12a
0668
(K12a
0668
)
A knot diagram
1
Linearized knot diagam
3 7 11 8 10 2 6 12 5 1 4 9
Solving Sequence
2,6
7 3 8
1,10
11 5 4 9 12
c
6
c
2
c
7
c
1
c
10
c
5
c
4
c
9
c
12
c
3
, c
8
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−192u
76
− 372u
75
+ ··· + 2304b + 776, −4833u
76
− 10270u
75
+ ··· + 52992a + 144168,
u
77
+ 4u
76
+ ··· − 32u + 46i
I
u
2
= hb + 1, −u
3
− 8u
2
+ 10a − 2u + 14, u
4
− 2u
2
+ 2i
I
u
3
= h−6a
2
u
2
+ 7a
2
u − u
2
a + 2a
2
− 2au + 2u
2
+ 19b + 13a + 4u + 12, a
3
+ 2a
2
u − 2a
2
+ au + u
2
+ a − u − 2,
u
3
− u
2
+ 1i
I
u
4
= h−b
2
u
2
a + b
2
au + b
3
+ u
2
b + u
2
a − bu − au − u
2
− b + u − 1, u
3
− u
2
+ 1i
I
v
1
= ha, b
3
− b + 1, v − 1i
I
v
2
= ha, b − 1, v − 1i
* 5 irreducible components of dim
C
= 0, with total 94 representations.
* 1 irreducible components of dim
C
= 1
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−192u
76
− 372u
75
+ · · · + 2304b + 776, −4833u
76
− 10270u
75
+ · · · +
52992a + 144168, u
77
+ 4u
76
+ · · · − 32u + 46i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
8
=
−u
2
+ 1
u
2
a
1
=
u
3
u
5
− u
3
+ u
a
10
=
0.0912024u
76
+ 0.193803u
75
+ ··· + 0.310198u − 2.72056
0.0833333u
76
+ 0.161458u
75
+ ··· + 3.90538u − 0.336806
a
11
=
−0.256454u
76
− 0.885624u
75
+ ··· + 7.09666u − 12.3039
0.375000u
76
+ 0.894531u
75
+ ··· + 9.90799u + 2.05903
a
5
=
−0.0515172u
76
− 0.0498188u
75
+ ··· − 4.84681u + 4.73970
−0.00520833u
76
− 0.0716146u
75
+ ··· + 3.14583u − 1.83073
a
4
=
0.0448370u
76
+ 0.127264u
75
+ ··· + 1.43965u + 2.90897
0.0234375u
76
+ 0.145833u
75
+ ··· − 5.97917u + 3.02083
a
9
=
−0.694407u
76
− 1.99580u
75
+ ··· + 9.07051u − 19.4422
0.222222u
76
+ 0.331597u
75
+ ··· + 20.5932u − 5.01620
a
12
=
0.672781u
76
+ 1.91826u
75
+ ··· − 11.6092u + 14.8391
−0.184028u
76
− 0.301215u
75
+ ··· − 14.0162u + 3.60880
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
677
576
u
76
−
233
54
u
75
+ ··· +
6815
72
u −
5585
108
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
77
+ 24u
76
+ ··· + 23104u + 2116
c
2
, c
6
u
77
+ 4u
76
+ ··· − 32u + 46
c
3
, c
11
27(27u
77
− 54u
76
+ ··· − 329u − 49)
c
4
64(64u
77
− 64u
76
+ ··· − 7193097u − 7328259)
c
5
, c
9
27(27u
77
+ 54u
76
+ ··· + 259u − 49)
c
8
, c
12
u
77
+ 8u
76
+ ··· + 288144u + 24982
c
10
64(64u
77
+ 192u
76
+ ··· + 2.57763 × 10
7
u − 4.07362 × 10
7
)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
77
+ 60y
76
+ ··· + 169876672y − 4477456
c
2
, c
6
y
77
− 24y
76
+ ··· + 23104y − 2116
c
3
, c
11
729(729y
77
− 49572y
76
+ ··· + 20923y − 2401)
c
4
4096
· (4096y
77
+ 192512y
76
+ ··· − 759954294636435y − 53703379971081)
c
5
, c
9
729(729y
77
− 26244y
76
+ ··· + 150283y − 2401)
c
8
, c
12
y
77
− 44y
76
+ ··· + 27288625256y − 624100324
c
10
4096(4096y
77
− 208896y
76
+ ··· + 3.66690 × 10
16
y − 1.65944 × 10
15
)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.945096 + 0.389047I
a = −0.050623 − 1.070080I
b = −1.137580 − 0.286797I
−2.25647 − 1.16651I 0
u = −0.945096 − 0.389047I
a = −0.050623 + 1.070080I
b = −1.137580 + 0.286797I
−2.25647 + 1.16651I 0
u = 1.013910 + 0.200035I
a = −1.38511 + 0.76890I
b = −1.264600 − 0.574600I
−3.31072 − 7.02161I 0. + 7.58050I
u = 1.013910 − 0.200035I
a = −1.38511 − 0.76890I
b = −1.264600 + 0.574600I
−3.31072 + 7.02161I 0. − 7.58050I
u = 0.990472 + 0.339163I
a = 0.261413 + 0.088872I
b = −0.074059 + 0.966853I
4.92520 − 6.64796I 0
u = 0.990472 − 0.339163I
a = 0.261413 − 0.088872I
b = −0.074059 − 0.966853I
4.92520 + 6.64796I 0
u = 1.038030 + 0.149643I
a = 1.43369 − 0.68788I
b = 1.211790 + 0.261921I
−5.95131 − 2.26654I 0
u = 1.038030 − 0.149643I
a = 1.43369 + 0.68788I
b = 1.211790 − 0.261921I
−5.95131 + 2.26654I 0
u = −0.731824 + 0.784777I
a = 0.776824 + 0.762762I
b = −1.101830 − 0.505985I
0.30848 − 1.84699I 0
u = −0.731824 − 0.784777I
a = 0.776824 − 0.762762I
b = −1.101830 + 0.505985I
0.30848 + 1.84699I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.853104 + 0.340329I
a = 0.213746 + 0.784050I
b = −0.255176 − 0.454138I
0.14615 − 3.23036I −1.33855 + 8.12824I
u = 0.853104 − 0.340329I
a = 0.213746 − 0.784050I
b = −0.255176 + 0.454138I
0.14615 + 3.23036I −1.33855 − 8.12824I
u = 0.914885
a = −3.07046
b = −0.837226
−0.405134 −11.6500
u = −0.796108 + 0.747982I
a = −0.10708 − 1.46416I
b = 0.752389 + 0.337777I
4.41618 + 1.21466I 0
u = −0.796108 − 0.747982I
a = −0.10708 + 1.46416I
b = 0.752389 − 0.337777I
4.41618 − 1.21466I 0
u = −1.090320 + 0.129993I
a = −1.262460 + 0.002905I
b = −0.404038 + 0.480402I
3.57275 − 0.38450I 0
u = −1.090320 − 0.129993I
a = −1.262460 − 0.002905I
b = −0.404038 − 0.480402I
3.57275 + 0.38450I 0
u = −0.745520 + 0.814582I
a = −1.16110 − 1.01142I
b = 1.19077 + 0.83255I
3.35457 − 6.24610I 0
u = −0.745520 − 0.814582I
a = −1.16110 + 1.01142I
b = 1.19077 − 0.83255I
3.35457 + 6.24610I 0
u = 0.143068 + 0.884102I
a = 0.168538 + 0.574471I
b = −0.837315 − 0.407607I
1.74848 − 1.75393I 4.13656 + 4.20451I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.143068 − 0.884102I
a = 0.168538 − 0.574471I
b = −0.837315 + 0.407607I
1.74848 + 1.75393I 4.13656 − 4.20451I
u = 0.672918 + 0.883872I
a = −0.114733 + 0.792856I
b = 0.786004 − 0.691152I
10.55870 + 0.08501I 0
u = 0.672918 − 0.883872I
a = −0.114733 − 0.792856I
b = 0.786004 + 0.691152I
10.55870 − 0.08501I 0
u = 0.833547 + 0.780072I
a = −1.32633 − 0.56782I
b = 0.720819 + 0.274380I
4.51444 − 3.95451I 0
u = 0.833547 − 0.780072I
a = −1.32633 + 0.56782I
b = 0.720819 − 0.274380I
4.51444 + 3.95451I 0
u = −1.033180 + 0.514901I
a = 0.384364 + 1.026620I
b = 1.154440 + 0.045045I
−3.87527 + 4.08507I 0
u = −1.033180 − 0.514901I
a = 0.384364 − 1.026620I
b = 1.154440 − 0.045045I
−3.87527 − 4.08507I 0
u = 0.726442 + 0.901497I
a = −0.783659 + 0.902156I
b = 1.31104 − 0.70021I
9.3431 + 11.9442I 0
u = 0.726442 − 0.901497I
a = −0.783659 − 0.902156I
b = 1.31104 + 0.70021I
9.3431 − 11.9442I 0
u = −1.127160 + 0.274111I
a = −1.36172 − 0.86647I
b = −1.258500 + 0.560387I
1.39680 + 12.09280I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.127160 − 0.274111I
a = −1.36172 + 0.86647I
b = −1.258500 − 0.560387I
1.39680 − 12.09280I 0
u = 0.711151 + 0.917982I
a = 0.585330 − 0.692040I
b = −1.136020 + 0.575865I
5.31393 + 5.84981I 0
u = 0.711151 − 0.917982I
a = 0.585330 + 0.692040I
b = −1.136020 − 0.575865I
5.31393 − 5.84981I 0
u = 0.044264 + 0.833776I
a = −0.543789 − 0.831187I
b = 1.135140 + 0.597942I
5.35934 − 8.41215I 3.55808 + 6.04760I
u = 0.044264 − 0.833776I
a = −0.543789 + 0.831187I
b = 1.135140 − 0.597942I
5.35934 + 8.41215I 3.55808 − 6.04760I
u = −0.786439 + 0.862035I
a = 0.31247 + 1.67394I
b = 0.280886 − 1.281690I
12.60540 − 5.08861I 0
u = −0.786439 − 0.862035I
a = 0.31247 − 1.67394I
b = 0.280886 + 1.281690I
12.60540 + 5.08861I 0
u = −0.943461 + 0.719095I
a = −0.73688 − 2.24689I
b = −0.844466 + 0.304206I
3.95903 + 4.37252I 0
u = −0.943461 − 0.719095I
a = −0.73688 + 2.24689I
b = −0.844466 − 0.304206I
3.95903 − 4.37252I 0
u = −0.811644 + 0.868166I
a = −0.333966 − 1.230620I
b = −0.301933 + 0.806219I
7.76103 − 0.71369I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.811644 − 0.868166I
a = −0.333966 + 1.230620I
b = −0.301933 − 0.806219I
7.76103 + 0.71369I 0
u = 0.915026 + 0.766565I
a = 1.55554 + 0.50366I
b = −0.741981 + 0.209231I
4.27170 − 1.87032I 0
u = 0.915026 − 0.766565I
a = 1.55554 − 0.50366I
b = −0.741981 − 0.209231I
4.27170 + 1.87032I 0
u = 1.148530 + 0.332767I
a = −0.499979 + 0.643714I
b = −1.058150 + 0.465766I
1.69031 + 4.35768I 0
u = 1.148530 − 0.332767I
a = −0.499979 − 0.643714I
b = −1.058150 − 0.465766I
1.69031 − 4.35768I 0
u = −0.796668 + 0.910957I
a = −0.184326 + 1.034980I
b = 0.801174 − 0.667508I
10.50630 + 5.08271I 0
u = −0.796668 − 0.910957I
a = −0.184326 − 1.034980I
b = 0.801174 + 0.667508I
10.50630 − 5.08271I 0
u = −1.194470 + 0.260364I
a = 1.063800 + 0.584697I
b = 1.043670 − 0.354227I
−2.80991 + 5.52160I 0
u = −1.194470 − 0.260364I
a = 1.063800 − 0.584697I
b = 1.043670 + 0.354227I
−2.80991 − 5.52160I 0
u = −0.986381 + 0.731464I
a = 0.24532 + 2.07161I
b = 1.195710 − 0.510507I
−0.45855 + 7.58020I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.986381 − 0.731464I
a = 0.24532 − 2.07161I
b = 1.195710 + 0.510507I
−0.45855 − 7.58020I 0
u = −0.987889 + 0.748346I
a = −0.27601 − 2.32525I
b = −1.27502 + 0.83398I
2.61638 + 12.11450I 0
u = −0.987889 − 0.748346I
a = −0.27601 + 2.32525I
b = −1.27502 − 0.83398I
2.61638 − 12.11450I 0
u = −0.973772 + 0.801220I
a = −0.577378 − 0.886916I
b = 0.197456 + 0.855837I
7.24965 + 6.91213I 0
u = −0.973772 − 0.801220I
a = −0.577378 + 0.886916I
b = 0.197456 − 0.855837I
7.24965 − 6.91213I 0
u = −0.986081 + 0.788378I
a = 1.18896 + 1.12839I
b = −0.210187 − 1.323810I
11.9837 + 11.2267I 0
u = −0.986081 − 0.788378I
a = 1.18896 − 1.12839I
b = −0.210187 + 1.323810I
11.9837 − 11.2267I 0
u = 0.129936 + 0.712668I
a = 0.383654 − 1.136340I
b = 0.355434 + 0.865456I
7.69892 + 3.04284I 6.83485 − 1.26769I
u = 0.129936 − 0.712668I
a = 0.383654 + 1.136340I
b = 0.355434 − 0.865456I
7.69892 − 3.04284I 6.83485 + 1.26769I
u = 1.186850 + 0.479807I
a = 0.484182 − 0.469768I
b = 0.778824 − 0.199434I
−1.52492 − 3.08303I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.186850 − 0.479807I
a = 0.484182 + 0.469768I
b = 0.778824 + 0.199434I
−1.52492 + 3.08303I 0
u = 1.034460 + 0.778303I
a = −0.40734 + 2.19878I
b = −1.35081 − 0.68759I
8.3818 − 18.1571I 0
u = 1.034460 − 0.778303I
a = −0.40734 − 2.19878I
b = −1.35081 + 0.68759I
8.3818 + 18.1571I 0
u = 1.057200 + 0.747188I
a = −0.82609 + 1.39760I
b = −0.869251 − 0.602242I
9.37216 − 6.14572I 0
u = 1.057200 − 0.747188I
a = −0.82609 − 1.39760I
b = −0.869251 + 0.602242I
9.37216 + 6.14572I 0
u = −1.000870 + 0.825577I
a = 0.649141 + 0.084858I
b = −0.699909 − 0.639831I
9.86775 + 1.31494I 0
u = −1.000870 − 0.825577I
a = 0.649141 − 0.084858I
b = −0.699909 + 0.639831I
9.86775 − 1.31494I 0
u = 1.048000 + 0.779868I
a = 0.36845 − 1.81774I
b = 1.201870 + 0.559633I
4.26339 − 12.11170I 0
u = 1.048000 − 0.779868I
a = 0.36845 + 1.81774I
b = 1.201870 − 0.559633I
4.26339 + 12.11170I 0
u = −0.301989 + 0.592941I
a = 0.659545 − 0.510641I
b = −1.038510 + 0.133211I
−1.93468 + 0.20175I −5.58615 − 0.28033I
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.301989 − 0.592941I
a = 0.659545 + 0.510641I
b = −1.038510 − 0.133211I
−1.93468 − 0.20175I −5.58615 + 0.28033I
u = −0.663608
a = 0.240549
b = −0.662103
−1.05112 −9.89690
u = −0.099805 + 0.595750I
a = −1.14595 + 0.84142I
b = 1.054840 − 0.512141I
0.13812 + 4.48157I 0.37821 − 5.73269I
u = −0.099805 − 0.595750I
a = −1.14595 − 0.84142I
b = 1.054840 + 0.512141I
0.13812 − 4.48157I 0.37821 + 5.73269I
u = 0.464539 + 0.371960I
a = −1.65664 − 0.13643I
b = 0.231873 − 0.001401I
1.401820 + 0.119914I 6.61037 + 0.17965I
u = 0.464539 − 0.371960I
a = −1.65664 + 0.13643I
b = 0.231873 + 0.001401I
1.401820 − 0.119914I 6.61037 − 0.17965I
u = 0.403195
a = −3.20121
b = 0.409741
1.30766 9.19350
12
II. I
u
2
= hb + 1, −u
3
− 8u
2
+ 10a − 2u + 14, u
4
− 2u
2
+ 2i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
3
+ u
a
8
=
−u
2
+ 1
u
2
a
1
=
u
3
u
3
− u
a
10
=
1
10
u
3
+
4
5
u
2
+
1
5
u −
7
5
−1
a
11
=
3
10
u
3
+
2
5
u
2
+
3
5
u −
1
5
2
5
u
3
+
1
5
u
2
−
1
5
u −
3
5
a
5
=
1
10
u
3
+
4
5
u
2
+
1
5
u −
2
5
−1
a
4
=
3
10
u
3
+
2
5
u
2
−
2
5
u −
1
5
−
3
5
u
3
+
1
5
u
2
+
4
5
u −
3
5
a
9
=
−1
0
a
12
=
u
u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 8
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
− 2u + 2)
2
c
2
, c
6
u
4
− 2u
2
+ 2
c
3
, c
5
(u + 1)
4
c
4
, c
10
5(5u
4
+ 8u
3
+ 8u
2
+ 4u + 1)
c
7
(u
2
+ 2u + 2)
2
c
8
, c
12
u
4
+ 2u
2
+ 2
c
9
, c
11
(u − 1)
4
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
2
+ 4)
2
c
2
, c
6
(y
2
− 2y + 2)
2
c
3
, c
5
, c
9
c
11
(y − 1)
4
c
4
, c
10
25(25y
4
+ 16y
3
+ 10y
2
+ 1)
c
8
, c
12
(y
2
+ 2y + 2)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.098680 + 0.455090I
a = −0.315904 + 1.046400I
b = −1.00000
−2.46740 − 3.66386I −4.00000 + 4.00000I
u = 1.098680 − 0.455090I
a = −0.315904 − 1.046400I
b = −1.00000
−2.46740 + 3.66386I −4.00000 − 4.00000I
u = −1.098680 + 0.455090I
a = −0.884096 − 0.553605I
b = −1.00000
−2.46740 + 3.66386I −4.00000 − 4.00000I
u = −1.098680 − 0.455090I
a = −0.884096 + 0.553605I
b = −1.00000
−2.46740 − 3.66386I −4.00000 + 4.00000I
16
III. I
u
3
=
h−6a
2
u
2
−u
2
a+· · ·+13a+12, a
3
+2a
2
u−2a
2
+au+u
2
+a−u−2, u
3
−u
2
+1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
8
=
−u
2
+ 1
u
2
a
1
=
u
2
− 1
−u
2
a
10
=
a
0.315789a
2
u
2
+ 0.0526316au
2
+ ··· − 0.684211a − 0.631579
a
11
=
0.210526a
2
u
2
+ 0.368421au
2
+ ··· + 0.210526a − 0.421053
0.578947a
2
u
2
− 0.736842au
2
+ ··· − 0.421053a − 1.15789
a
5
=
−0.789474a
2
u
2
+ 0.368421au
2
+ ··· + 0.210526a + 1.57895
0.473684a
2
u
2
− 0.421053au
2
+ ··· − 0.526316a − 0.947368
a
4
=
−0.315789a
2
u
2
+ 0.947368au
2
+ ··· − 0.315789a + 0.631579
0.315789a
2
u
2
− 0.947368au
2
+ ··· − 0.684211a − 0.631579
a
9
=
u
2
− 1
−u
2
a
12
=
u
2
− 1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u − 6
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
3
+ u
2
+ 2u + 1)
3
c
2
, c
6
(u
3
− u
2
+ 1)
3
c
3
, c
4
, c
5
c
9
, c
11
u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
+ u
3
− u
2
− 2u − 1
c
8
, c
12
u
9
c
10
u
9
− 6u
8
+ 15u
7
− 17u
6
+ 3u
5
+ 12u
4
− 9u
3
− u
2
+ 2u − 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
3
+ 3y
2
+ 2y − 1)
3
c
2
, c
6
(y
3
− y
2
+ 2y − 1)
3
c
3
, c
4
, c
5
c
9
, c
11
y
9
− 6y
8
+ 15y
7
− 17y
6
+ 3y
5
+ 12y
4
− 9y
3
− y
2
+ 2y − 1
c
8
, c
12
y
9
c
10
y
9
− 6y
8
+ 27y
7
− 73y
6
+ 139y
5
− 184y
4
+ 83y
3
− 13y
2
+ 2y − 1
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.739154 − 0.526270I
b = −1.180080 + 0.437737I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = 0.877439 + 0.744862I
a = −0.523982 + 1.249570I
b = −0.073457 − 0.802780I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = 0.877439 + 0.744862I
a = 0.02995 − 2.21303I
b = 1.253530 + 0.365043I
3.02413 − 2.82812I −2.49024 + 2.97945I
u = 0.877439 − 0.744862I
a = 0.739154 + 0.526270I
b = −1.180080 − 0.437737I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = 0.877439 − 0.744862I
a = −0.523982 − 1.249570I
b = −0.073457 + 0.802780I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = 0.877439 − 0.744862I
a = 0.02995 + 2.21303I
b = 1.253530 − 0.365043I
3.02413 + 2.82812I −2.49024 − 2.97945I
u = −0.754878
a = 0.007426 + 0.439504I
b = −0.606217 − 0.320153I
−1.11345 −9.01950
u = −0.754878
a = 0.007426 − 0.439504I
b = −0.606217 + 0.320153I
−1.11345 −9.01950
u = −0.754878
a = 3.49490
b = 1.21243
−1.11345 −9.01950
20
IV.
I
u
4
= h−b
2
u
2
a + b
2
au + b
3
+ u
2
b + u
2
a − bu − au − u
2
− b + u − 1, u
3
− u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
−u
−u
2
+ u + 1
a
8
=
−u
2
+ 1
u
2
a
1
=
u
2
− 1
−u
2
a
10
=
a
b
a
11
=
−u
2
b − bu − au
−u
2
a + bu + au + 2b + a
a
5
=
−ba + 1
−b
2
a
4
=
b
2
u
2
+ b
2
u + bau − u
u
2
ba − b
2
u − bau − 2b
2
− ba − u
2
+ u + 1
a
9
=
−b
2
a + b + a
−b
2
u
2
a + b
2
au + u
2
b + u
2
a − bu − au − u
2
+ u − 1
a
12
=
−b
2
a + u
2
+ b + a − 1
−b
2
u
2
a + b
2
au + u
2
b + u
2
a − bu − au − 2u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
(iv) u-Polynomials at the component : It cannot be defined for a positive
dimension component.
(v) Riley Polynomials at the component : It cannot be defined for a positive
dimension component.
21
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0.531480 3.50976 + 2.97945I
22
V. I
v
1
= ha, b
3
− b + 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
1
0
a
7
=
1
0
a
3
=
1
0
a
8
=
1
0
a
1
=
1
0
a
10
=
0
b
a
11
=
b
b
a
5
=
1
−b
2
a
4
=
−b
2
+ 1
−b
2
a
9
=
b
1
a
12
=
b + 1
1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
23
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
u
3
c
3
, c
5
, c
9
c
10
, c
11
u
3
− u + 1
c
4
u
3
+ 2u
2
+ u + 1
c
8
, c
12
(u − 1)
3
24
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
y
3
c
3
, c
5
, c
9
c
10
, c
11
y
3
− 2y
2
+ y − 1
c
4
y
3
− 2y
2
− 3y − 1
c
8
, c
12
(y − 1)
3
25
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 0.662359 + 0.562280I
1.64493 6.00000
v = 1.00000
a = 0
b = 0.662359 − 0.562280I
1.64493 6.00000
v = 1.00000
a = 0
b = −1.32472
1.64493 6.00000
26
VI. I
v
2
= ha, b − 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
1
0
a
7
=
1
0
a
3
=
1
0
a
8
=
1
0
a
1
=
1
0
a
10
=
0
1
a
11
=
1
1
a
5
=
1
−1
a
4
=
0
−1
a
9
=
1
0
a
12
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
27
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
8
, c
12
u
c
3
, c
4
, c
5
u − 1
c
9
, c
10
, c
11
u + 1
28
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
8
, c
12
y
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y − 1
29
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
2
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
30
VII. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
4
(u
2
− 2u + 2)
2
(u
3
+ u
2
+ 2u + 1)
3
· (u
77
+ 24u
76
+ ··· + 23104u + 2116)
c
2
, c
6
u
4
(u
3
− u
2
+ 1)
3
(u
4
− 2u
2
+ 2)(u
77
+ 4u
76
+ ··· − 32u + 46)
c
3
27(u − 1)(u + 1)
4
(u
3
− u + 1)
· (u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
+ u
3
− u
2
− 2u − 1)
· (27u
77
− 54u
76
+ ··· − 329u − 49)
c
4
320(u − 1)(u
3
+ 2u
2
+ u + 1)(5u
4
+ 8u
3
+ 8u
2
+ 4u + 1)
· (u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
+ u
3
− u
2
− 2u − 1)
· (64u
77
− 64u
76
+ ··· − 7193097u − 7328259)
c
5
27(u − 1)(u + 1)
4
(u
3
− u + 1)
· (u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
+ u
3
− u
2
− 2u − 1)
· (27u
77
+ 54u
76
+ ··· + 259u − 49)
c
7
u
4
(u
2
+ 2u + 2)
2
(u
3
+ u
2
+ 2u + 1)
3
· (u
77
+ 24u
76
+ ··· + 23104u + 2116)
c
8
, c
12
u
10
(u − 1)
3
(u
4
+ 2u
2
+ 2)(u
77
+ 8u
76
+ ··· + 288144u + 24982)
c
9
27(u − 1)
4
(u + 1)(u
3
− u + 1)
· (u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
+ u
3
− u
2
− 2u − 1)
· (27u
77
+ 54u
76
+ ··· + 259u − 49)
c
10
320(u + 1)(u
3
− u + 1)(5u
4
+ 8u
3
+ 8u
2
+ 4u + 1)
· (u
9
− 6u
8
+ 15u
7
− 17u
6
+ 3u
5
+ 12u
4
− 9u
3
− u
2
+ 2u − 1)
· (64u
77
+ 192u
76
+ ··· + 25776279u − 40736169)
c
11
27(u − 1)
4
(u + 1)(u
3
− u + 1)
· (u
9
− 3u
7
− u
6
+ 3u
5
+ 2u
4
+ u
3
− u
2
− 2u − 1)
· (27u
77
− 54u
76
+ ··· − 329u − 49)
31
VIII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
4
(y
2
+ 4)
2
(y
3
+ 3y
2
+ 2y − 1)
3
· (y
77
+ 60y
76
+ ··· + 169876672y − 4477456)
c
2
, c
6
y
4
(y
2
− 2y + 2)
2
(y
3
− y
2
+ 2y − 1)
3
· (y
77
− 24y
76
+ ··· + 23104y − 2116)
c
3
, c
11
729(y − 1)
5
(y
3
− 2y
2
+ y − 1)
· (y
9
− 6y
8
+ 15y
7
− 17y
6
+ 3y
5
+ 12y
4
− 9y
3
− y
2
+ 2y − 1)
· (729y
77
− 49572y
76
+ ··· + 20923y − 2401)
c
4
102400(y − 1)(y
3
− 2y
2
− 3y − 1)(25y
4
+ 16y
3
+ 10y
2
+ 1)
· (y
9
− 6y
8
+ 15y
7
− 17y
6
+ 3y
5
+ 12y
4
− 9y
3
− y
2
+ 2y − 1)
· (4096y
77
+ 192512y
76
+ ··· − 759954294636435y − 53703379971081)
c
5
, c
9
729(y − 1)
5
(y
3
− 2y
2
+ y − 1)
· (y
9
− 6y
8
+ 15y
7
− 17y
6
+ 3y
5
+ 12y
4
− 9y
3
− y
2
+ 2y − 1)
· (729y
77
− 26244y
76
+ ··· + 150283y − 2401)
c
8
, c
12
y
10
(y − 1)
3
(y
2
+ 2y + 2)
2
· (y
77
− 44y
76
+ ··· + 27288625256y − 624100324)
c
10
102400(y − 1)(y
3
− 2y
2
+ y − 1)(25y
4
+ 16y
3
+ 10y
2
+ 1)
· (y
9
− 6y
8
+ 27y
7
− 73y
6
+ 139y
5
− 184y
4
+ 83y
3
− 13y
2
+ 2y − 1)
· (4096y
77
− 2.09 × 10
5
y
76
+ ··· + 3.67 × 10
16
y − 1.66 × 10
15
)
32
