11n
72
(K11n
72
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 11 9 4 6 8 5 10
Solving Sequence
5,10
11 6
1,2 4,8
3 7 9
c
10
c
5
c
11
c
4
c
3
c
7
c
9
c
1
, c
2
, c
6
, c
8
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
2
+ d, −u
6
− u
5
+ u
3
− 2u
2
+ 2c − 2u − 1, −u
6
− u
5
+ u
3
− 2u
2
+ 2b − 2u + 1, a − 1,
u
8
+ u
7
− u
6
− 2u
5
+ 2u
4
+ 3u
3
+ u
2
− 2u + 1i
I
u
2
= hu
2
+ d, −u
10
− 2u
9
− u
8
+ 2u
7
+ u
6
− 2u
5
− 4u
4
+ u
2
+ c + u − 1, b − 1,
u
10
+ 2u
9
− u
8
− 5u
7
− u
6
+ 6u
5
+ 4u
4
− 4u
3
− 5u
2
+ a + u + 4,
u
11
+ 2u
10
− 4u
8
− 2u
7
+ 4u
6
+ 5u
5
− 2u
4
− 5u
3
− u
2
+ 3u + 1i
I
u
3
= h−u
10
− u
9
+ u
8
+ 2u
7
− u
6
− 2u
5
+ 2u
3
+ d − u + 1,
u
10
+ 2u
9
− u
8
− 5u
7
− u
6
+ 6u
5
+ 4u
4
− 4u
3
− 5u
2
+ c + u + 3,
− u
10
− 2u
9
− u
8
+ 2u
7
+ u
6
− 2u
5
− 4u
4
+ u
2
+ b + u, a − 1,
u
11
+ 2u
10
− 4u
8
− 2u
7
+ 4u
6
+ 5u
5
− 2u
4
− 5u
3
− u
2
+ 3u + 1i
I
u
4
= h−17u
10
+ 8u
9
+ 27u
8
− 4u
7
− 39u
6
+ 6u
5
+ 68u
4
+ 2u
3
− 75u
2
+ 86d − 41u + 80,
− 41u
10
− 6u
9
+ 55u
8
+ 132u
7
− 3u
6
− 198u
5
− 180u
4
+ 106u
3
+ 325u
2
+ 344c − 109u − 318, b − 1,
− 41u
10
− 6u
9
+ 55u
8
+ 132u
7
− 3u
6
− 198u
5
− 180u
4
+ 106u
3
+ 325u
2
+ 344a − 109u + 26,
u
11
− 3u
9
− 2u
8
+ 3u
7
+ 4u
6
− 2u
4
− u
3
+ 3u
2
− 4i
I
u
5
= hd, c − 1, b − 1, a + 1, u + 1i
I
u
6
= hd + 1, c, b − 1, a, u − 1i
I
u
7
= hu
2
a + d + 1, c + a, b − 1, a
2
+ u
2
+ a − u, u
3
− u − 1i
I
u
8
= hd + 1, cb − c − 1, a + 1, u + 1i
I
v
1
= ha, d + 1, c + a − 1, b − 1, v + 1i
* 8 irreducible components of dim
C
= 0, with total 50 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
2
+d, −u
6
−u
5
+· · ·+2c−1, −u
6
−u
5
+· · ·+2b+1, a−1, u
8
+u
7
+· · ·−2u+1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
u
2
a
2
=
1
1
2
u
6
+
1
2
u
5
+ ··· + u −
1
2
a
4
=
u
1
2
u
7
+
1
2
u
6
+ ··· + u
2
+
1
2
u
a
8
=
1
2
u
6
+
1
2
u
5
+ ··· + u +
1
2
−u
2
a
3
=
u
4
− u
2
+ 1
1
2
u
6
+
1
2
u
5
+ ··· + u −
1
2
a
7
=
1
2
u
7
+
1
2
u
6
−
1
2
u
4
+ u
2
+
3
2
u
−u
5
+ u
3
− u
a
9
=
1
2
u
6
+
1
2
u
5
−
1
2
u
3
+ u +
1
2
−u
4
a
9
=
1
2
u
6
+
1
2
u
5
−
1
2
u
3
+ u +
1
2
−u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
7
+ 6u
6
− u
5
− 7u
4
+ 3u
3
+ 16u
2
+ 7u − 9
3
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
8
, c
10
u
8
− u
7
− u
6
+ 2u
5
+ 2u
4
− 3u
3
+ u
2
+ 2u + 1
c
2
, c
9
, c
11
u
8
+ 3u
7
+ 9u
6
+ 12u
5
+ 20u
4
+ 15u
3
+ 17u
2
+ 2u + 1
c
3
, c
7
u
8
− u
7
− u
6
+ 5u
5
− 4u
4
+ 8u
2
− 4u + 4
4
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
8
, c
10
y
8
− 3y
7
+ 9y
6
− 12y
5
+ 20y
4
− 15y
3
+ 17y
2
− 2y + 1
c
2
, c
9
, c
11
y
8
+ 9y
7
+ 49y
6
+ 160y
5
+ 336y
4
+ 425y
3
+ 269y
2
+ 30y + 1
c
3
, c
7
y
8
− 3y
7
+ 3y
6
− y
5
− 32y
3
+ 32y
2
+ 48y + 16
5
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.725725 + 0.895340I
a = 1.00000
b = −1.23064 − 0.78420I
c = −0.230638 − 0.784197I
d = 0.274957 + 1.299540I
6.13361 + 3.53925I −3.48597 − 4.52491I
u = −0.725725 − 0.895340I
a = 1.00000
b = −1.23064 + 0.78420I
c = −0.230638 + 0.784197I
d = 0.274957 − 1.299540I
6.13361 − 3.53925I −3.48597 + 4.52491I
u = 1.052770 + 0.635427I
a = 1.00000
b = −1.68524 + 1.42536I
c = −0.68524 + 1.42536I
d = −0.70455 − 1.33791I
−1.61416 − 7.63502I −9.74769 + 6.83193I
u = 1.052770 − 0.635427I
a = 1.00000
b = −1.68524 − 1.42536I
c = −0.68524 − 1.42536I
d = −0.70455 + 1.33791I
−1.61416 + 7.63502I −9.74769 − 6.83193I
u = −1.213440 + 0.663590I
a = 1.00000
b = −2.02473 − 1.24139I
c = −1.02473 − 1.24139I
d = −1.03209 + 1.61046I
0.8567 + 14.6934I −9.31845 − 9.04054I
u = −1.213440 − 0.663590I
a = 1.00000
b = −2.02473 + 1.24139I
c = −1.02473 + 1.24139I
d = −1.03209 − 1.61046I
0.8567 − 14.6934I −9.31845 + 9.04054I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.386400 + 0.333144I
a = 1.00000
b = −0.059390 + 0.519525I
c = 0.940610 + 0.519525I
d = −0.038320 − 0.257454I
−0.441338 − 1.103720I −5.44788 + 6.54224I
u = 0.386400 − 0.333144I
a = 1.00000
b = −0.059390 − 0.519525I
c = 0.940610 − 0.519525I
d = −0.038320 + 0.257454I
−0.441338 + 1.103720I −5.44788 − 6.54224I
7
II. I
u
2
= hu
2
+ d, −u
10
− 2u
9
+ · · · + c − 1, b − 1, u
10
+ 2u
9
+ · · · + a +
4, u
11
+ 2u
10
+ · · · + 3u + 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
u
2
a
2
=
−u
10
− 2u
9
+ u
8
+ 5u
7
+ u
6
− 6u
5
− 4u
4
+ 4u
3
+ 5u
2
− u − 4
1
a
4
=
−u
10
− 3u
9
− u
8
+ 5u
7
+ 4u
6
− 5u
5
− 7u
4
+ 2u
3
+ 7u
2
+ u − 4
u
9
+ u
8
− u
7
− 2u
6
+ u
5
+ 2u
4
− 2u
2
+ 1
a
8
=
u
10
+ 2u
9
+ u
8
− 2u
7
− u
6
+ 2u
5
+ 4u
4
− u
2
− u + 1
−u
2
a
3
=
−u
10
− 3u
9
+ u
8
+ 7u
7
+ 3u
6
− 9u
5
− 7u
4
+ 5u
3
+ 10u
2
− 2u − 6
u
10
+ 2u
9
− u
8
− 4u
7
− u
6
+ 5u
5
+ 3u
4
− 3u
3
− 5u
2
+ 2u + 2
a
7
=
u
9
+ 2u
8
+ u
7
− 2u
6
− u
5
+ 2u
4
+ 3u
3
− u − 1
−u
5
+ u
3
− u
a
9
=
u
10
+ 2u
9
+ u
8
− 2u
7
− u
6
+ 2u
5
+ 4u
4
− 2u
2
− u + 1
−u
4
a
9
=
u
10
+ 2u
9
+ u
8
− 2u
7
− u
6
+ 2u
5
+ 4u
4
− 2u
2
− u + 1
−u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
10
+ 6u
9
− 10u
7
− 4u
6
+ 6u
5
+ 12u
4
− 4u
3
− 8u
2
− 6u − 4
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
11
− 3u
9
+ 2u
8
+ 3u
7
− 4u
6
+ 2u
4
− u
3
− 3u
2
+ 4
c
2
u
11
+ 6u
10
+ ··· + 24u + 16
c
3
, c
7
u
11
− 2u
10
− u
9
+ 3u
8
+ u
7
− 2u
6
+ 4u
5
− 11u
4
+ 9u
3
− u
2
− 2u + 2
c
5
, c
6
, c
8
c
10
u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1
c
9
, c
11
u
11
+ 4u
10
+ ··· + 11u + 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
11
− 6y
10
+ ··· + 24y −16
c
2
y
11
− 6y
10
+ ··· − 224y −256
c
3
, c
7
y
11
− 6y
10
+ ··· + 8y −4
c
5
, c
6
, c
8
c
10
y
11
− 4y
10
+ ··· + 11y −1
c
9
, c
11
y
11
+ 8y
10
+ ··· + 67y −1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.952018 + 0.226513I
a = −1.085970 + 0.401284I
b = 1.00000
c = 0.40050 + 4.16652I
d = −0.855030 − 0.431288I
−5.02081 − 0.74196I −15.5393 + 1.1191I
u = 0.952018 − 0.226513I
a = −1.085970 − 0.401284I
b = 1.00000
c = 0.40050 − 4.16652I
d = −0.855030 + 0.431288I
−5.02081 + 0.74196I −15.5393 − 1.1191I
u = 0.850023 + 0.614930I
a = 0.007368 − 0.850380I
b = 1.00000
c = −0.138893 + 1.373110I
d = −0.344399 − 1.045410I
−0.08426 − 2.41892I −7.07184 + 2.88947I
u = 0.850023 − 0.614930I
a = 0.007368 + 0.850380I
b = 1.00000
c = −0.138893 − 1.373110I
d = −0.344399 + 1.045410I
−0.08426 + 2.41892I −7.07184 − 2.88947I
u = −0.523691 + 0.948055I
a = −0.184008 + 1.141810I
b = 1.00000
c = −0.103739 − 0.547821I
d = 0.624556 + 0.992977I
5.32590 − 2.58451I −3.80806 + 1.01660I
u = −0.523691 − 0.948055I
a = −0.184008 − 1.141810I
b = 1.00000
c = −0.103739 + 0.547821I
d = 0.624556 − 0.992977I
5.32590 + 2.58451I −3.80806 − 1.01660I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.978643 + 0.595733I
a = −0.939343 − 0.770160I
b = 1.00000
c = −0.47651 − 1.53693I
d = −0.602844 + 1.166020I
−2.61864 + 4.69742I −9.08124 − 5.88322I
u = −0.978643 − 0.595733I
a = −0.939343 + 0.770160I
b = 1.00000
c = −0.47651 + 1.53693I
d = −0.602844 − 1.166020I
−2.61864 − 4.69742I −9.08124 + 5.88322I
u = −1.126060 + 0.711355I
a = −0.175044 + 0.783251I
b = 1.00000
c = −0.81852 − 1.22144I
d = −0.76197 + 1.60205I
3.47965 + 8.65115I −6.21430 − 5.57892I
u = −1.126060 − 0.711355I
a = −0.175044 − 0.783251I
b = 1.00000
c = −0.81852 + 1.22144I
d = −0.76197 − 1.60205I
3.47965 − 8.65115I −6.21430 + 5.57892I
u = −0.347303
a = −3.24600
b = 1.00000
c = 1.27433
d = −0.120619
−2.16369 −2.57060
12
III. I
u
3
= h−u
10
− u
9
+ · · · + d + 1, u
10
+ 2u
9
+ · · · + c + 3, −u
10
− 2u
9
+
· · · + b + u, a − 1, u
11
+ 2u
10
+ · · · + 3u + 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
u
2
a
2
=
1
u
10
+ 2u
9
+ u
8
− 2u
7
− u
6
+ 2u
5
+ 4u
4
− u
2
− u
a
4
=
u
u
9
+ 2u
8
+ u
7
− 2u
6
− u
5
+ 2u
4
+ 4u
3
− 2u − 1
a
8
=
−u
10
− 2u
9
+ u
8
+ 5u
7
+ u
6
− 6u
5
− 4u
4
+ 4u
3
+ 5u
2
− u − 3
u
10
+ u
9
− u
8
− 2u
7
+ u
6
+ 2u
5
− 2u
3
+ u − 1
a
3
=
u
4
− u
2
+ 1
u
10
+ 2u
9
+ u
8
− 2u
7
− u
6
+ 2u
5
+ 3u
4
− u
2
− u
a
7
=
−2u
10
− 3u
9
+ 2u
8
+ 7u
7
− 9u
5
− 5u
4
+ 7u
3
+ 7u
2
− 3u − 4
u
10
− u
9
− 3u
8
− u
7
+ 5u
6
+ 2u
5
− 3u
4
− 5u
3
+ 3u
2
+ 3u − 1
a
9
=
−2u
10
− 3u
9
+ 2u
8
+ 7u
7
− 8u
5
− 4u
4
+ 6u
3
+ 6u
2
− 2u − 3
u
10
− u
8
+ 3u
6
− u
5
− 2u
4
− u
3
+ 3u
2
− 2
a
9
=
−2u
10
− 3u
9
+ 2u
8
+ 7u
7
− 8u
5
− 4u
4
+ 6u
3
+ 6u
2
− 2u − 3
u
10
− u
8
+ 3u
6
− u
5
− 2u
4
− u
3
+ 3u
2
− 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
10
+ 6u
9
− 10u
7
− 4u
6
+ 6u
5
+ 12u
4
− 4u
3
− 8u
2
− 6u − 4
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
10
u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1
c
2
, c
11
u
11
+ 4u
10
+ ··· + 11u + 1
c
3
, c
7
u
11
− 2u
10
− u
9
+ 3u
8
+ u
7
− 2u
6
+ 4u
5
− 11u
4
+ 9u
3
− u
2
− 2u + 2
c
6
, c
8
u
11
− 3u
9
+ 2u
8
+ 3u
7
− 4u
6
+ 2u
4
− u
3
− 3u
2
+ 4
c
9
u
11
+ 6u
10
+ ··· + 24u + 16
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
10
y
11
− 4y
10
+ ··· + 11y −1
c
2
, c
11
y
11
+ 8y
10
+ ··· + 67y −1
c
3
, c
7
y
11
− 6y
10
+ ··· + 8y −4
c
6
, c
8
y
11
− 6y
10
+ ··· + 24y −16
c
9
y
11
− 6y
10
+ ··· − 224y −256
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.952018 + 0.226513I
a = 1.00000
b = −0.59950 + 4.16652I
c = −0.085971 + 0.401284I
d = −1.246580 + 0.306031I
−5.02081 − 0.74196I −15.5393 + 1.1191I
u = 0.952018 − 0.226513I
a = 1.00000
b = −0.59950 − 4.16652I
c = −0.085971 − 0.401284I
d = −1.246580 − 0.306031I
−5.02081 + 0.74196I −15.5393 − 1.1191I
u = 0.850023 + 0.614930I
a = 1.00000
b = −1.13889 + 1.37311I
c = 1.007370 − 0.850380I
d = 0.235931 + 0.760242I
−0.08426 − 2.41892I −7.07184 + 2.88947I
u = 0.850023 − 0.614930I
a = 1.00000
b = −1.13889 − 1.37311I
c = 1.007370 + 0.850380I
d = 0.235931 − 0.760242I
−0.08426 + 2.41892I −7.07184 − 2.88947I
u = −0.523691 + 0.948055I
a = 1.00000
b = −1.103740 − 0.547821I
c = 0.815992 + 1.141810I
d = −0.37585 − 1.52338I
5.32590 − 2.58451I −3.80806 + 1.01660I
u = −0.523691 − 0.948055I
a = 1.00000
b = −1.103740 + 0.547821I
c = 0.815992 − 1.141810I
d = −0.37585 + 1.52338I
5.32590 + 2.58451I −3.80806 − 1.01660I
16
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.978643 + 0.595733I
a = 1.00000
b = −1.47651 − 1.53693I
c = 0.060657 − 0.770160I
d = −1.86145 − 0.53501I
−2.61864 + 4.69742I −9.08124 − 5.88322I
u = −0.978643 − 0.595733I
a = 1.00000
b = −1.47651 + 1.53693I
c = 0.060657 + 0.770160I
d = −1.86145 + 0.53501I
−2.61864 − 4.69742I −9.08124 + 5.88322I
u = −1.126060 + 0.711355I
a = 1.00000
b = −1.81852 − 1.22144I
c = 0.824956 + 0.783251I
d = 0.883402 − 0.724805I
3.47965 + 8.65115I −6.21430 − 5.57892I
u = −1.126060 − 0.711355I
a = 1.00000
b = −1.81852 + 1.22144I
c = 0.824956 − 0.783251I
d = 0.883402 + 0.724805I
3.47965 − 8.65115I −6.21430 + 5.57892I
u = −0.347303
a = 1.00000
b = 0.274328
c = −2.24600
d = −1.27091
−2.16369 −2.57060
17
IV. I
u
4
= h−17u
10
+ 8u
9
+ · · · + 86d + 80, −41u
10
− 6u
9
+ · · · + 344c −
318, b − 1, −41u
10
− 6u
9
+ · · · + 344a + 26, u
11
− 3u
9
+ · · · + 3u
2
− 4i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−u
3
+ u
a
1
=
−u
2
+ 1
u
2
a
2
=
0.119186u
10
+ 0.0174419u
9
+ ··· + 0.316860u − 0.0755814
1
a
4
=
0.232558u
10
− 0.197674u
9
+ ··· − 0.174419u − 0.476744
0.0174419u
10
+ 0.197674u
9
+ ··· + 0.924419u + 0.476744
a
8
=
0.119186u
10
+ 0.0174419u
9
+ ··· + 0.316860u + 0.924419
0.197674u
10
− 0.0930233u
9
+ ··· + 0.476744u − 0.930233
a
3
=
0.0377907u
10
− 0.238372u
9
+ ··· + 0.00290698u − 0.633721
0.279070u
10
+ 0.162791u
9
+ ··· + 0.790698u + 0.627907
a
7
=
0.156977u
10
+ 0.279070u
9
+ ··· − 0.180233u + 0.790698
0.238372u
10
+ 0.0348837u
9
+ ··· + 0.633721u − 0.151163
a
9
=
−0.0784884u
10
+ 0.110465u
9
+ ··· − 0.159884u + 0.854651
0.116279u
10
− 0.348837u
9
+ ··· + 0.162791u − 0.488372
a
9
=
−0.0784884u
10
+ 0.110465u
9
+ ··· − 0.159884u + 0.854651
0.116279u
10
− 0.348837u
9
+ ··· + 0.162791u − 0.488372
(ii) Obstruction class = −1
(iii) Cusp Shapes
=
54
43
u
10
+
10
43
u
9
−
106
43
u
8
−
134
43
u
7
−
38
43
u
6
+
158
43
u
5
+
128
43
u
4
+
24
43
u
3
−
126
43
u
2
+
196
43
u −
244
43
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
8
u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1
c
2
, c
9
u
11
+ 4u
10
+ ··· + 11u + 1
c
3
, c
7
u
11
− 2u
10
− u
9
+ 3u
8
+ u
7
− 2u
6
+ 4u
5
− 11u
4
+ 9u
3
− u
2
− 2u + 2
c
5
, c
10
u
11
− 3u
9
+ 2u
8
+ 3u
7
− 4u
6
+ 2u
4
− u
3
− 3u
2
+ 4
c
11
u
11
+ 6u
10
+ ··· + 24u + 16
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
8
y
11
− 4y
10
+ ··· + 11y −1
c
2
, c
9
y
11
+ 8y
10
+ ··· + 67y −1
c
3
, c
7
y
11
− 6y
10
+ ··· + 8y −4
c
5
, c
10
y
11
− 6y
10
+ ··· + 24y −16
c
11
y
11
− 6y
10
+ ··· − 224y −256
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.360061 + 1.006500I
a = −0.271755 + 1.216000I
b = 1.00000
c = 0.72825 + 1.21600I
d = −0.76197 − 1.60205I
3.47965 − 8.65115I −6.21430 + 5.57892I
u = −0.360061 − 1.006500I
a = −0.271755 − 1.216000I
b = 1.00000
c = 0.72825 − 1.21600I
d = −0.76197 + 1.60205I
3.47965 + 8.65115I −6.21430 − 5.57892I
u = 0.529187 + 0.718311I
a = 0.010188 − 1.175860I
b = 1.00000
c = 1.01019 − 1.17586I
d = −0.344399 + 1.045410I
−0.08426 + 2.41892I −7.07184 − 2.88947I
u = 0.529187 − 0.718311I
a = 0.010188 + 1.175860I
b = 1.00000
c = 1.01019 + 1.17586I
d = −0.344399 − 1.045410I
−0.08426 − 2.41892I −7.07184 + 2.88947I
u = 1.12735
a = −0.308071
b = 1.00000
c = 0.691929
d = −0.120619
−2.16369 −2.57060
u = −1.124760 + 0.136043I
a = −0.810207 − 0.299385I
b = 1.00000
c = 0.189793 − 0.299385I
d = −0.855030 − 0.431288I
−5.02081 − 0.74196I −15.5393 + 1.1191I
21
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.124760 − 0.136043I
a = −0.810207 + 0.299385I
b = 1.00000
c = 0.189793 + 0.299385I
d = −0.855030 + 0.431288I
−5.02081 + 0.74196I −15.5393 − 1.1191I
u = −0.986131 + 0.772404I
a = −0.137568 + 0.853636I
b = 1.00000
c = 0.862432 + 0.853636I
d = 0.624556 − 0.992977I
5.32590 + 2.58451I −3.80806 − 1.01660I
u = −0.986131 − 0.772404I
a = −0.137568 − 0.853636I
b = 1.00000
c = 0.862432 − 0.853636I
d = 0.624556 + 0.992977I
5.32590 − 2.58451I −3.80806 + 1.01660I
u = 1.378090 + 0.194114I
a = −0.636622 + 0.521961I
b = 1.00000
c = 0.363378 + 0.521961I
d = −0.602844 + 1.166020I
−2.61864 + 4.69742I −9.08124 − 5.88322I
u = 1.378090 − 0.194114I
a = −0.636622 − 0.521961I
b = 1.00000
c = 0.363378 − 0.521961I
d = −0.602844 − 1.166020I
−2.61864 − 4.69742I −9.08124 + 5.88322I
22
V. I
u
5
= hd, c − 1, b − 1, a + 1, u + 1i
(i) Arc colorings
a
5
=
0
−1
a
10
=
1
0
a
11
=
1
1
a
6
=
1
0
a
1
=
0
1
a
2
=
−1
1
a
4
=
−1
0
a
8
=
1
0
a
3
=
−1
0
a
7
=
1
0
a
9
=
1
0
a
9
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
23
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u − 1
c
2
, c
4
, c
10
c
11
u + 1
c
3
, c
6
, c
7
c
8
, c
9
u
24
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
10
, c
11
y −1
c
3
, c
6
, c
7
c
8
, c
9
y
25
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.00000
b = 1.00000
c = 1.00000
d = 0
−3.28987 −12.0000
26
VI. I
u
6
= hd + 1, c, b − 1, a, u − 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
1
0
a
11
=
1
1
a
6
=
−1
0
a
1
=
0
1
a
2
=
0
1
a
4
=
0
1
a
8
=
0
−1
a
3
=
0
1
a
7
=
0
−1
a
9
=
1
−1
a
9
=
1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
27
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
u
c
5
, c
8
, c
9
c
11
u + 1
c
6
, c
10
u − 1
28
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
7
y
c
5
, c
6
, c
8
c
9
, c
10
, c
11
y −1
29
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = 1.00000
c = 0
d = −1.00000
−3.28987 −12.0000
30
VII. I
u
7
= hu
2
a + d + 1, c + a, b − 1, a
2
+ u
2
+ a − u, u
3
− u − 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
6
=
−u
−1
a
1
=
−u
2
+ 1
u
2
a
2
=
a
1
a
4
=
−au + u
2
− u − 1
au + u
a
8
=
−a
−u
2
a − 1
a
3
=
−au + u
2
+ a − u − 1
u
2
a + au + u + 1
a
7
=
−au + u
2
− a − u − 1
−u
2
a + au + u − 1
a
9
=
u
2
a + u
2
− a
−u
2
a + au + u − 1
a
9
=
u
2
a + u
2
− a
−u
2
a + au + u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
31
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
8
, c
10
(u
3
− u + 1)
2
c
2
, c
9
, c
11
(u
3
+ 2u
2
+ u + 1)
2
c
3
, c
7
(u + 1)
6
32
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
8
, c
10
(y
3
− 2y
2
+ y −1)
2
c
2
, c
9
, c
11
(y
3
− 2y
2
− 3y −1)
2
c
3
, c
7
(y −1)
6
33
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = −0.662359 + 0.562280I
a = 0.162359 + 0.986732I
b = 1.00000
c = −0.162359 − 0.986732I
d = −1.75488
−1.64493 −6.00000
u = −0.662359 + 0.562280I
a = −1.16236 − 0.98673I
b = 1.00000
c = 1.16236 + 0.98673I
d = −0.122561 − 0.744862I
−1.64493 −6.00000
u = −0.662359 − 0.562280I
a = 0.162359 − 0.986732I
b = 1.00000
c = −0.162359 + 0.986732I
d = −1.75488
−1.64493 −6.00000
u = −0.662359 − 0.562280I
a = −1.16236 + 0.98673I
b = 1.00000
c = 1.16236 − 0.98673I
d = −0.122561 + 0.744862I
−1.64493 −6.00000
u = 1.32472
a = −0.500000 + 0.424452I
b = 1.00000
c = 0.500000 − 0.424452I
d = −0.122561 − 0.744862I
−1.64493 −6.00000
u = 1.32472
a = −0.500000 − 0.424452I
b = 1.00000
c = 0.500000 + 0.424452I
d = −0.122561 + 0.744862I
−1.64493 −6.00000
34
VIII. I
u
8
= hd + 1, cb − c − 1, a + 1, u + 1i
(i) Arc colorings
a
5
=
0
−1
a
10
=
1
0
a
11
=
1
1
a
6
=
1
0
a
1
=
0
1
a
2
=
−1
b
a
4
=
−1
b − 1
a
8
=
c
−1
a
3
=
−1
b − 1
a
7
=
c
−1
a
9
=
c + 1
−1
a
9
=
c + 1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −c
2
− b
2
+ 2b − 17
(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.
35
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
c = ···
d = ···
−4.93480 −18.1451 − 0.9948I
36
IX. I
v
1
= ha, d + 1, c + a − 1, b − 1, v + 1i
(i) Arc colorings
a
5
=
−1
0
a
10
=
1
0
a
11
=
1
0
a
6
=
−1
0
a
1
=
1
0
a
2
=
0
1
a
4
=
−1
1
a
8
=
1
−1
a
3
=
−1
1
a
7
=
1
−1
a
9
=
2
−1
a
9
=
2
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
37
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u − 1
c
2
, c
4
, c
8
c
9
u + 1
c
3
, c
5
, c
7
c
10
, c
11
u
38
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
8
, c
9
y −1
c
3
, c
5
, c
7
c
10
, c
11
y
39
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
c = 1.00000
d = −1.00000
−3.28987 −12.0000
40
X. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u(u − 1)
2
(u
3
− u + 1)
2
(u
8
− u
7
+ ··· + 2u + 1)
· (u
11
− 3u
9
+ 2u
8
+ 3u
7
− 4u
6
+ 2u
4
− u
3
− 3u
2
+ 4)
· (u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1)
2
c
2
, c
9
, c
11
u(u + 1)
2
(u
3
+ 2u
2
+ u + 1)
2
· (u
8
+ 3u
7
+ 9u
6
+ 12u
5
+ 20u
4
+ 15u
3
+ 17u
2
+ 2u + 1)
· ((u
11
+ 4u
10
+ ··· + 11u + 1)
2
)(u
11
+ 6u
10
+ ··· + 24u + 16)
c
3
, c
7
u
3
(u + 1)
6
(u
8
− u
7
− u
6
+ 5u
5
− 4u
4
+ 8u
2
− 4u + 4)
· (u
11
− 2u
10
− u
9
+ 3u
8
+ u
7
− 2u
6
+ 4u
5
− 11u
4
+ 9u
3
− u
2
− 2u + 2)
3
c
4
, c
8
u(u + 1)
2
(u
3
− u + 1)
2
(u
8
− u
7
+ ··· + 2u + 1)
· (u
11
− 3u
9
+ 2u
8
+ 3u
7
− 4u
6
+ 2u
4
− u
3
− 3u
2
+ 4)
· (u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1)
2
c
5
, c
10
u(u − 1)(u + 1)(u
3
− u + 1)
2
(u
8
− u
7
+ ··· + 2u + 1)
· (u
11
− 3u
9
+ 2u
8
+ 3u
7
− 4u
6
+ 2u
4
− u
3
− 3u
2
+ 4)
· (u
11
− 2u
10
+ 4u
8
− 2u
7
− 4u
6
+ 5u
5
+ 2u
4
− 5u
3
+ u
2
+ 3u − 1)
2
41
XI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
6
, c
8
, c
10
y(y − 1)
2
(y
3
− 2y
2
+ y −1)
2
· (y
8
− 3y
7
+ 9y
6
− 12y
5
+ 20y
4
− 15y
3
+ 17y
2
− 2y + 1)
· (y
11
− 6y
10
+ ··· + 24y −16)(y
11
− 4y
10
+ ··· + 11y −1)
2
c
2
, c
9
, c
11
y(y − 1)
2
(y
3
− 2y
2
− 3y −1)
2
· (y
8
+ 9y
7
+ 49y
6
+ 160y
5
+ 336y
4
+ 425y
3
+ 269y
2
+ 30y + 1)
· (y
11
− 6y
10
+ ··· − 224y −256)(y
11
+ 8y
10
+ ··· + 67y −1)
2
c
3
, c
7
y
3
(y −1)
6
(y
8
− 3y
7
+ 3y
6
− y
5
− 32y
3
+ 32y
2
+ 48y + 16)
· (y
11
− 6y
10
+ ··· + 8y −4)
3
42
