12a
0427
(K12a
0427
)
A knot diagram
1
Linearized knot diagam
3 6 10 7 9 2 12 5 1 4 8 11
Solving Sequence
8,12 4,7
5 9 11 1 10 3 2 6
c
7
c
4
c
8
c
11
c
12
c
10
c
3
c
1
c
6
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= ⟨575260u
67
+ 2605705u
66
+ ··· + 248832b − 26005926,
563467900u
67
+ 2831326125u
66
+ ··· + 212253696a + 11136298543,
5u
68
+ 30u
67
+ ··· + 5054u + 853⟩
I
u
2
= ⟨−a
2
u + au + b − 2a + 2, a
3
− 2a
2
u + 2a
2
+ au − 2a + 3u − 1, u
2
− u + 1⟩
I
u
3
= ⟨u
4
+ b, −u
2
+ a − 1, u
5
+ u
3
+ u + 1⟩
I
u
4
= ⟨b + u, a + u, u
5
+ u
3
+ u − 1⟩
I
u
5
= ⟨b
2
au − 2a
2
bu + a
3
u + b
3
− 3b
2
a + 3a
2
b − a
3
+ 2bu − au − a + u − 1, u
2
− u + 1⟩
I
u
6
= ⟨bau − a
2
u + b
2
− 2ba + bu + a
2
− au − b + u, u
2
− u + 1⟩
I
u
7
= ⟨u
2
a + au + b, u
3
a + u
2
a + au − 1⟩
I
u
8
= ⟨b − a − u + 1, u
2
− u + 1⟩
I
v
1
= ⟨a, b
6
− 2b
4
− b
3
+ b
2
+ b + 1, v − 1⟩
* 5 irreducible components of dim
C
= 0, with total 90 representations.
* 4 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
= ⟨5.75×10
5
u
67
+2.61×10
6
u
66
+· · ·+2.49×10
5
b−2.60×10
7
, 5.63×10
8
u
67
+
2.83×10
9
u
66
+· · ·+2.12×10
8
a+1.11×10
10
, 5u
68
+30u
67
+· · ·+5054u+853⟩
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
−2.65469u
67
− 13.3393u
66
+ ··· − 582.171u − 52.4669
−2.31184u
67
− 10.4717u
66
+ ··· + 255.745u + 104.512
a
7
=
1
−u
2
a
5
=
−1.24736u
67
− 3.35352u
66
+ ··· + 1325.95u + 284.669
1.92813u
67
+ 11.8675u
66
+ ··· + 2054.31u + 367.547
a
9
=
−0.313259u
67
− 1.69428u
66
+ ··· − 190.456u − 28.1702
0.0108507u
67
+ 0.221354u
66
+ ··· + 85.6145u + 16.6233
a
11
=
u
u
a
1
=
u
3
u
3
+ u
a
10
=
0.103137u
67
+ 0.458229u
66
+ ··· − 55.5265u − 13.1298
0.158691u
67
+ 0.674371u
66
+ ··· − 92.1597u − 23.0834
a
3
=
−1.97519u
67
− 5.35349u
66
+ ··· + 1993.77u + 439.517
2.40346u
67
+ 18.6888u
66
+ ··· + 4616.08u + 863.329
a
2
=
3.36495u
67
+ 16.0548u
66
+ ··· + 184.845u − 49.3617
1.33590u
67
+ 2.51799u
66
+ ··· − 2210.43u − 464.485
a
6
=
2.83094u
67
+ 10.9767u
66
+ ··· − 906.631u − 247.880
−2.22960u
67
− 16.9831u
66
+ ··· − 4002.13u − 746.999
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
327295
186624
u
67
−
1808905
124416
u
66
+ ··· −
177580613
46656
u −
269328401
373248
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
25(25u
68
+ 680u
67
+ ··· + 7437476u + 727609)
c
2
, c
6
5(5u
68
+ 30u
67
+ ··· + 5054u + 853)
c
3
, c
10
81(81u
68
+ 648u
67
+ ··· + 29832u + 4477)
c
4
64(64u
68
− 256u
67
+ ··· − 4.94845 × 10
7
u + 9687600)
c
5
, c
8
81(81u
68
− 648u
67
+ ··· − 29832u + 4477)
c
7
, c
11
5(5u
68
− 30u
67
+ ··· − 5054u + 853)
c
9
64(64u
68
+ 256u
67
+ ··· + 4.94845 × 10
7
u + 9687600)
c
12
25(25u
68
− 680u
67
+ ··· − 7437476u + 727609)
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
12
625
· (625y
68
+ 16300y
67
+ ··· + 11609525524248y + 529414856881)
c
2
, c
6
, c
7
c
11
25(25y
68
+ 680y
67
+ ··· + 7437476y + 727609)
c
3
, c
5
, c
8
c
10
6561(6561y
68
− 279936y
67
+ ··· − 3.61575 × 10
7
y + 2.00435 × 10
7
)
c
4
, c
9
4096
· (4096y
68
− 8192y
67
+ ··· − 815469037963200y + 93849593760000)
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.702519 + 0.717277I
a = −0.75644 − 1.31522I
b = −1.296900 − 0.451791I
1.29111 − 5.34461I 0
u = 0.702519 − 0.717277I
a = −0.75644 + 1.31522I
b = −1.296900 + 0.451791I
1.29111 + 5.34461I 0
u = −0.772722 + 0.614454I
a = −0.904625 − 0.687185I
b = −0.34299 − 1.42396I
−3.93086 − 2.25762I 0
u = −0.772722 − 0.614454I
a = −0.904625 + 0.687185I
b = −0.34299 + 1.42396I
−3.93086 + 2.25762I 0
u = −0.787880 + 0.588901I
a = 1.094050 + 0.575479I
b = 0.45000 + 1.53766I
−5.97017 − 7.33663I 0
u = −0.787880 − 0.588901I
a = 1.094050 − 0.575479I
b = 0.45000 − 1.53766I
−5.97017 + 7.33663I 0
u = 0.112405 + 1.036780I
a = −0.685771 − 0.811514I
b = −0.008091 − 0.465537I
−0.02177 − 6.74730I 0
u = 0.112405 − 1.036780I
a = −0.685771 + 0.811514I
b = −0.008091 + 0.465537I
−0.02177 + 6.74730I 0
u = −0.833820 + 0.630175I
a = 0.718590 + 0.298717I
b = 0.549137 + 1.181960I
−8.97416 + 0.22766I 0
u = −0.833820 − 0.630175I
a = 0.718590 − 0.298717I
b = 0.549137 − 1.181960I
−8.97416 − 0.22766I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.078667 + 0.947690I
a = 0.769173 + 0.818647I
b = 0.215986 + 0.247360I
1.67199 − 2.07344I 4.37435 + 4.03078I
u = 0.078667 − 0.947690I
a = 0.769173 − 0.818647I
b = 0.215986 − 0.247360I
1.67199 + 2.07344I 4.37435 − 4.03078I
u = 0.937214 + 0.495255I
a = 0.95339 − 1.23785I
b = −0.408666 − 1.113570I
−1.95572 + 13.26270I 0
u = 0.937214 − 0.495255I
a = 0.95339 + 1.23785I
b = −0.408666 + 1.113570I
−1.95572 − 13.26270I 0
u = 0.984482 + 0.423718I
a = 0.934550 − 0.721368I
b = −0.123632 − 0.785388I
−7.11233 + 4.71108I 0
u = 0.984482 − 0.423718I
a = 0.934550 + 0.721368I
b = −0.123632 + 0.785388I
−7.11233 − 4.71108I 0
u = 0.957820 + 0.502893I
a = −0.811297 + 1.154980I
b = 0.446758 + 0.985756I
7.23221I 0
u = 0.957820 − 0.502893I
a = −0.811297 − 1.154980I
b = 0.446758 − 0.985756I
− 7.23221I 0
u = 0.526643 + 0.956182I
a = 0.65049 + 1.37033I
b = 1.06139 + 1.06704I
−0.45309 − 3.04384I 0
u = 0.526643 − 0.956182I
a = 0.65049 − 1.37033I
b = 1.06139 − 1.06704I
−0.45309 + 3.04384I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.749817 + 0.415369I
a = −1.04809 − 1.17549I
b = 0.620776 − 1.056180I
2.68807 − 7.69227I 0.67631 + 5.85863I
u = −0.749817 − 0.415369I
a = −1.04809 + 1.17549I
b = 0.620776 + 1.056180I
2.68807 + 7.69227I 0.67631 − 5.85863I
u = −0.670154 + 0.937244I
a = −1.45327 + 0.06264I
b = −1.234960 − 0.202287I
−1.29111 + 5.34461I 0
u = −0.670154 − 0.937244I
a = −1.45327 − 0.06264I
b = −1.234960 + 0.202287I
−1.29111 − 5.34461I 0
u = −0.117684 + 1.148120I
a = 0.066271 + 0.344259I
b = −0.348380 − 0.848238I
7.75434 − 5.46492I 0
u = −0.117684 − 1.148120I
a = 0.066271 − 0.344259I
b = −0.348380 + 0.848238I
7.75434 + 5.46492I 0
u = −0.753289 + 0.380335I
a = 0.98818 + 1.05811I
b = −0.609854 + 0.844847I
3.93086 − 2.25762I 3.33892 + 0.57210I
u = −0.753289 − 0.380335I
a = 0.98818 − 1.05811I
b = −0.609854 − 0.844847I
3.93086 + 2.25762I 3.33892 − 0.57210I
u = −0.341284 + 0.760500I
a = −2.29641 − 0.77204I
b = −1.73365 − 0.13982I
0.45309 + 3.04384I 3.47317 + 0.77959I
u = −0.341284 − 0.760500I
a = −2.29641 + 0.77204I
b = −1.73365 + 0.13982I
0.45309 − 3.04384I 3.47317 − 0.77959I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.145225 + 0.813062I
a = 1.28772 + 0.60761I
b = 0.749993 − 0.175588I
1.65241 − 1.05941I 6.63964 + 4.57024I
u = −0.145225 − 0.813062I
a = 1.28772 − 0.60761I
b = 0.749993 + 0.175588I
1.65241 + 1.05941I 6.63964 − 4.57024I
u = 0.780665 + 0.269621I
a = 1.108650 + 0.142041I
b = 0.637811 − 0.389139I
−4.47467 − 4.39199I −7.74935 + 3.75955I
u = 0.780665 − 0.269621I
a = 1.108650 − 0.142041I
b = 0.637811 + 0.389139I
−4.47467 + 4.39199I −7.74935 − 3.75955I
u = −0.132701 + 1.176080I
a = 0.023962 − 0.238727I
b = 0.328720 + 0.927847I
8.97416 + 0.22766I 0
u = −0.132701 − 1.176080I
a = 0.023962 + 0.238727I
b = 0.328720 − 0.927847I
8.97416 − 0.22766I 0
u = −0.567585 + 1.053430I
a = −1.51541 − 0.77084I
b = −1.55197 − 1.72836I
0.02177 + 6.74730I 0
u = −0.567585 − 1.053430I
a = −1.51541 + 0.77084I
b = −1.55197 + 1.72836I
0.02177 − 6.74730I 0
u = −1.059300 + 0.560745I
a = 0.104792 − 0.286003I
b = 0.754560 + 0.357556I
−2.18567 + 7.77249I 0
u = −1.059300 − 0.560745I
a = 0.104792 + 0.286003I
b = 0.754560 − 0.357556I
−2.18567 − 7.77249I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.662387 + 1.025070I
a = −1.40530 − 0.89343I
b = −0.72294 − 1.44985I
−2.68807 + 7.69227I 0
u = −0.662387 − 1.025070I
a = −1.40530 + 0.89343I
b = −0.72294 + 1.44985I
−2.68807 − 7.69227I 0
u = −0.663727 + 1.037980I
a = 1.38765 + 1.08315I
b = 0.54606 + 1.68325I
−4.61883 + 12.81070I 0
u = −0.663727 − 1.037980I
a = 1.38765 − 1.08315I
b = 0.54606 − 1.68325I
−4.61883 − 12.81070I 0
u = −0.603158 + 1.083480I
a = −1.75432 − 0.31678I
b = −1.84792 − 1.60763I
4.61883 + 12.81070I 0
u = −0.603158 − 1.083480I
a = −1.75432 + 0.31678I
b = −1.84792 + 1.60763I
4.61883 − 12.81070I 0
u = −0.692945 + 1.028870I
a = 1.002200 + 0.844351I
b = 0.244445 + 1.103960I
−7.75434 + 5.46492I 0
u = −0.692945 − 1.028870I
a = 1.002200 − 0.844351I
b = 0.244445 − 1.103960I
−7.75434 − 5.46492I 0
u = −0.599314 + 0.464063I
a = −1.33435 − 1.16820I
b = −0.107178 − 1.031260I
−1.67199 − 2.07344I −4.37435 + 4.03078I
u = −0.599314 − 0.464063I
a = −1.33435 + 1.16820I
b = −0.107178 + 1.031260I
−1.67199 + 2.07344I −4.37435 − 4.03078I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.595289 + 1.091030I
a = 1.59309 + 0.27206I
b = 1.76899 + 1.48931I
5.97017 + 7.33663I 0
u = −0.595289 − 1.091030I
a = 1.59309 − 0.27206I
b = 1.76899 − 1.48931I
5.97017 − 7.33663I 0
u = −0.532947 + 1.124330I
a = 0.912459 + 0.425272I
b = 1.19050 + 1.36318I
4.47467 + 4.39199I 0
u = −0.532947 − 1.124330I
a = 0.912459 − 0.425272I
b = 1.19050 − 1.36318I
4.47467 − 4.39199I 0
u = −0.029081 + 1.297740I
a = −0.435981 − 0.038934I
b = −0.178672 − 1.072860I
4.86026 + 10.75690I 0
u = −0.029081 − 1.297740I
a = −0.435981 + 0.038934I
b = −0.178672 + 1.072860I
4.86026 − 10.75690I 0
u = −0.073579 + 1.312560I
a = 0.338952 + 0.014268I
b = 0.224066 + 1.023900I
7.11233 + 4.71108I 0
u = −0.073579 − 1.312560I
a = 0.338952 − 0.014268I
b = 0.224066 − 1.023900I
7.11233 − 4.71108I 0
u = 0.687648 + 1.131730I
a = 1.67548 − 0.61053I
b = 1.85706 − 1.75220I
− 19.2093I 0
u = 0.687648 − 1.131730I
a = 1.67548 + 0.61053I
b = 1.85706 + 1.75220I
19.2093I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.696018 + 1.135630I
a = −1.57600 + 0.50092I
b = −1.79380 + 1.58426I
1.95572 − 13.26270I 0
u = 0.696018 − 1.135630I
a = −1.57600 − 0.50092I
b = −1.79380 − 1.58426I
1.95572 + 13.26270I 0
u = 0.458183 + 0.480361I
a = −0.807160 − 1.060200I
b = −0.930879 − 0.136305I
−1.65241 − 1.05941I −6.63964 + 4.57024I
u = 0.458183 − 0.480361I
a = −0.807160 + 1.060200I
b = −0.930879 + 0.136305I
−1.65241 + 1.05941I −6.63964 − 4.57024I
u = 0.686607 + 1.164600I
a = 1.238110 − 0.659956I
b = 1.34714 − 1.61050I
−4.86026 − 10.75690I 0
u = 0.686607 − 1.164600I
a = 1.238110 + 0.659956I
b = 1.34714 + 1.61050I
−4.86026 + 10.75690I 0
u = 0.77502 + 1.18600I
a = −0.943171 + 0.128059I
b = −1.25289 + 0.90407I
2.18567 − 7.77249I 0
u = 0.77502 − 1.18600I
a = −0.943171 − 0.128059I
b = −1.25289 − 0.90407I
2.18567 + 7.77249I 0
11
II.
I
u
2
= ⟨−a
2
u + au + b − 2a + 2, a
3
− 2a
2
u + 2a
2
+ au − 2a + 3u − 1, u
2
− u + 1⟩
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
a
a
2
u − au + 2a − 2
a
7
=
1
−u + 1
a
5
=
−a
2
u + 2
a
2
u − a
2
+ a − 2u
a
9
=
−a
2
u + 2
−a
2
u − a + 2
a
11
=
u
u
a
1
=
−1
u − 1
a
10
=
−2a
2
u + a
2
− a + 2u + 2
−2a
2
u + au − 2a + 4
a
3
=
1
−u + 1
a
2
=
−1
u − 1
a
6
=
1
−u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u − 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
u
6
c
3
, c
5
, c
8
c
9
, c
10
u
6
− 2u
4
+ u
3
+ u
2
− u + 1
c
4
u
6
+ 4u
5
+ 6u
4
+ 3u
3
− u
2
− u + 1
c
7
, c
11
(u
2
+ u + 1)
3
c
12
(u
2
− u + 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
y
6
c
3
, c
5
, c
8
c
9
, c
10
y
6
− 4y
5
+ 6y
4
− 3y
3
− y
2
+ y + 1
c
4
y
6
− 4y
5
+ 10y
4
− 11y
3
+ 19y
2
− 3y + 1
c
7
, c
11
, c
12
(y
2
+ y + 1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 1.137010 − 0.340420I
b = 0.669552 − 0.863143I
− 2.02988I 0. + 3.46410I
u = 0.500000 + 0.866025I
a = −1.072830 + 0.640783I
b = −1.49343 + 1.84400I
− 2.02988I 0. + 3.46410I
u = 0.500000 + 0.866025I
a = −1.06417 + 1.43169I
b = −0.176126 + 0.751194I
− 2.02988I 0. + 3.46410I
u = 0.500000 − 0.866025I
a = 1.137010 + 0.340420I
b = 0.669552 + 0.863143I
2.02988I 0. − 3.46410I
u = 0.500000 − 0.866025I
a = −1.072830 − 0.640783I
b = −1.49343 − 1.84400I
2.02988I 0. − 3.46410I
u = 0.500000 − 0.866025I
a = −1.06417 − 1.43169I
b = −0.176126 − 0.751194I
2.02988I 0. − 3.46410I
15
III. I
u
3
= ⟨u
4
+ b, −u
2
+ a − 1, u
5
+ u
3
+ u + 1⟩
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
u
2
+ 1
−u
4
a
7
=
1
−u
2
a
5
=
1
0
a
9
=
1
0
a
11
=
u
u
a
1
=
u
3
u
3
+ u
a
10
=
u
2
+ u + 1
−u
4
+ u
a
3
=
−u
−u
a
2
=
0
u
a
6
=
1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = 6
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ u − 1
c
2
, c
6
, c
7
c
11
u
5
+ u
3
+ u − 1
c
3
, c
10
(u − 1)
5
c
5
, c
8
u
5
c
9
u
5
+ u
3
+ 2u
2
− u − 2
c
12
u
5
− 2u
4
+ 3u
3
− 2u
2
+ u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
12
y
5
+ 2y
4
+ 3y
3
+ 6y
2
+ 5y − 1
c
2
, c
6
, c
7
c
11
y
5
+ 2y
4
+ 3y
3
+ 2y
2
+ y − 1
c
3
, c
10
(y − 1)
5
c
5
, c
8
y
5
c
9
y
5
+ 2y
4
− y
3
− 6y
2
+ 9y − 4
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.707729 + 0.841955I
a = 0.79199 + 1.19175I
b = 1.37700 + 0.49579I
1.64493 6.00000
u = 0.707729 − 0.841955I
a = 0.79199 − 1.19175I
b = 1.37700 − 0.49579I
1.64493 6.00000
u = −0.389287 + 1.070680I
a = 0.005198 − 0.833601I
b = −0.29474 − 1.65854I
1.64493 6.00000
u = −0.389287 − 1.070680I
a = 0.005198 + 0.833601I
b = −0.29474 + 1.65854I
1.64493 6.00000
u = −0.636883
a = 1.40562
b = −0.164527
1.64493 6.00000
19
IV. I
u
4
= ⟨b + u, a + u, u
5
+ u
3
+ u − 1⟩
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
−u
−u
a
7
=
1
−u
2
a
5
=
u
3
−1
a
9
=
−u
3
+ 1
1
a
11
=
u
u
a
1
=
u
3
u
3
+ u
a
10
=
u
u
a
3
=
−u
−u
a
2
=
0
u
a
6
=
1
0
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
20
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ u − 1
c
2
, c
6
, c
7
c
11
u
5
+ u
3
+ u + 1
c
3
, c
10
u
5
c
4
u
5
+ u
3
− 2u
2
− u + 2
c
5
, c
8
(u + 1)
5
c
9
, c
12
u
5
− 2u
4
+ 3u
3
− 2u
2
+ u + 1
21
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
9
, c
12
y
5
+ 2y
4
+ 3y
3
+ 6y
2
+ 5y − 1
c
2
, c
6
, c
7
c
11
y
5
+ 2y
4
+ 3y
3
+ 2y
2
+ y − 1
c
3
, c
10
y
5
c
4
y
5
+ 2y
4
− y
3
− 6y
2
+ 9y − 4
c
5
, c
8
(y − 1)
5
22
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.707729 + 0.841955I
a = 0.707729 − 0.841955I
b = 0.707729 − 0.841955I
−1.64493 −6.00000
u = −0.707729 − 0.841955I
a = 0.707729 + 0.841955I
b = 0.707729 + 0.841955I
−1.64493 −6.00000
u = 0.389287 + 1.070680I
a = −0.389287 − 1.070680I
b = −0.389287 − 1.070680I
−1.64493 −6.00000
u = 0.389287 − 1.070680I
a = −0.389287 + 1.070680I
b = −0.389287 + 1.070680I
−1.64493 −6.00000
u = 0.636883
a = −0.636883
b = −0.636883
−1.64493 −6.00000
23
V.
I
u
5
= ⟨b
2
au−2a
2
bu+a
3
u+b
3
−3b
2
a+3a
2
b−a
3
+2bu−au−a+u−1, u
2
−u+1⟩
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
a
b
a
7
=
1
−u + 1
a
5
=
−au − b + 2a
bu − au
a
9
=
−b
2
u + 2bau − a
2
u + ba − a
2
+ 1
b
2
u − 2bau + a
2
u − b
2
+ 2ba − a
2
a
11
=
u
u
a
1
=
−1
u − 1
a
10
=
bau − a
2
u + u
b
2
u − bau + u
a
3
=
−b
2
au + 2a
2
bu − a
3
u + b
2
a − 2a
2
b + a
3
− bu + au + b
−b
2
au + 2a
2
bu − a
3
u − bu + a − u
a
2
=
b
2
a − 2a
2
b + a
3
+ au + b − a − 1
−b
2
au + 2a
2
bu − a
3
u + b
2
a − 2a
2
b + a
3
− bu + au + b
a
6
=
−b
2
au + 2a
2
bu − a
3
u + b
2
a − 2a
2
b + a
3
− bu + au + b + u
−b
2
au + 2a
2
bu − a
3
u − bu + a − u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u − 4
(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.
24
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
4.05977I 6.92820I
25
VI. I
u
6
= ⟨bau − a
2
u + b
2
− 2ba + bu + a
2
− au − b + u, u
2
− u + 1⟩
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
a
b
a
7
=
1
−u + 1
a
5
=
−au − b + 2a
bu − au
a
9
=
bau − a
2
u − au − b + a + u
ba + bu − a
2
− a + 1
a
11
=
u
u
a
1
=
−1
u − 1
a
10
=
bau − a
2
u + u
ba − a
2
+ au + b − a + 1
a
3
=
−a
2
b − bau + a
3
− bu + a
2
+ au + b − a
a
2
bu − a
3
u − a
2
b + a
3
− a
2
u − ba − bu + a
2
+ au + u − 1
a
2
=
a
2
bu − a
3
u − a
2
b + a
3
− a
2
u − ba − bu + a
2
+ au − 1
a
2
bu − a
3
u + bau − a
2
u − ba − b + a + 2u − 1
a
6
=
−a
2
b − bau + a
3
− bu + a
2
+ au + b − a − u + 1
a
2
bu − a
3
u − a
2
b + a
3
− a
2
u − ba − bu + a
2
+ au − 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.
26
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
27
VII. I
u
7
= ⟨u
2
a + au + b, u
3
a + u
2
a + au − 1⟩
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
a
−u
2
a − au
a
7
=
1
−u
2
a
5
=
au + a
−1
a
9
=
−au − a + 1
1
a
11
=
u
u
a
1
=
u
3
u
3
+ u
a
10
=
−a + u
u
2
a + au + u
a
3
=
u
u
a
2
=
0
u
a
6
=
1
0
(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.
28
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
7
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
29
VIII. I
u
8
= ⟨b − a − u + 1, u
2
− u + 1⟩
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
4
=
a
a + u − 1
a
7
=
1
−u + 1
a
5
=
−au + a − u + 1
−1
a
9
=
au − a + u
1
a
11
=
u
u
a
1
=
−1
u − 1
a
10
=
−a + u
−a + 1
a
3
=
u
u
a
2
=
0
u
a
6
=
1
0
(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.
30
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
8
√
−1(vol +
√
−1CS) Cusp shape
u = ···
a = ···
b = ···
0 0
31
IX. I
v
1
= ⟨a, b
6
− 2b
4
− b
3
+ b
2
+ b + 1, v − 1⟩
(i) Arc colorings
a
8
=
1
0
a
12
=
1
0
a
4
=
0
b
a
7
=
1
0
a
5
=
−b
b
a
9
=
−b
2
+ 1
b
2
a
11
=
1
0
a
1
=
1
0
a
10
=
1
b
2
a
3
=
−b
−b
3
+ b
a
2
=
b
4
− b
2
+ 1
b
3
− b − 1
a
6
=
b
3
− 2b
−b
3
+ b
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4b
3
− 4b − 2
32
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
2
+ u + 1)
3
c
2
, c
6
(u
2
− u + 1)
3
c
3
, c
4
, c
5
c
8
, c
10
u
6
− 2u
4
− u
3
+ u
2
+ u + 1
c
7
, c
11
, c
12
u
6
c
9
u
6
− 4u
5
+ 6u
4
− 3u
3
− u
2
+ u + 1
33
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
(y
2
+ y + 1)
3
c
3
, c
4
, c
5
c
8
, c
10
y
6
− 4y
5
+ 6y
4
− 3y
3
− y
2
+ y + 1
c
7
, c
11
, c
12
y
6
c
9
y
6
− 4y
5
+ 10y
4
− 11y
3
+ 19y
2
− 3y + 1
34
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.033350 + 0.428825I
− 2.02988I 0. + 3.46410I
v = 1.00000
a = 0
b = −1.033350 − 0.428825I
2.02988I 0. − 3.46410I
v = 1.00000
a = 0
b = 1.252310 + 0.237364I
− 2.02988I 0. + 3.46410I
v = 1.00000
a = 0
b = 1.252310 − 0.237364I
2.02988I 0. − 3.46410I
v = 1.00000
a = 0
b = −0.218964 + 0.666188I
2.02988I 0. − 3.46410I
v = 1.00000
a = 0
b = −0.218964 − 0.666188I
− 2.02988I 0. + 3.46410I
35
X. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
25u
6
(u
2
+ u + 1)
3
(u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ u − 1)
2
· (25u
68
+ 680u
67
+ ··· + 7437476u + 727609)
c
2
, c
6
5u
6
(u
2
− u + 1)
3
(u
5
+ u
3
+ u − 1)(u
5
+ u
3
+ u + 1)
· (5u
68
+ 30u
67
+ ··· + 5054u + 853)
c
3
, c
10
81u
5
(u − 1)
5
(u
6
− 2u
4
+ ··· + u + 1)(u
6
− 2u
4
+ ··· − u + 1)
· (81u
68
+ 648u
67
+ ··· + 29832u + 4477)
c
4
64(u
5
+ u
3
− 2u
2
− u + 2)(u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ u − 1)
· (u
6
− 2u
4
− u
3
+ u
2
+ u + 1)(u
6
+ 4u
5
+ 6u
4
+ 3u
3
− u
2
− u + 1)
· (64u
68
− 256u
67
+ ··· − 49484520u + 9687600)
c
5
, c
8
81u
5
(u + 1)
5
(u
6
− 2u
4
+ ··· + u + 1)(u
6
− 2u
4
+ ··· − u + 1)
· (81u
68
− 648u
67
+ ··· − 29832u + 4477)
c
7
, c
11
5u
6
(u
2
+ u + 1)
3
(u
5
+ u
3
+ u − 1)(u
5
+ u
3
+ u + 1)
· (5u
68
− 30u
67
+ ··· − 5054u + 853)
c
9
64(u
5
+ u
3
+ 2u
2
− u − 2)(u
5
− 2u
4
+ 3u
3
− 2u
2
+ u + 1)
· (u
6
− 2u
4
+ u
3
+ u
2
− u + 1)(u
6
− 4u
5
+ 6u
4
− 3u
3
− u
2
+ u + 1)
· (64u
68
+ 256u
67
+ ··· + 49484520u + 9687600)
c
12
25u
6
(u
2
− u + 1)
3
(u
5
− 2u
4
+ 3u
3
− 2u
2
+ u + 1)
2
· (25u
68
− 680u
67
+ ··· − 7437476u + 727609)
36
XI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
12
625y
6
(y
2
+ y + 1)
3
(y
5
+ 2y
4
+ 3y
3
+ 6y
2
+ 5y − 1)
2
· (625y
68
+ 16300y
67
+ ··· + 11609525524248y + 529414856881)
c
2
, c
6
, c
7
c
11
25y
6
(y
2
+ y + 1)
3
(y
5
+ 2y
4
+ 3y
3
+ 2y
2
+ y − 1)
2
· (25y
68
+ 680y
67
+ ··· + 7437476y + 727609)
c
3
, c
5
, c
8
c
10
6561y
5
(y − 1)
5
(y
6
− 4y
5
+ 6y
4
− 3y
3
− y
2
+ y + 1)
2
· (6561y
68
− 279936y
67
+ ··· − 36157462y + 20043529)
c
4
, c
9
4096(y
5
+ 2y
4
− y
3
− 6y
2
+ 9y − 4)(y
5
+ 2y
4
+ 3y
3
+ 6y
2
+ 5y − 1)
· (y
6
− 4y
5
+ 6y
4
− 3y
3
− y
2
+ y + 1)
· (y
6
− 4y
5
+ 10y
4
− 11y
3
+ 19y
2
− 3y + 1)
· (4096y
68
− 8192y
67
+ ··· − 815469037963200y + 93849593760000)
37
