12n
0155
(K12n
0155
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 9 11 4 5 6 12 7 10
Solving Sequence
6,11 3,7
4 12 10 1 9 5 2 8
c
6
c
3
c
11
c
10
c
12
c
9
c
5
c
2
c
8
c
1
, c
4
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
51
− u
50
+ ··· + b − 1, −2u
51
− 2u
50
+ ··· + a − 3, u
52
+ 2u
51
+ ··· + 2u + 1i
I
u
2
= hu
7
− u
5
+ 2u
3
+ b − u + 1, u
7
− u
5
+ u
4
+ 2u
3
− u
2
+ a + 2, u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 60 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
51
−u
50
+· · ·+b−1, −2u
51
−2u
50
+· · ·+a−3, u
52
+2u
51
+· · ·+2u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
2u
51
+ 2u
50
+ ··· − 3u + 3
u
51
+ u
50
+ ··· − 2u
2
+ 1
a
7
=
1
u
2
a
4
=
u
51
− 8u
49
+ ··· − 5u + 2
2u
51
+ u
50
+ ··· + u + 1
a
12
=
−u
−u
3
+ u
a
10
=
u
3
u
5
− u
3
+ u
a
1
=
−u
5
− u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
a
5
=
−u
10
+ u
8
− 2u
6
+ u
4
− u
2
+ 1
−u
10
+ 2u
8
− 3u
6
+ 2u
4
− u
2
a
2
=
u
51
+ u
50
+ ··· − 4u + 3
u
51
+ u
50
+ ··· + u + 1
a
8
=
−u
15
+ 2u
13
− 4u
11
+ 4u
9
− 4u
7
+ 4u
5
− 2u
3
+ 2u
−u
15
+ 3u
13
− 6u
11
+ 7u
9
− 6u
7
+ 4u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
51
+ 2u
50
+ ··· − 14u − 3
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
52
+ 15u
51
+ ··· + 34u + 1
c
2
, c
4
u
52
− 9u
51
+ ··· − 10u + 1
c
3
, c
7
u
52
− u
51
+ ··· + 640u + 256
c
5
, c
8
, c
9
u
52
+ 2u
51
+ ··· + 336u + 49
c
6
, c
11
u
52
− 2u
51
+ ··· − 2u + 1
c
10
, c
12
u
52
+ 18u
51
+ ··· + 14u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
52
+ 53y
51
+ ··· − 706y + 1
c
2
, c
4
y
52
− 15y
51
+ ··· − 34y + 1
c
3
, c
7
y
52
− 51y
51
+ ··· − 1622016y + 65536
c
5
, c
8
, c
9
y
52
− 26y
51
+ ··· − 65170y + 2401
c
6
, c
11
y
52
− 18y
51
+ ··· − 14y + 1
c
10
, c
12
y
52
+ 34y
51
+ ··· − 14y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.661298 + 0.748302I
a = −0.320274 + 0.754666I
b = −0.79199 − 1.48215I
0.80822 + 2.47111I −6.26058 − 3.26854I
u = 0.661298 − 0.748302I
a = −0.320274 − 0.754666I
b = −0.79199 + 1.48215I
0.80822 − 2.47111I −6.26058 + 3.26854I
u = −0.636633 + 0.816820I
a = 0.268660 + 0.935747I
b = 2.46939 − 1.06759I
6.10406 − 8.66203I −5.62163 + 4.32258I
u = −0.636633 − 0.816820I
a = 0.268660 − 0.935747I
b = 2.46939 + 1.06759I
6.10406 + 8.66203I −5.62163 − 4.32258I
u = −1.036160 + 0.045183I
a = 0.56922 − 2.81424I
b = 0.23556 − 2.03154I
−4.76535 + 2.18839I −14.6770 − 3.6633I
u = −1.036160 − 0.045183I
a = 0.56922 + 2.81424I
b = 0.23556 + 2.03154I
−4.76535 − 2.18839I −14.6770 + 3.6633I
u = 0.723846 + 0.632329I
a = 0.194221 − 1.378370I
b = 1.63877 + 0.76596I
−0.43554 − 1.57909I −9.64188 + 1.77235I
u = 0.723846 − 0.632329I
a = 0.194221 + 1.378370I
b = 1.63877 − 0.76596I
−0.43554 + 1.57909I −9.64188 − 1.77235I
u = 1.04031
a = 0.375292
b = −0.518107
−6.36986 −13.6620
u = −0.657608 + 0.692008I
a = 1.36055 − 0.47884I
b = 0.330991 − 0.978148I
−1.297150 − 0.510740I −6.51784 − 0.83295I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.657608 − 0.692008I
a = 1.36055 + 0.47884I
b = 0.330991 + 0.978148I
−1.297150 + 0.510740I −6.51784 + 0.83295I
u = −0.666205 + 0.808385I
a = −0.331437 − 0.469968I
b = −2.08963 + 0.58134I
7.27982 − 1.78274I −3.98925 − 0.13082I
u = −0.666205 − 0.808385I
a = −0.331437 + 0.469968I
b = −2.08963 − 0.58134I
7.27982 + 1.78274I −3.98925 + 0.13082I
u = −0.806239 + 0.701530I
a = −0.587597 + 0.348457I
b = −0.210379 + 0.054742I
2.80086 + 2.09505I −2.50659 − 3.48544I
u = −0.806239 − 0.701530I
a = −0.587597 − 0.348457I
b = −0.210379 − 0.054742I
2.80086 − 2.09505I −2.50659 + 3.48544I
u = 1.063220 + 0.121546I
a = −2.29627 + 1.25450I
b = −1.38281 + 0.68102I
0.93793 − 1.58244I −10.84568 + 1.37730I
u = 1.063220 − 0.121546I
a = −2.29627 − 1.25450I
b = −1.38281 − 0.68102I
0.93793 + 1.58244I −10.84568 − 1.37730I
u = 0.549487 + 0.745372I
a = −0.521065 − 0.048868I
b = 0.799139 − 0.412000I
−1.81149 + 1.27627I −3.31406 − 1.04966I
u = 0.549487 − 0.745372I
a = −0.521065 + 0.048868I
b = 0.799139 + 0.412000I
−1.81149 − 1.27627I −3.31406 + 1.04966I
u = −0.973478 + 0.497457I
a = −1.35010 − 1.41893I
b = −0.920949 + 0.416007I
3.12207 + 4.60453I −8.72425 − 5.65975I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.973478 − 0.497457I
a = −1.35010 + 1.41893I
b = −0.920949 − 0.416007I
3.12207 − 4.60453I −8.72425 + 5.65975I
u = 1.099660 + 0.097397I
a = 2.11231 − 2.23460I
b = 1.53001 − 1.70626I
−0.24824 − 8.05886I −12.40052 + 5.70179I
u = 1.099660 − 0.097397I
a = 2.11231 + 2.23460I
b = 1.53001 + 1.70626I
−0.24824 + 8.05886I −12.40052 − 5.70179I
u = −1.11145
a = 1.48524
b = 1.21896
−7.36266 −8.96100
u = −0.905213 + 0.681246I
a = −0.523152 − 0.438777I
b = −0.1179880 − 0.0609129I
2.49405 + 3.22105I −3.32606 − 3.39208I
u = −0.905213 − 0.681246I
a = −0.523152 + 0.438777I
b = −0.1179880 + 0.0609129I
2.49405 − 3.22105I −3.32606 + 3.39208I
u = 0.856090 + 0.776847I
a = −0.786022 − 0.932466I
b = −0.57351 − 1.38717I
10.43740 + 0.66966I −3.01264 + 0.I
u = 0.856090 − 0.776847I
a = −0.786022 + 0.932466I
b = −0.57351 + 1.38717I
10.43740 − 0.66966I −3.01264 + 0.I
u = −1.015300 + 0.553492I
a = 1.73668 + 0.33530I
b = 0.221364 − 1.300870I
2.49928 − 1.45715I −9.46738 + 0.I
u = −1.015300 − 0.553492I
a = 1.73668 − 0.33530I
b = 0.221364 + 1.300870I
2.49928 + 1.45715I −9.46738 + 0.I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.972910 + 0.643628I
a = −1.12101 − 2.08154I
b = 1.67842 − 1.59230I
−1.23183 − 3.44766I −10.73445 + 3.25709I
u = 0.972910 − 0.643628I
a = −1.12101 + 2.08154I
b = 1.67842 + 1.59230I
−1.23183 + 3.44766I −10.73445 − 3.25709I
u = 0.884417 + 0.769320I
a = 1.04291 + 1.44163I
b = −0.05794 + 1.42188I
10.35090 − 6.48169I −3.29234 + 5.33033I
u = 0.884417 − 0.769320I
a = 1.04291 − 1.44163I
b = −0.05794 − 1.42188I
10.35090 + 6.48169I −3.29234 − 5.33033I
u = −0.995994 + 0.663488I
a = −0.447793 + 0.743729I
b = −0.197171 + 1.217400I
−2.30244 + 5.77043I −8.73703 − 4.68081I
u = −0.995994 − 0.663488I
a = −0.447793 − 0.743729I
b = −0.197171 − 1.217400I
−2.30244 − 5.77043I −8.73703 + 4.68081I
u = 1.005230 + 0.684675I
a = 1.66351 + 1.36728I
b = −0.93378 + 1.89685I
−0.22005 − 7.93959I −8.35668 + 7.99048I
u = 1.005230 − 0.684675I
a = 1.66351 − 1.36728I
b = −0.93378 − 1.89685I
−0.22005 + 7.93959I −8.35668 − 7.99048I
u = 1.040490 + 0.652647I
a = 0.837700 − 0.810359I
b = 1.025560 + 0.617772I
−3.23709 − 6.60394I 0
u = 1.040490 − 0.652647I
a = 0.837700 + 0.810359I
b = 1.025560 − 0.617772I
−3.23709 + 6.60394I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.020440 + 0.711207I
a = −0.20228 − 2.51624I
b = −2.10957 − 0.88029I
6.20768 + 7.49853I 0
u = −1.020440 − 0.711207I
a = −0.20228 + 2.51624I
b = −2.10957 + 0.88029I
6.20768 − 7.49853I 0
u = −1.036210 + 0.704214I
a = −0.19781 + 3.00542I
b = 2.71533 + 1.45235I
4.8974 + 14.3736I 0
u = −1.036210 − 0.704214I
a = −0.19781 − 3.00542I
b = 2.71533 − 1.45235I
4.8974 − 14.3736I 0
u = −0.316269 + 0.673477I
a = 0.152937 − 1.157570I
b = 0.964431 + 0.772467I
4.38303 + 5.94973I −5.70519 − 4.98093I
u = −0.316269 − 0.673477I
a = 0.152937 + 1.157570I
b = 0.964431 − 0.772467I
4.38303 − 5.94973I −5.70519 + 4.98093I
u = 0.702232
a = −0.797304
b = 0.0429664
−1.05113 −9.14920
u = −0.237887 + 0.635480I
a = −0.196476 + 0.636369I
b = −1.285860 − 0.135549I
5.12115 − 0.59863I −4.16704 − 0.03207I
u = −0.237887 − 0.635480I
a = −0.196476 − 0.636369I
b = −1.285860 + 0.135549I
5.12115 + 0.59863I −4.16704 + 0.03207I
u = 0.311034 + 0.368798I
a = −0.691540 − 1.071180I
b = 0.258704 + 0.649617I
−0.684234 − 1.109730I −7.41214 + 5.86160I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.311034 − 0.368798I
a = −0.691540 + 1.071180I
b = 0.258704 − 0.649617I
−0.684234 + 1.109730I −7.41214 − 5.86160I
u = −0.359182
a = 3.20503
b = 0.863998
−2.10063 0.990710
10
II. I
u
2
= hu
7
− u
5
+ 2u
3
+ b − u + 1, u
7
− u
5
+ u
4
+ 2u
3
− u
2
+ a + 2, u
8
−
u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
3
=
−u
7
+ u
5
− u
4
− 2u
3
+ u
2
− 2
−u
7
+ u
5
− 2u
3
+ u − 1
a
7
=
1
u
2
a
4
=
−u
7
+ u
5
− u
4
− 2u
3
+ u
2
− 2
−u
7
+ u
5
− 2u
3
+ u − 1
a
12
=
−u
−u
3
+ u
a
10
=
u
3
u
5
− u
3
+ u
a
1
=
−u
5
− u
−u
7
+ u
5
− 2u
3
+ u
a
9
=
u
5
+ u
u
5
− u
3
+ u
a
5
=
u
5
+ u
u
7
− u
5
+ 2u
3
− u
a
2
=
−u
7
− u
4
− 2u
3
+ u
2
− u − 2
−2u
7
+ 2u
5
− 4u
3
+ 2u − 1
a
8
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −6u
7
+ u
6
+ 11u
5
− 8u
4
− 11u
3
+ 7u
2
+ 4u − 23
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
7
u
8
c
4
(u + 1)
8
c
5
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
6
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
8
, c
9
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
10
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
11
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
12
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
7
y
8
c
5
, c
8
, c
9
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
6
, c
11
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
10
, c
12
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.570868 + 0.730671I
a = −0.805639 − 0.183365I
b = 0.320534 − 0.633953I
−2.68559 + 1.13123I −13.35119 − 0.17229I
u = 0.570868 − 0.730671I
a = −0.805639 + 0.183365I
b = 0.320534 + 0.633953I
−2.68559 − 1.13123I −13.35119 + 0.17229I
u = −0.855237 + 0.665892I
a = −0.189481 − 1.310380I
b = −1.54709 − 0.16160I
0.51448 + 2.57849I −6.04880 − 3.90294I
u = −0.855237 − 0.665892I
a = −0.189481 + 1.310380I
b = −1.54709 + 0.16160I
0.51448 − 2.57849I −6.04880 + 3.90294I
u = −1.09818
a = 0.729394
b = 0.879647
−8.14766 −20.2760
u = 1.031810 + 0.655470I
a = 0.708845 − 0.169402I
b = 0.679246 + 0.851242I
−4.02461 − 6.44354I −15.5815 + 4.6831I
u = 1.031810 − 0.655470I
a = 0.708845 + 0.169402I
b = 0.679246 − 0.851242I
−4.02461 + 6.44354I −15.5815 − 4.6831I
u = 0.603304
a = −2.15684
b = −0.785038
−2.48997 −20.7610
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
52
+ 15u
51
+ ··· + 34u + 1)
c
2
((u − 1)
8
)(u
52
− 9u
51
+ ··· − 10u + 1)
c
3
, c
7
u
8
(u
52
− u
51
+ ··· + 640u + 256)
c
4
((u + 1)
8
)(u
52
− 9u
51
+ ··· − 10u + 1)
c
5
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
52
+ 2u
51
+ ··· + 336u + 49)
c
6
(u
8
− u
7
+ ··· + 2u − 1)(u
52
− 2u
51
+ ··· − 2u + 1)
c
8
, c
9
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)(u
52
+ 2u
51
+ ··· + 336u + 49)
c
10
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
52
+ 18u
51
+ ··· + 14u + 1)
c
11
(u
8
+ u
7
+ ··· − 2u − 1)(u
52
− 2u
51
+ ··· − 2u + 1)
c
12
(u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1)
· (u
52
+ 18u
51
+ ··· + 14u + 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
52
+ 53y
51
+ ··· − 706y + 1)
c
2
, c
4
((y −1)
8
)(y
52
− 15y
51
+ ··· − 34y + 1)
c
3
, c
7
y
8
(y
52
− 51y
51
+ ··· − 1622016y + 65536)
c
5
, c
8
, c
9
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
52
− 26y
51
+ ··· − 65170y + 2401)
c
6
, c
11
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
52
− 18y
51
+ ··· − 14y + 1)
c
10
, c
12
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
52
+ 34y
51
+ ··· − 14y + 1)
16
