12a
1288
(K12a
1288
)
A knot diagram
1
Linearized knot diagam
6 7 11 9 10 2 12 1 5 3 4 8
Solving Sequence
1,6
2 7
3,10
11 5 9 4 8 12
c
1
c
6
c
2
c
10
c
5
c
9
c
4
c
8
c
12
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
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).
1
I
u
1
= hu
11
+ u
10
− 6u
9
− 5u
8
+ 11u
7
+ 5u
6
− 7u
5
+ 5u
4
+ 3u
3
− 4u
2
+ 4b − 2u,
− 3u
11
− 8u
10
+ 9u
9
+ 33u
8
− 4u
7
− 30u
6
+ 20u
5
− 42u
3
+ u
2
+ 4a + 10u − 2,
u
12
+ 3u
11
− 3u
10
− 14u
9
+ u
8
+ 18u
7
− 8u
6
− 6u
5
+ 24u
4
− 3u
3
− 13u
2
+ 6u − 2i
I
u
2
= h1999u
15
+ 3106u
14
+ ··· + 2878b − 22827, −1627u
15
− 1680u
14
+ ··· + 15829a + 36947,
u
16
+ 3u
15
+ ··· − 2u − 11i
I
u
3
= hu
7
a − u
7
− 4u
5
a − u
4
a + 4u
5
+ 4u
3
a − u
4
+ u
2
a − 4u
3
+ 2au + u
2
+ 2b + a + 1,
− u
7
a + u
6
a + 2u
7
+ 3u
5
a − 3u
6
− 3u
4
a − 7u
5
− 2u
3
a + 10u
4
+ 2u
2
a + 7u
3
+ a
2
− au − 7u
2
+ a − 5,
u
8
− u
7
− 4u
6
+ 3u
5
+ 5u
4
− u
3
− u
2
− 3u − 1i
I
u
4
= h−u
11
a − u
11
+ ··· + a + 1,
u
11
− 5u
9
+ u
7
a − 2u
8
+ 9u
7
− 2u
5
a + 8u
6
− u
4
a − 4u
5
− 10u
4
+ u
2
a − 6u
3
+ a
2
+ 2au + 2u
2
+ a + 6u + 2,
u
12
− u
11
− 4u
10
+ 2u
9
+ 7u
8
+ u
7
− 5u
6
− 5u
5
− u
4
+ 3u
3
+ 2u
2
+ 1i
I
u
5
= h2b − a − 1, a
2
− 3, u − 1i
I
u
6
= h2b − u + 1, 3a − u, u
2
− 3i
I
u
7
= hb + 1, a, u − 1i
I
u
8
= h4b
2
− 4b + 5, 2ba − 2b − 3a + 4u + 7, 2bu + 2b − 2a + u + 3, a
2
− 2a + 1, au + a − u − 1, u
2
+ 2u + 1i
I
u
9
= ha − 1, u + 1i
I
v
1
= ha, b − 1, v + 1i
* 9 irreducible components of dim
C
= 0, with total 78 representations.
* 1 irreducible components of dim
C
= 1
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
2
I. I
u
1
=
hu
11
+u
10
+· · ·+4b −2u, −3u
11
−8u
10
+· · ·+4a −2, u
12
+3u
11
+· · ·+6u −2i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
−u
−u
3
+ u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
3
4
u
11
+ 2u
10
+ ··· −
5
2
u +
1
2
−
1
4
u
11
−
1
4
u
10
+ ··· + u
2
+
1
2
u
a
11
=
1
4
u
11
+ u
10
+ ··· −
7
2
u +
1
2
−
1
4
u
10
−
3
4
u
9
+ ··· +
3
2
u −
1
2
a
5
=
−
1
4
u
11
− u
10
+ ··· +
3
2
u −
1
2
−
1
4
u
11
−
1
4
u
10
+ ··· + u
2
+
1
2
u
a
9
=
−
1
2
u
10
−
3
2
u
9
+ ··· −
15
2
u
2
− 1
u
a
4
=
−
1
4
u
11
+
7
4
u
9
+ ··· −
3
2
u +
3
2
−u
11
−
7
4
u
10
+ ··· −
5
2
u +
1
2
a
8
=
−
1
2
u
10
−
3
2
u
9
+ ··· + u − 1
u
a
12
=
1
2
u
11
+
3
2
u
10
+ ··· + u + 1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
1
2
u
11
+ u
10
−
3
2
u
9
−
1
2
u
8
+ 3u
7
− 13u
6
− 8u
5
+ 22u
4
− 11u
3
−
19
2
u
2
+ 32u − 7
3
(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
12
− 3u
11
+ ··· − 6u − 2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
12
+ 3u
11
+ ··· + 6u − 2
4
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y
12
− 15y
11
+ ··· + 16y + 4
5
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.369891 + 0.895594I
a = −0.40167 + 1.49298I
b = −0.02653 − 1.85453I
11.33630 − 5.25212I 7.30257 + 4.65184I
u = 0.369891 − 0.895594I
a = −0.40167 − 1.49298I
b = −0.02653 + 1.85453I
11.33630 + 5.25212I 7.30257 − 4.65184I
u = 1.22438
a = −1.51331
b = −1.43561
8.64658 −5.83920
u = 0.740748
a = 2.25319
b = 0.277533
10.7798 10.0010
u = −1.48568 + 0.19251I
a = −0.168040 − 0.624592I
b = −0.695418 + 0.595170I
−11.33630 + 5.25212I −7.30257 − 4.65184I
u = −1.48568 − 0.19251I
a = −0.168040 + 0.624592I
b = −0.695418 − 0.595170I
−11.33630 − 5.25212I −7.30257 + 4.65184I
u = −1.46094 + 0.44342I
a = 0.831296 + 0.555829I
b = 1.16548 − 2.08317I
15.2352I 0. − 7.62682I
u = −1.46094 − 0.44342I
a = 0.831296 − 0.555829I
b = 1.16548 + 2.08317I
− 15.2352I 0. + 7.62682I
u = 0.186071 + 0.332496I
a = −0.523030 − 0.852314I
b = 0.072588 + 0.300683I
− 0.761015I 0. + 9.12858I
u = 0.186071 − 0.332496I
a = −0.523030 + 0.852314I
b = 0.072588 − 0.300683I
0.761015I 0. − 9.12858I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.66904
a = 0.443816
b = 0.958809
−10.7798 −10.0010
u = −1.85286
a = −0.660804
b = −0.832979
−8.64658 5.83920
7
II. I
u
2
= h1999u
15
+ 3106u
14
+ · · · + 2878b − 22827, −1627u
15
− 1680u
14
+
· · · + 15829a + 36947, u
16
+ 3u
15
+ · · · − 2u − 11i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
−u
−u
3
+ u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
0.102786u
15
+ 0.106134u
14
+ ··· − 0.0633647u − 2.33413
−0.694580u
15
− 1.07922u
14
+ ··· − 5.24913u + 7.93155
a
11
=
0.0812433u
15
+ 0.190220u
14
+ ··· + 0.503696u + 0.181502
−1.01355u
15
− 1.80195u
14
+ ··· − 3.62717u + 5.92113
a
5
=
0.0504138u
15
− 0.0794744u
14
+ ··· + 1.94864u + 1.81092
0.312370u
15
+ 0.280751u
14
+ ··· + 1.27762u − 4.97672
a
9
=
0.503254u
15
+ 0.504896u
14
+ ··· + 3.85571u − 3.97524
−0.594163u
15
− 0.777623u
14
+ ··· − 6.03753u + 4.15705
a
4
=
0.434645u
15
+ 0.605534u
14
+ ··· + 2.21884u − 1.80814
0.0309243u
15
− 0.298124u
14
+ ··· + 0.944058u − 3.69180
a
8
=
−0.0909091u
15
− 0.272727u
14
+ ··· − 2.18182u + 0.181818
−0.594163u
15
− 0.777623u
14
+ ··· − 6.03753u + 4.15705
a
12
=
0.377914u
15
+ 0.539579u
14
+ ··· + 1.33679u − 5.79335
1.00486u
15
+ 1.85198u
14
+ ··· + 2.96873u − 7.53579
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
3610
1439
u
15
−
5684
1439
u
14
+ ··· −
43398
1439
u +
24298
1439
8
(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
16
− 3u
15
+ ··· + 2u − 11
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(u
8
− u
7
− 4u
6
+ 3u
5
+ 5u
4
− u
3
− u
2
− 3u − 1)
2
9
(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
16
− 13y
15
+ ··· − 532y + 121
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(y
8
− 9y
7
+ 32y
6
− 53y
5
+ 31y
4
+ 15y
3
− 15y
2
− 7y + 1)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.694226 + 0.667719I
a = 1.12920 − 1.08608I
b = 0.276681 + 1.311520I
10.1546 6.33746 + 0.I
u = 0.694226 − 0.667719I
a = 1.12920 + 1.08608I
b = 0.276681 − 1.311520I
10.1546 6.33746 + 0.I
u = 0.262333 + 1.058630I
a = 0.019128 − 1.343770I
b = 0.19190 + 2.13545I
5.44991 − 9.88301I 3.28252 + 6.06963I
u = 0.262333 − 1.058630I
a = 0.019128 + 1.343770I
b = 0.19190 − 2.13545I
5.44991 + 9.88301I 3.28252 − 6.06963I
u = 1.15427
a = −0.296909
b = −0.236501
−2.57083 2.16010
u = 0.524313 + 0.657146I
a = 0.513557 + 0.640043I
b = −0.598451 − 0.154997I
−4.77492 − 2.26376I −6.05872 + 4.53378I
u = 0.524313 − 0.657146I
a = 0.513557 − 0.640043I
b = −0.598451 + 0.154997I
−4.77492 + 2.26376I −6.05872 − 4.53378I
u = −1.345930 + 0.090134I
a = 0.078599 + 0.505339I
b = 0.329421 − 1.036490I
−4.77492 + 2.26376I −6.05872 − 4.53378I
u = −1.345930 − 0.090134I
a = 0.078599 − 0.505339I
b = 0.329421 + 1.036490I
−4.77492 − 2.26376I −6.05872 + 4.53378I
u = 1.125920 + 0.800303I
a = −0.889080 + 0.447838I
b = −0.77247 − 1.44681I
2.93531 + 3.55755I 2.52739 − 2.62489I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.125920 − 0.800303I
a = −0.889080 − 0.447838I
b = −0.77247 + 1.44681I
2.93531 − 3.55755I 2.52739 + 2.62489I
u = −0.604309
a = 0.567118
b = 1.07322
−2.57083 2.16010
u = −1.47759 + 0.37462I
a = −0.854231 − 0.441448I
b = −0.72115 + 1.92612I
5.44991 + 9.88301I 3.28252 − 6.06963I
u = −1.47759 − 0.37462I
a = −0.854231 + 0.441448I
b = −0.72115 − 1.92612I
5.44991 − 9.88301I 3.28252 + 6.06963I
u = −1.55826 + 0.27885I
a = 0.822270 + 0.280177I
b = 0.375716 − 1.326680I
2.93531 + 3.55755I 2.52739 − 2.62489I
u = −1.55826 − 0.27885I
a = 0.822270 − 0.280177I
b = 0.375716 + 1.326680I
2.93531 − 3.55755I 2.52739 + 2.62489I
12
III. I
u
3
= hu
7
a − u
7
+ · · · + a + 1, −u
7
a + 2u
7
+ · · · + a − 5, u
8
− u
7
− 4u
6
+
3u
5
+ 5u
4
− u
3
− u
2
− 3u − 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
−u
−u
3
+ u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
a
−
1
2
u
7
a +
1
2
u
7
+ ··· −
1
2
a −
1
2
a
11
=
u
7
− 4u
5
− u
3
a + 4u
3
+ au + a + u
−
1
2
u
7
a +
1
2
u
7
+ ··· −
1
2
a −
1
2
a
5
=
−u
6
a − u
7
+ ··· − a + 2
−
1
2
u
7
a +
1
2
u
7
+ ··· −
1
2
a −
1
2
a
9
=
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ 1
u
a
4
=
−u
7
a − u
7
+ ··· − a + 2
−
1
2
u
7
a +
1
2
u
7
+ ··· −
1
2
a −
1
2
a
8
=
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u + 1
u
a
12
=
−u
7
+ u
6
+ 3u
5
− 2u
4
− 2u
3
− u
2
− u + 1
−u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
6
− 2u
5
+ 10u
4
+ 8u
3
− 12u
2
− 10u − 4
13
(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
8
+ u
7
− 4u
6
− 3u
5
+ 5u
4
+ u
3
− u
2
+ 3u − 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
16
+ 3u
15
+ ··· − 2u − 11
14
(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
8
− 9y
7
+ 32y
6
− 53y
5
+ 31y
4
+ 15y
3
− 15y
2
− 7y + 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y
16
− 13y
15
+ ··· − 532y + 121
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.151337 + 0.673064I
a = 0.762637 − 0.950471I
b = 0.076801 + 0.408443I
4.77492 + 2.26376I 6.05872 − 4.53378I
u = −0.151337 + 0.673064I
a = 0.30052 + 1.93213I
b = −0.06507 − 1.92874I
4.77492 + 2.26376I 6.05872 − 4.53378I
u = −0.151337 − 0.673064I
a = 0.762637 + 0.950471I
b = 0.076801 − 0.408443I
4.77492 − 2.26376I 6.05872 + 4.53378I
u = −0.151337 − 0.673064I
a = 0.30052 − 1.93213I
b = −0.06507 + 1.92874I
4.77492 − 2.26376I 6.05872 + 4.53378I
u = −1.359440 + 0.207304I
a = −0.897134 + 0.451895I
b = −0.592239 − 0.125436I
−2.93531 + 3.55755I −2.52739 − 2.62489I
u = −1.359440 + 0.207304I
a = 1.089640 + 0.371279I
b = 1.80045 − 1.27064I
−2.93531 + 3.55755I −2.52739 − 2.62489I
u = −1.359440 − 0.207304I
a = −0.897134 − 0.451895I
b = −0.592239 + 0.125436I
−2.93531 − 3.55755I −2.52739 + 2.62489I
u = −1.359440 − 0.207304I
a = 1.089640 − 0.371279I
b = 1.80045 + 1.27064I
−2.93531 − 3.55755I −2.52739 + 2.62489I
u = 1.42757 + 0.33227I
a = −0.923905 + 0.477454I
b = −1.51269 − 1.88228I
−5.44991 − 9.88301I −3.28252 + 6.06963I
u = 1.42757 + 0.33227I
a = 0.010591 − 0.744025I
b = 0.534351 + 0.711339I
−5.44991 − 9.88301I −3.28252 + 6.06963I
16
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.42757 − 0.33227I
a = −0.923905 − 0.477454I
b = −1.51269 + 1.88228I
−5.44991 + 9.88301I −3.28252 − 6.06963I
u = 1.42757 − 0.33227I
a = 0.010591 + 0.744025I
b = 0.534351 − 0.711339I
−5.44991 + 9.88301I −3.28252 − 6.06963I
u = 1.50912
a = 0.460021 + 0.442457I
b = 0.770987 − 0.303857I
−10.1546 −6.33750
u = 1.50912
a = 0.460021 − 0.442457I
b = 0.770987 + 0.303857I
−10.1546 −6.33750
u = −0.342714
a = 1.76330
b = −0.866117
2.57083 −2.16010
u = −0.342714
a = −3.36804
b = −0.159086
2.57083 −2.16010
17
IV.
I
u
4
= h−u
11
a−u
11
+· · ·+a+1, u
11
−5u
9
+· · ·+a+2, u
12
−u
11
+· · ·+2u
2
+1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
u
2
a
7
=
−u
−u
3
+ u
a
3
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
a
u
11
a + u
11
+ ··· − a − 1
a
11
=
−u
10
+ 3u
8
+ u
7
− 2u
6
− 2u
5
− u
3
a − 2u
4
+ au + 2u
2
+ a + u
u
11
a − 4u
9
a + ··· + u
2
− a
a
5
=
u
11
− u
10
+ ··· + 2u − 1
u
11
a − u
11
+ ··· − a − u
a
9
=
−u
10
+ 3u
8
+ 2u
7
− 2u
6
− 4u
5
− 3u
4
+ 3u
2
+ 3u + 1
u
11
− 4u
9
− u
8
+ 5u
7
+ 3u
6
− u
5
− 2u
4
− u
3
− u − 1
a
4
=
u
11
a + u
11
+ ··· + 2u − 1
u
10
a − u
11
+ ··· + 5u
3
− u
a
8
=
u
11
− u
10
− 4u
9
+ 2u
8
+ 7u
7
+ u
6
− 5u
5
− 5u
4
− u
3
+ 3u
2
+ 2u
u
11
− 4u
9
− u
8
+ 5u
7
+ 3u
6
− u
5
− 2u
4
− u
3
− u − 1
a
12
=
u
11
− 4u
9
− 2u
8
+ 6u
7
+ 6u
6
− 2u
5
− 6u
4
− 3u
3
+ 2u
2
+ 2u
u
11
− 3u
9
− 2u
8
+ 2u
7
+ 4u
6
+ 3u
5
− 3u
3
− 2u
2
− u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
8
+ 12u
6
+ 4u
5
− 8u
4
− 8u
3
− 4u
2
+ 2
18
(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
12
+ u
11
− 4u
10
− 2u
9
+ 7u
8
− u
7
− 5u
6
+ 5u
5
− u
4
− 3u
3
+ 2u
2
+ 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(u
12
− u
11
− 4u
10
+ 2u
9
+ 7u
8
+ u
7
− 5u
6
− 5u
5
− u
4
+ 3u
3
+ 2u
2
+ 1)
2
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
(y
12
− 9y
11
+ ··· + 4y + 1)
2
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.895235 + 0.524661I
a = 1.053870 + 0.403232I
b = 1.02100 − 1.56444I
−1.89061 − 0.92430I −3.71672 + 0.79423I
u = −0.895235 + 0.524661I
a = −0.364606 + 0.330843I
b = 0.939757 − 0.425557I
−1.89061 − 0.92430I −3.71672 + 0.79423I
u = −0.895235 − 0.524661I
a = 1.053870 − 0.403232I
b = 1.02100 + 1.56444I
−1.89061 + 0.92430I −3.71672 − 0.79423I
u = −0.895235 − 0.524661I
a = −0.364606 − 0.330843I
b = 0.939757 + 0.425557I
−1.89061 + 0.92430I −3.71672 − 0.79423I
u = −0.282166 + 0.828798I
a = −0.792263 + 0.610180I
b = 0.383261 − 0.056485I
5.69302I 0. − 5.51057I
u = −0.282166 + 0.828798I
a = −0.20722 − 1.56570I
b = −0.47925 + 2.17825I
5.69302I 0. − 5.51057I
u = −0.282166 − 0.828798I
a = −0.792263 − 0.610180I
b = 0.383261 + 0.056485I
− 5.69302I 0. + 5.51057I
u = −0.282166 − 0.828798I
a = −0.20722 + 1.56570I
b = −0.47925 − 2.17825I
− 5.69302I 0. + 5.51057I
u = −1.155020 + 0.191936I
a = 0.827710 − 0.316699I
b = −0.087686 + 0.786615I
1.89061 + 0.92430I 3.71672 − 0.79423I
u = −1.155020 + 0.191936I
a = −1.095480 − 0.303138I
b = −1.53926 + 1.57073I
1.89061 + 0.92430I 3.71672 − 0.79423I
21
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.155020 − 0.191936I
a = 0.827710 + 0.316699I
b = −0.087686 − 0.786615I
1.89061 − 0.92430I 3.71672 + 0.79423I
u = −1.155020 − 0.191936I
a = −1.095480 + 0.303138I
b = −1.53926 − 1.57073I
1.89061 − 0.92430I 3.71672 + 0.79423I
u = 1.323480 + 0.139870I
a = −0.847918 + 0.234635I
b = −0.01623 − 1.47826I
−1.89061 − 0.92430I −3.71672 + 0.79423I
u = 1.323480 + 0.139870I
a = 0.075702 − 0.376331I
b = −0.831407 + 1.047810I
−1.89061 − 0.92430I −3.71672 + 0.79423I
u = 1.323480 − 0.139870I
a = −0.847918 − 0.234635I
b = −0.01623 + 1.47826I
−1.89061 + 0.92430I −3.71672 − 0.79423I
u = 1.323480 − 0.139870I
a = 0.075702 + 0.376331I
b = −0.831407 − 1.047810I
−1.89061 + 0.92430I −3.71672 − 0.79423I
u = 1.356120 + 0.270046I
a = 0.923718 − 0.383073I
b = 0.95599 + 1.99574I
− 5.69302I 0. + 5.51057I
u = 1.356120 + 0.270046I
a = −0.083074 + 0.627698I
b = −0.127208 − 1.130510I
− 5.69302I 0. + 5.51057I
u = 1.356120 − 0.270046I
a = 0.923718 + 0.383073I
b = 0.95599 − 1.99574I
5.69302I 0. − 5.51057I
u = 1.356120 − 0.270046I
a = −0.083074 − 0.627698I
b = −0.127208 + 1.130510I
5.69302I 0. − 5.51057I
22
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.152828 + 0.487477I
a = −1.50418 + 1.36489I
b = −0.820813 − 0.942146I
1.89061 − 0.92430I 3.71672 + 0.79423I
u = 0.152828 + 0.487477I
a = 0.51374 − 2.55389I
b = 1.10185 + 1.67160I
1.89061 − 0.92430I 3.71672 + 0.79423I
u = 0.152828 − 0.487477I
a = −1.50418 − 1.36489I
b = −0.820813 + 0.942146I
1.89061 + 0.92430I 3.71672 − 0.79423I
u = 0.152828 − 0.487477I
a = 0.51374 + 2.55389I
b = 1.10185 − 1.67160I
1.89061 + 0.92430I 3.71672 − 0.79423I
23
V. I
u
5
= h2b − a − 1, a
2
− 3, u − 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
1
a
2
=
1
1
a
7
=
−1
0
a
3
=
0
1
a
10
=
a
1
2
a +
1
2
a
11
=
a
−
1
2
a +
1
2
a
5
=
−3
−
1
2
a −
1
2
a
9
=
−2a
−1
a
4
=
3
1
2
a −
1
2
a
8
=
−2a − 1
−1
a
12
=
−2a
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
24
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
8
(u − 1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
2
− 3
c
6
, c
12
(u + 1)
2
25
(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 −1)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(y −3)
2
26
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.73205
b = 1.36603
9.86960 0
u = 1.00000
a = −1.73205
b = −0.366025
9.86960 0
27
VI. I
u
6
= h2b − u + 1, 3a − u, u
2
− 3i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
3
a
7
=
−u
−2u
a
3
=
−2
−3
a
10
=
1
3
u
1
2
u −
1
2
a
11
=
1
3
u − 2
1
2
u −
7
2
a
5
=
−
1
3
u
1
2
u +
1
2
a
9
=
0
u
a
4
=
−
1
3
u
−
1
2
u +
1
2
a
8
=
u
u
a
12
=
−2
−3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
28
(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
2
− 3
c
3
, c
9
(u − 1)
2
c
4
, c
5
, c
10
c
11
(u + 1)
2
29
(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 −3)
2
c
3
, c
4
, c
5
c
9
, c
10
, c
11
(y −1)
2
30
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.73205
a = 0.577350
b = 0.366025
−9.86960 0
u = −1.73205
a = −0.577350
b = −1.36603
−9.86960 0
31
VII. I
u
7
= hb + 1, a, u − 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
1
a
2
=
1
1
a
7
=
−1
0
a
3
=
0
1
a
10
=
0
−1
a
11
=
0
−1
a
5
=
0
1
a
9
=
0
−1
a
4
=
0
1
a
8
=
−1
−1
a
12
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
32
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
8
u − 1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
u
c
6
, c
12
u + 1
33
(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 −1
c
3
, c
4
, c
5
c
9
, c
10
, c
11
y
34
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
35
VIII. I
u
8
= h4b
2
− 4b + 5, 2ba + 4u + · · · − 3a + 7, 2bu + u + · · · − 2a +
3, a
2
− 2a + 1, au + a − u − 1, u
2
+ 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
u
a
2
=
1
−2u − 1
a
7
=
−u
−2u − 2
a
3
=
2u + 2
1
a
10
=
a
b
a
11
=
a − 2u − 2
b + a − u − 3
a
5
=
2a − u − 2
b +
1
2
a −
1
2
u − 2
a
9
=
−2a + 2u + 4
u + 2
a
4
=
u + 2
b +
3
2
a −
1
2
u − 2
a
8
=
−2a + 3u + 6
u + 2
a
12
=
2a − 4u − 6
−2u − 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
36
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
7
, c
8
, c
9
(u + 1)
4
c
4
, c
5
, c
6
c
10
, c
11
, c
12
(u − 1)
4
37
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
(y −1)
4
38
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = 0.500000 + 1.000000I
0 0
u = −1.00000
a = 1.00000
b = 0.500000 + 1.000000I
0 0
u = −1.00000
a = 1.00000
b = 0.500000 − 1.000000I
0 0
u = −1.00000
a = 1.00000
b = 0.500000 − 1.000000I
0 0
39
IX. I
u
9
= ha − 1, u + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
0
−1
a
2
=
1
1
a
7
=
1
0
a
3
=
0
1
a
10
=
1
b
a
11
=
1
b − 1
a
5
=
1
b − 1
a
9
=
0
1
a
4
=
1
b
a
8
=
1
1
a
12
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
(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.
40
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
9
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
41
X. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
1
=
1
0
a
6
=
−1
0
a
2
=
1
0
a
7
=
−1
0
a
3
=
1
0
a
10
=
0
1
a
11
=
1
1
a
5
=
−1
−1
a
9
=
−1
0
a
4
=
0
−1
a
8
=
−1
0
a
12
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
42
(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
9
u − 1
c
4
, c
5
, c
10
c
11
u + 1
43
(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
44
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
3.28987 12.0000
45
XI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
c
8
u(u − 1)
3
(u + 1)
4
(u
2
− 3)
· (u
8
+ u
7
− 4u
6
− 3u
5
+ 5u
4
+ u
3
− u
2
+ 3u − 1)
2
· (u
12
− 3u
11
+ ··· − 6u − 2)
· (u
12
+ u
11
− 4u
10
− 2u
9
+ 7u
8
− u
7
− 5u
6
+ 5u
5
− u
4
− 3u
3
+ 2u
2
+ 1)
2
· (u
16
− 3u
15
+ ··· + 2u − 11)
c
3
, c
9
u(u − 1)
3
(u + 1)
4
(u
2
− 3)
· (u
8
− u
7
− 4u
6
+ 3u
5
+ 5u
4
− u
3
− u
2
− 3u − 1)
2
· (u
12
− u
11
− 4u
10
+ 2u
9
+ 7u
8
+ u
7
− 5u
6
− 5u
5
− u
4
+ 3u
3
+ 2u
2
+ 1)
2
· (u
12
+ 3u
11
+ ··· + 6u − 2)(u
16
+ 3u
15
+ ··· − 2u − 11)
c
4
, c
5
, c
10
c
11
u(u − 1)
4
(u + 1)
3
(u
2
− 3)
· (u
8
− u
7
− 4u
6
+ 3u
5
+ 5u
4
− u
3
− u
2
− 3u − 1)
2
· (u
12
− u
11
− 4u
10
+ 2u
9
+ 7u
8
+ u
7
− 5u
6
− 5u
5
− u
4
+ 3u
3
+ 2u
2
+ 1)
2
· (u
12
+ 3u
11
+ ··· + 6u − 2)(u
16
+ 3u
15
+ ··· − 2u − 11)
c
6
, c
12
u(u − 1)
4
(u + 1)
3
(u
2
− 3)
· (u
8
+ u
7
− 4u
6
− 3u
5
+ 5u
4
+ u
3
− u
2
+ 3u − 1)
2
· (u
12
− 3u
11
+ ··· − 6u − 2)
· (u
12
+ u
11
− 4u
10
− 2u
9
+ 7u
8
− u
7
− 5u
6
+ 5u
5
− u
4
− 3u
3
+ 2u
2
+ 1)
2
· (u
16
− 3u
15
+ ··· + 2u − 11)
46
XII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y(y − 3)
2
(y −1)
7
· (y
8
− 9y
7
+ 32y
6
− 53y
5
+ 31y
4
+ 15y
3
− 15y
2
− 7y + 1)
2
· (y
12
− 15y
11
+ ··· + 16y + 4)(y
12
− 9y
11
+ ··· + 4y + 1)
2
· (y
16
− 13y
15
+ ··· − 532y + 121)
47
