u
8
+ 10u
6
− 5u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes =
29
2
u
75
+ 24u
74
+ ··· +
13
2
u + 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
76
+ 35u
75
+ ··· − 34u + 1
c
2
, c
5
u
76
+ 3u
75
+ ··· − 2u + 1
c
3
u
76
− 3u
75
+ ··· − 458u + 41
c
4
, c
9
u
76
+ u
75
+ ··· − 124u
2
− 16
c
6
, c
7
, c
11
c
12
u
76
− 3u
75
+ ··· − 2u − 1
c
8
u
76
+ 25u
75
+ ··· + 3968u + 256
c
10
u
76
+ 23u
75
+ ··· − 53922u − 8023
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
76
+ 15y
75
+ ··· − 1370y + 1
c
2
, c
5
y
76
+ 35y
75
+ ··· − 34y + 1
c
3
y
76
− 5y
75
+ ··· − 135554y + 1681
c
4
, c
9
y
76
+ 25y
75
+ ··· + 3968y + 256
c
6
, c
7
, c
11
c
12
y
76
− 89y
75
+ ··· − 14y + 1
c
8
y
76
+ 45y
75
+ ··· − 4202496y + 65536
c
10
y
76
− 29y
75
+ ··· − 1605176402y + 64368529
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.885990 + 0.333160I
a = 0.872634 − 0.542118I
b = −0.67077 + 1.24857I
1.97205 + 5.49296I 0
u = −0.885990 − 0.333160I
a = 0.872634 + 0.542118I
b = −0.67077 − 1.24857I
1.97205 − 5.49296I 0
u = 0.751786 + 0.507447I
a = −1.107940 − 0.282272I
b = −0.99521 − 1.10936I
0.67458 + 12.58000I 0
u = 0.751786 − 0.507447I
a = −1.107940 + 0.282272I
b = −0.99521 + 1.10936I
0.67458 − 12.58000I 0
u = −0.836373 + 0.339607I
a = −0.755203 + 0.432767I
b = −0.004969 − 0.761767I
3.83321 + 0.59439I 0
u = −0.836373 − 0.339607I
a = −0.755203 − 0.432767I
b = −0.004969 + 0.761767I
3.83321 − 0.59439I 0
u = 0.747341 + 0.484979I
a = 0.910922 + 0.390103I
b = 0.844895 + 0.477346I
2.79991 + 7.37350I 0
u = 0.747341 − 0.484979I
a = 0.910922 − 0.390103I
b = 0.844895 − 0.477346I
2.79991 − 7.37350I 0
u = −0.867041 + 0.154083I
a = 0.590113 + 0.095531I
b = −0.737271 − 0.597559I
−0.213421 − 1.034600I 0
u = −0.867041 − 0.154083I
a = 0.590113 − 0.095531I
b = −0.737271 + 0.597559I
−0.213421 + 1.034600I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.694227 + 0.488418I
a = −0.862705 − 0.739564I
b = 0.268689 − 0.485835I
−2.20786 + 5.00695I 0
u = 0.694227 − 0.488418I
a = −0.862705 + 0.739564I
b = 0.268689 + 0.485835I
−2.20786 − 5.00695I 0
u = 0.740484 + 0.411101I
a = 0.765867 + 0.569492I
b = 0.136180 − 0.867389I
3.76796 + 4.58951I 0
u = 0.740484 − 0.411101I
a = 0.765867 − 0.569492I
b = 0.136180 + 0.867389I
3.76796 − 4.58951I 0
u = −0.745942 + 0.396837I
a = −1.022130 + 0.377114I
b = −0.892114 + 0.438279I
3.86486 − 1.58644I 0
u = −0.745942 − 0.396837I
a = −1.022130 − 0.377114I
b = −0.892114 − 0.438279I
3.86486 + 1.58644I 0
u = −0.719638 + 0.432491I
a = 1.39607 − 0.37314I
b = 1.09423 − 1.13795I
2.04591 − 6.50107I 0
u = −0.719638 − 0.432491I
a = 1.39607 + 0.37314I
b = 1.09423 + 1.13795I
2.04591 + 6.50107I 0
u = 0.731033 + 0.364174I
a = −0.925187 − 0.555287I
b = 0.510713 + 1.186120I
2.50490 − 0.55456I 9.61533 − 2.91728I
u = 0.731033 − 0.364174I
a = −0.925187 + 0.555287I
b = 0.510713 − 1.186120I
2.50490 + 0.55456I 9.61533 + 2.91728I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.505111 + 0.534579I
a = −1.202510 + 0.531702I
b = −0.942344 − 0.780413I
−5.36892 + 5.69274I 0.38490 − 7.89158I
u = 0.505111 − 0.534579I
a = −1.202510 − 0.531702I
b = −0.942344 + 0.780413I
−5.36892 − 5.69274I 0.38490 + 7.89158I
u = −0.596247 + 0.341693I
a = 0.55214 − 1.30810I
b = −0.441166 − 0.807337I
−0.315070 − 0.076640I 5.64186 + 2.36469I
u = −0.596247 − 0.341693I
a = 0.55214 + 1.30810I
b = −0.441166 + 0.807337I
−0.315070 + 0.076640I 5.64186 − 2.36469I
u = 0.404061 + 0.546779I
a = −1.53472 − 0.82950I
b = −0.103415 − 0.768073I
−5.65998 − 1.97285I −0.963134 + 0.338595I
u = 0.404061 − 0.546779I
a = −1.53472 + 0.82950I
b = −0.103415 + 0.768073I
−5.65998 + 1.97285I −0.963134 − 0.338595I
u = 0.456233 + 0.476408I
a = 0.807953 + 0.259709I
b = 0.581862 + 0.454575I
−2.40945 + 1.67574I 3.56432 − 4.46273I
u = 0.456233 − 0.476408I
a = 0.807953 − 0.259709I
b = 0.581862 − 0.454575I
−2.40945 − 1.67574I 3.56432 + 4.46273I
u = 0.121254 + 0.636401I
a = −1.82213 − 0.10923I
b = −0.352065 − 0.838752I
−1.18944 − 8.72098I 3.35149 + 6.41865I
u = 0.121254 − 0.636401I
a = −1.82213 + 0.10923I
b = −0.352065 + 0.838752I
−1.18944 + 8.72098I 3.35149 − 6.41865I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.105876 + 0.603977I
a = 0.938724 + 0.288370I
b = 0.264725 + 0.759368I
0.91644 − 3.68473I 6.38251 + 2.47980I
u = 0.105876 − 0.603977I
a = 0.938724 − 0.288370I
b = 0.264725 − 0.759368I
0.91644 + 3.68473I 6.38251 − 2.47980I
u = 0.187489 + 0.573478I
a = −0.431525 + 1.144150I
b = −0.402625 − 0.455153I
−3.68972 − 1.37288I −0.612694 + 0.508979I
u = 0.187489 − 0.573478I
a = −0.431525 − 1.144150I
b = −0.402625 + 0.455153I
−3.68972 + 1.37288I −0.612694 − 0.508979I
u = −1.46039 + 0.08103I
a = 1.46326 − 1.34448I
b = −2.15619 + 2.33527I
0.265608 − 0.146440I 0
u = −1.46039 − 0.08103I
a = 1.46326 + 1.34448I
b = −2.15619 − 2.33527I
0.265608 + 0.146440I 0
u = 0.013313 + 0.514387I
a = −1.138510 + 0.344338I
b = −0.070209 + 0.768340I
1.70461 − 1.46252I 7.24578 + 3.00546I
u = 0.013313 − 0.514387I
a = −1.138510 − 0.344338I
b = −0.070209 − 0.768340I
1.70461 + 1.46252I 7.24578 − 3.00546I
u = −0.082708 + 0.494197I
a = 2.30376 − 0.01004I
b = 0.338632 − 0.907658I
0.24315 + 3.28654I 4.66258 − 2.10063I
u = −0.082708 − 0.494197I
a = 2.30376 + 0.01004I
b = 0.338632 + 0.907658I
0.24315 − 3.28654I 4.66258 + 2.10063I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.50687 + 0.12545I
a = 2.62859 − 0.35369I
b = −3.79993 + 0.65614I
1.23477 − 8.00671I 0
u = −1.50687 − 0.12545I
a = 2.62859 + 0.35369I
b = −3.79993 − 0.65614I
1.23477 + 8.00671I 0
u = −1.50985 + 0.09146I
a = −1.82204 + 0.35941I
b = 2.53314 − 0.79410I
4.06697 − 3.56308I 0
u = −1.50985 − 0.09146I
a = −1.82204 − 0.35941I
b = 2.53314 + 0.79410I
4.06697 + 3.56308I 0
u = −0.485277
a = −0.607470
b = −0.408478
0.739641 13.5000
u = −0.332468 + 0.348514I
a = 1.46781 + 1.80245I
b = 1.019820 − 0.128661I
−1.06070 − 2.50916I 3.18547 + 6.68598I
u = −0.332468 − 0.348514I
a = 1.46781 − 1.80245I
b = 1.019820 + 0.128661I
−1.06070 + 2.50916I 3.18547 − 6.68598I
u = 1.51846 + 0.03177I
a = −3.24068 + 1.10209I
b = 4.94954 − 1.73737I
5.15855 + 3.46829I 0
u = 1.51846 − 0.03177I
a = −3.24068 − 1.10209I
b = 4.94954 + 1.73737I
5.15855 − 3.46829I 0
u = −1.54556 + 0.01387I
a = −0.439333 − 0.753773I
b = 0.502805 + 0.325969I
7.25856 − 2.56091I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.54556 − 0.01387I
a = −0.439333 + 0.753773I
b = 0.502805 − 0.325969I
7.25856 + 2.56091I 0
u = 1.55696
a = 2.02517
b = −3.03221
7.77426 0
u = 0.398202 + 0.093446I
a = −0.14340 + 1.82967I
b = 0.405564 − 0.876916I
0.51331 + 2.24257I −1.43172 − 6.93184I
u = 0.398202 − 0.093446I
a = −0.14340 − 1.82967I
b = 0.405564 + 0.876916I
0.51331 − 2.24257I −1.43172 + 6.93184I
u = 1.59191 + 0.09160I
a = −0.24009 − 2.06938I
b = 0.42940 + 3.38884I
7.23380 + 1.61258I 0
u = 1.59191 − 0.09160I
a = −0.24009 + 2.06938I
b = 0.42940 − 3.38884I
7.23380 − 1.61258I 0
u = −1.60359 + 0.14007I
a = 0.34694 − 1.56141I
b = −0.66188 + 2.68266I
5.58997 − 7.33867I 0
u = −1.60359 − 0.14007I
a = 0.34694 + 1.56141I
b = −0.66188 − 2.68266I
5.58997 + 7.33867I 0
u = 1.61266 + 0.12405I
a = −2.90463 − 1.94112I
b = 4.28379 + 3.40715I
10.00480 + 8.58426I 0
u = 1.61266 − 0.12405I
a = −2.90463 + 1.94112I
b = 4.28379 − 3.40715I
10.00480 − 8.58426I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.61463 + 0.10575I
a = −0.383616 + 0.816434I
b = −0.096006 − 0.643122I
10.53420 − 1.22209I 0
u = −1.61463 − 0.10575I
a = −0.383616 − 0.816434I
b = −0.096006 + 0.643122I
10.53420 + 1.22209I 0
u = −1.61807 + 0.11768I
a = −0.465116 − 0.496373I
b = 1.131700 + 0.081107I
11.83030 − 6.57887I 0
u = −1.61807 − 0.11768I
a = −0.465116 + 0.496373I
b = 1.131700 − 0.081107I
11.83030 + 6.57887I 0
u = 1.61915 + 0.11332I
a = 2.22775 + 1.07424I
b = −3.39037 − 2.13749I
11.95540 + 3.50791I 0
u = 1.61915 − 0.11332I
a = 2.22775 − 1.07424I
b = −3.39037 + 2.13749I
11.95540 − 3.50791I 0
u = −1.62160 + 0.14108I
a = −1.95698 + 1.11347I
b = 2.90889 − 2.25295I
10.8692 − 9.7340I 0
u = −1.62160 − 0.14108I
a = −1.95698 − 1.11347I
b = 2.90889 + 2.25295I
10.8692 + 9.7340I 0
u = −1.62326 + 0.14883I
a = 2.44619 − 1.87526I
b = −3.49846 + 3.32410I
8.7544 − 15.0591I 0
u = −1.62326 − 0.14883I
a = 2.44619 + 1.87526I
b = −3.49846 − 3.32410I
8.7544 + 15.0591I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.64269 + 0.01976I
a = 0.543002 − 1.001810I
b = −0.599317 + 1.275470I
8.41907 + 1.56630I 0
u = 1.64269 − 0.01976I
a = 0.543002 + 1.001810I
b = −0.599317 − 1.275470I
8.41907 − 1.56630I 0
u = 1.64045 + 0.09059I
a = 0.559579 − 0.461479I
b = −1.254220 + 0.186662I
12.34950 + 1.02749I 0
u = 1.64045 − 0.09059I
a = 0.559579 + 0.461479I
b = −1.254220 − 0.186662I
12.34950 − 1.02749I 0
u = 1.65268 + 0.08154I
a = 0.368277 + 0.846944I
b = 0.084298 − 0.838477I
10.73130 − 3.95211I 0
u = 1.65268 − 0.08154I
a = 0.368277 − 0.846944I
b = 0.084298 + 0.838477I
10.73130 + 3.95211I 0
12
II. I
u
2
= hb + a, a
2
− a + 1, 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
−a
a
2
=
a + u − 1
−a − u
a
11
=
−u
u
a
6
=
−u
u + 1
a
4
=
−au + a
au
a
5
=
−au + a
au
a
10
=
1
−u − 1
a
9
=
1
−u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2au + 3a − u + 12
13
(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
8
, c
9
u
4
c
6
, c
7
, c
10
(u
2
− u − 1)
2
c
11
, c
12
(u
2
+ u − 1)
2
14
(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
8
, c
9
y
4
c
6
, c
7
, c
10
c
11
, c
12
(y
2
− 3y + 1)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 0.500000 + 0.866025I
b = −0.500000 − 0.866025I
0.98696 − 2.02988I 13.50000 + 1.52761I
u = −0.618034
a = 0.500000 − 0.866025I
b = −0.500000 + 0.866025I
0.98696 + 2.02988I 13.50000 − 1.52761I
u = 1.61803
a = 0.500000 + 0.866025I
b = −0.500000 − 0.866025I
8.88264 − 2.02988I 13.5000 + 5.4006I
u = 1.61803
a = 0.500000 − 0.866025I
b = −0.500000 + 0.866025I
8.88264 + 2.02988I 13.5000 − 5.4006I
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
2
)(u
76
+ 35u
75
+ ··· − 34u + 1)
c
2
((u
2
+ u + 1)
2
)(u
76
+ 3u
75
+ ··· − 2u + 1)
c
3
((u
2
− u + 1)
2
)(u
76
− 3u
75
+ ··· − 458u + 41)
c
4
, c
9
u
4
(u
76
+ u
75
+ ··· − 124u
2
− 16)
c
5
((u
2
− u + 1)
2
)(u
76
+ 3u
75
+ ··· − 2u + 1)
c
6
, c
7
((u
2
− u − 1)
2
)(u
76
− 3u
75
+ ··· − 2u − 1)
c
8
u
4
(u
76
+ 25u
75
+ ··· + 3968u + 256)
c
10
((u
2
− u − 1)
2
)(u
76
+ 23u
75
+ ··· − 53922u − 8023)
c
11
, c
12
((u
2
+ u − 1)
2
)(u
76
− 3u
75
+ ··· − 2u − 1)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
2
)(y
76
+ 15y
75
+ ··· − 1370y + 1)
c
2
, c
5
((y
2
+ y + 1)
2
)(y
76
+ 35y
75
+ ··· − 34y + 1)
c
3
((y
2
+ y + 1)
2
)(y
76
− 5y
75
+ ··· − 135554y + 1681)
c
4
, c
9
y
4
(y
76
+ 25y
75
+ ··· + 3968y + 256)
c
6
, c
7
, c
11
c
12
((y
2
− 3y + 1)
2
)(y
76
− 89y
75
+ ··· − 14y + 1)
c
8
y
4
(y
76
+ 45y
75
+ ··· − 4202496y + 65536)
c
10
((y
2
− 3y + 1)
2
)(y
76
− 29y
75
+ ··· − 1.60518 × 10
9
y + 6.43685 × 10
7
)
18
