12a
0167
(K12a
0167
)
A knot diagram
1
Linearized knot diagam
3 5 9 2 12 1 10 11 4 8 7 6
Solving Sequence
4,9 8,10
11
1,3
2 5 7 6 12
c
9
c
10
c
3
c
1
c
4
c
7
c
6
c
12
c
2
, c
5
, c
8
, 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
= h−179878036u
22
− 223200067u
21
+ ··· + 15409974654d − 115530422,
487430093u
22
+ 2626639316u
21
+ ··· + 184919695848c − 182635192736,
6729099212u
22
+ 11896606493u
21
+ ··· + 92459847924b + 39193550776,
− 321494617u
22
+ 1287799250u
21
+ ··· + 92459847924a − 91427201564,
u
23
+ 2u
22
+ ··· − 4u
2
+ 8i
I
u
2
= hu
2
c − u
3
− cu + u
2
+ d + c − 1, −2u
3
c + 3u
2
c + u
3
+ c
2
− 2cu − u
2
+ u, u
2
+ b + 1, u
3
− 2u
2
+ a + u − 1,
u
4
− 2u
3
+ 2u
2
− u + 1i
I
u
3
= h−u
7
+ u
5
− 2u
3
+ d + u, u
5
+ c + u, −u
7
− u
6
+ u
4
− au + b + 1,
− u
7
a + u
7
+ 2u
5
a + 2u
4
a − 2u
5
− 2u
3
a − u
4
− 2u
2
a + 3u
3
+ a
2
+ u
2
+ 2a − 2,
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1i
I
u
4
= h−u
6
− u
5
+ u
4
− u
2
a + 2u
3
− au + d − u − 1,
2u
7
a + 2u
6
a − u
7
− 2u
5
a − u
6
− 4u
4
a + u
5
+ 2u
3
a + 3u
4
+ 3u
2
a − au − 2u
2
+ c − 3a − u + 1,
− u
7
− u
6
+ u
4
− au + b + 1, −u
7
a + u
7
+ 2u
5
a + 2u
4
a − 2u
5
− 2u
3
a − u
4
− 2u
2
a + 3u
3
+ a
2
+ u
2
+ 2a − 2,
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1i
I
u
5
= h−u
7
+ u
5
− 2u
3
+ d + u, u
5
+ c + u, −u
7
− u
6
+ 2u
5
+ 3u
4
− 2u
3
− 4u
2
+ b + 2u + 3,
u
7
− 2u
5
− 2u
4
+ 2u
3
+ 2u
2
+ a − 2u − 2, u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1i
I
u
6
= hu
5
c − u
4
c − 2u
3
c + 3u
2
c + u
3
+ cu + d − 2c − u,
− 3u
5
c − u
4
c + u
5
+ 5u
3
c + u
4
− 3u
2
c − 3u
3
+ 2c
2
− 5cu + u
2
+ 6c + 3u − 4, −u
4
+ u
2
+ b − u − 1,
− u
5
+ u
4
+ 3u
3
− 3u
2
+ 2a − u + 4, u
6
− u
5
− u
4
+ 3u
3
− u
2
− 2u + 2i
I
v
1
= hc, d − 1, b, a + 1, v + 1i
I
v
2
= ha, d, c − 1, b + 1, v − 1i
I
v
3
= ha, d − 1, c + a − 1, b + 1, v − 1i
I
v
4
= hc, d − 1, av + c + v −1, bv + 1i
* 9 irreducible components of dim
C
= 0, with total 86 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
= h−1.80 × 10
8
u
22
− 2.23 × 10
8
u
21
+ · · · + 1.54 × 10
10
d − 1.16 ×
10
8
, 4.87 × 10
8
u
22
+ 2.63 × 10
9
u
21
+ · · · + 1.85 × 10
11
c − 1.83 × 10
11
, 6.73 ×
10
9
u
22
+ 1.19 × 10
10
u
21
+ · · · + 9.25 × 10
10
b + 3.92 × 10
10
, −3.21 × 10
8
u
22
+
1.29 × 10
9
u
21
+ · · · + 9.25 × 10
10
a − 9.14 × 10
10
, u
23
+ 2u
22
+ · · · − 4u
2
+ 8i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
8
=
−0.00263590u
22
− 0.0142042u
21
+ ··· − 0.0222392u + 0.987646
0.0116728u
22
+ 0.0144841u
21
+ ··· − 0.162782u + 0.00749712
a
10
=
1
−u
2
a
11
=
−0.00263590u
22
− 0.0142042u
21
+ ··· − 0.0222392u + 0.987646
−0.0165392u
22
− 0.0250589u
21
+ ··· + 0.141695u − 0.0789564
a
1
=
0.00347713u
22
− 0.0139282u
21
+ ··· − 0.640963u + 0.988831
−0.0727786u
22
− 0.128668u
21
+ ··· + 0.850224u − 0.423898
a
3
=
−u
u
a
2
=
0.0160451u
22
− 0.0132319u
21
+ ··· − 1.19538u + 0.956887
−0.0853466u
22
− 0.129364u
21
+ ··· + 1.40464u − 0.391954
a
5
=
0.0693015u
22
+ 0.142596u
21
+ ··· − 0.209260u − 0.564933
−0.0853466u
22
− 0.129364u
21
+ ··· + 1.40464u − 0.391954
a
7
=
−0.0191751u
22
− 0.0392631u
21
+ ··· + 0.119456u + 0.908690
0.0174433u
22
+ 0.0237607u
21
+ ··· − 0.295096u + 0.0716533
a
6
=
−0.0462554u
22
− 0.0937250u
21
+ ··· + 0.159204u + 0.719639
0.0587694u
22
+ 0.0800767u
21
+ ··· − 0.985189u + 0.264336
a
12
=
−0.00173177u
22
− 0.0155024u
21
+ ··· − 0.175640u + 0.980343
−0.0358544u
22
− 0.0587621u
21
+ ··· + 0.281242u − 0.167964
(ii) Obstruction class = −1
(iii) Cusp Shapes =
15567855023
46229923962
u
22
+
8703838979
46229923962
u
21
+ ··· −
168604101146
23114961981
u +
87470148380
23114961981
3
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
23
+ 10u
22
+ ··· + 88u + 16
c
2
, c
4
u
23
− 2u
22
+ ··· + 8u − 4
c
3
, c
9
u
23
− 2u
22
+ ··· + 4u
2
− 8
c
5
, c
6
, c
7
c
8
, c
10
, c
12
u
23
+ 2u
22
+ ··· − u − 1
c
11
u
23
− 6u
22
+ ··· + 64u − 64
4
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
23
+ 6y
22
+ ··· + 1824y −256
c
2
, c
4
y
23
− 10y
22
+ ··· + 88y −16
c
3
, c
9
y
23
− 6y
22
+ ··· + 64y −64
c
5
, c
6
, c
7
c
8
, c
10
, c
12
y
23
− 24y
22
+ ··· − 9y −1
c
11
y
23
+ 10y
22
+ ··· − 6144y −4096
5
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.758227 + 0.807207I
a = −1.18253 + 1.19256I
b = −0.21429 − 1.43213I
c = 0.632217 + 0.500472I
d = −0.591761 + 0.042510I
−5.90461 + 1.36538I −0.279938 − 0.826772I
u = −0.758227 − 0.807207I
a = −1.18253 − 1.19256I
b = −0.21429 + 1.43213I
c = 0.632217 − 0.500472I
d = −0.591761 − 0.042510I
−5.90461 − 1.36538I −0.279938 + 0.826772I
u = 0.830705 + 0.204801I
a = −0.258089 + 0.246069I
b = 0.666656 − 1.123170I
c = 0.629069 − 0.215069I
d = −0.057606 + 0.411947I
0.25505 + 3.01929I 7.24264 − 9.08374I
u = 0.830705 − 0.204801I
a = −0.258089 − 0.246069I
b = 0.666656 + 1.123170I
c = 0.629069 + 0.215069I
d = −0.057606 − 0.411947I
0.25505 − 3.01929I 7.24264 + 9.08374I
u = 0.112218 + 1.144740I
a = −0.511153 − 0.296391I
b = 1.40432 + 0.22505I
c = 0.417690 + 0.009308I
d = 1.93741 − 0.14856I
8.23677 − 2.50119I 13.28602 + 3.12140I
u = 0.112218 − 1.144740I
a = −0.511153 + 0.296391I
b = 1.40432 − 0.22505I
c = 0.417690 − 0.009308I
d = 1.93741 + 0.14856I
8.23677 + 2.50119I 13.28602 − 3.12140I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.561270 + 1.026650I
a = −0.390779 − 1.343850I
b = −0.15516 + 1.60423I
c = 0.423044 + 0.049165I
d = 1.70161 − 0.72226I
5.56899 − 4.43236I 12.33564 + 2.61344I
u = 0.561270 − 1.026650I
a = −0.390779 + 1.343850I
b = −0.15516 − 1.60423I
c = 0.423044 − 0.049165I
d = 1.70161 + 0.72226I
5.56899 + 4.43236I 12.33564 − 2.61344I
u = −0.972761 + 0.735330I
a = −1.43485 + 0.61545I
b = 1.26169 − 1.93679I
c = 0.564139 + 0.426386I
d = −0.710629 − 0.218537I
−5.23569 − 7.16228I 1.72036 + 6.58026I
u = −0.972761 − 0.735330I
a = −1.43485 − 0.61545I
b = 1.26169 + 1.93679I
c = 0.564139 − 0.426386I
d = −0.710629 + 0.218537I
−5.23569 + 7.16228I 1.72036 − 6.58026I
u = −0.701924 + 1.071670I
a = −0.21749 + 1.49119I
b = −1.06220 − 1.62478I
c = 0.415821 − 0.060496I
d = 1.71873 + 0.92854I
2.90411 + 9.45510I 9.09507 − 6.28090I
u = −0.701924 − 1.071670I
a = −0.21749 − 1.49119I
b = −1.06220 + 1.62478I
c = 0.415821 + 0.060496I
d = 1.71873 − 0.92854I
2.90411 − 9.45510I 9.09507 + 6.28090I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.324650 + 0.201985I
a = 1.23370 + 0.73880I
b = −1.082960 + 0.067487I
c = −2.00719 − 0.40410I
d = 2.17756 − 0.28510I
13.75320 − 2.16453I 16.4022 + 0.8027I
u = −1.324650 − 0.201985I
a = 1.23370 − 0.73880I
b = −1.082960 − 0.067487I
c = −2.00719 + 0.40410I
d = 2.17756 + 0.28510I
13.75320 + 2.16453I 16.4022 − 0.8027I
u = 1.140080 + 0.732610I
a = −1.76969 − 0.67662I
b = 1.39872 + 1.26269I
c = −1.39259 + 1.27651I
d = 1.80476 + 0.99453I
7.42067 + 10.78250I 12.9034 − 6.4003I
u = 1.140080 − 0.732610I
a = −1.76969 + 0.67662I
b = 1.39872 − 1.26269I
c = −1.39259 − 1.27651I
d = 1.80476 − 0.99453I
7.42067 − 10.78250I 12.9034 + 6.4003I
u = 1.315590 + 0.366431I
a = 0.530331 − 1.007860I
b = −0.511464 − 0.076354I
c = −1.85311 + 0.68498I
d = 2.14413 + 0.51761I
12.6616 + 7.9478I 14.6243 − 6.1519I
u = 1.315590 − 0.366431I
a = 0.530331 + 1.007860I
b = −0.511464 + 0.076354I
c = −1.85311 − 0.68498I
d = 2.14413 − 0.51761I
12.6616 − 7.9478I 14.6243 + 6.1519I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618010
a = 0.764150
b = −1.38261
c = 0.651263
d = 0.286737
0.841351 11.7320
u = −1.130850 + 0.817356I
a = −2.01990 + 0.20234I
b = 1.93555 − 1.52159I
c = −1.24564 − 1.29190I
d = 1.76222 − 1.11255I
4.3220 − 16.2949I 9.65915 + 9.61437I
u = −1.130850 − 0.817356I
a = −2.01990 − 0.20234I
b = 1.93555 + 1.52159I
c = −1.24564 + 1.29190I
d = 1.76222 + 1.11255I
4.3220 + 16.2949I 9.65915 − 9.61437I
u = 0.237558 + 0.464767I
a = 1.13837 − 1.02403I
b = 0.050451 − 0.233290I
c = 1.090930 − 0.283409I
d = −0.0297983 − 0.0630426I
−1.63449 − 0.53093I −3.85466 + 0.92872I
u = 0.237558 − 0.464767I
a = 1.13837 + 1.02403I
b = 0.050451 + 0.233290I
c = 1.090930 + 0.283409I
d = −0.0297983 + 0.0630426I
−1.63449 + 0.53093I −3.85466 − 0.92872I
9
II. I
u
2
= hu
2
c − u
3
+ · · · + c − 1, −2u
3
c + u
3
+ · · · + c
2
+ u, u
2
+ b + 1, u
3
−
2u
2
+ a + u − 1, u
4
− 2u
3
+ 2u
2
− u + 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
8
=
c
−u
2
c + u
3
+ cu − u
2
− c + 1
a
10
=
1
−u
2
a
11
=
c
−u
3
− cu + u
2
+ c − 1
a
1
=
−u
3
+ 2u
2
− u + 1
−u
2
− 1
a
3
=
−u
u
a
2
=
u
2
− u
−u
3
a
5
=
u
3
− u
2
+ u
−u
3
a
7
=
−u
3
− cu + u
2
+ 2c − 1
u
3
c − 2u
2
c + u
3
+ cu − u
2
− c
a
6
=
u
3
c − u
2
c − 2u
3
+ 3u
2
+ 2c − u
−u
2
c + u
3
− 2u
2
− 1
a
12
=
u
3
c − 2u
2
c + c − 1
u
3
c − u
3
+ 2u
2
+ c
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
3
+ 4u
2
− 8u + 10
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
4
+ 3u
3
+ 5u
2
+ 3u + 1)
2
c
2
, c
4
(u
4
− u
3
− u
2
+ u + 1)
2
c
3
, c
9
(u
4
+ 2u
3
+ 2u
2
+ u + 1)
2
c
5
, c
6
, c
7
c
8
, c
10
, c
12
u
8
+ u
7
− 2u
6
− 2u
5
− u
3
+ u
2
+ 2u + 1
c
11
(u
4
+ 2u
2
+ 3u + 1)
2
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
4
+ y
3
+ 9y
2
+ y + 1)
2
c
2
, c
4
(y
4
− 3y
3
+ 5y
2
− 3y + 1)
2
c
3
, c
9
(y
4
+ 2y
2
+ 3y + 1)
2
c
5
, c
6
, c
7
c
8
, c
10
, c
12
y
8
− 5y
7
+ 8y
6
− 10y
4
+ 3y
3
+ 5y
2
− 2y + 1
c
11
(y
4
+ 4y
3
+ 6y
2
− 5y + 1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.070696 + 0.758745I
a = −0.192440 − 0.547877I
b = −0.429304 + 0.107280I
c = 0.451634 − 0.006403I
d = 1.47217 + 0.07618I
2.21227 + 1.41376I 7.79581 − 4.79737I
u = −0.070696 + 0.758745I
a = −0.192440 − 0.547877I
b = −0.429304 + 0.107280I
c = 1.36255 + 0.99488I
d = 0.149577 + 0.364417I
2.21227 + 1.41376I 7.79581 − 4.79737I
u = −0.070696 − 0.758745I
a = −0.192440 + 0.547877I
b = −0.429304 − 0.107280I
c = 0.451634 + 0.006403I
d = 1.47217 − 0.07618I
2.21227 − 1.41376I 7.79581 + 4.79737I
u = −0.070696 − 0.758745I
a = −0.192440 + 0.547877I
b = −0.429304 − 0.107280I
c = 1.36255 − 0.99488I
d = 0.149577 − 0.364417I
2.21227 − 1.41376I 7.79581 + 4.79737I
u = 1.070700 + 0.758745I
a = 1.69244 + 0.31815I
b = −1.57070 − 1.62477I
c = 0.529061 − 0.418553I
d = −0.819448 + 0.298973I
−0.56734 + 11.56320I 6.20419 − 8.26147I
u = 1.070700 + 0.758745I
a = 1.69244 + 0.31815I
b = −1.57070 − 1.62477I
c = −1.34325 + 1.40703I
d = 1.69770 + 1.00765I
−0.56734 + 11.56320I 6.20419 − 8.26147I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.070700 − 0.758745I
a = 1.69244 − 0.31815I
b = −1.57070 + 1.62477I
c = 0.529061 + 0.418553I
d = −0.819448 − 0.298973I
−0.56734 − 11.56320I 6.20419 + 8.26147I
u = 1.070700 − 0.758745I
a = 1.69244 − 0.31815I
b = −1.57070 + 1.62477I
c = −1.34325 − 1.40703I
d = 1.69770 − 1.00765I
−0.56734 − 11.56320I 6.20419 + 8.26147I
14
III. I
u
3
= h−u
7
+ u
5
− 2u
3
+ d + u, u
5
+ c + u, −u
7
− u
6
+ · · · + b +
1, −u
7
a + u
7
+ · · · + 2a − 2, u
8
+ u
7
+ · · · − 2u − 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
8
=
−u
5
− u
u
7
− u
5
+ 2u
3
− u
a
10
=
1
−u
2
a
11
=
−u
5
− u
u
5
− u
3
+ u
a
1
=
a
u
7
+ u
6
− u
4
+ au − 1
a
3
=
−u
u
a
2
=
−u
7
− u
6
+ u
5
− u
3
a + 2u
4
− u
2
a − u
2
+ a − u
2u
7
+ 2u
6
− u
5
+ u
3
a − 3u
4
+ u
2
a + au + u
2
+ u − 1
a
5
=
−u
7
− u
6
+ u
4
− au − a + 1
2u
7
+ 2u
6
− u
5
+ u
3
a − 3u
4
+ u
2
a + au + u
2
+ u − 1
a
7
=
−u
3
u
3
− u
a
6
=
u
7
− u
5
a + u
6
− u
4
a + u
5
+ u
2
a − u
3
+ au − 2u
2
+ a − u − 1
−2u
7
+ u
5
a − 2u
6
+ u
4
a − u
3
a + 2u
4
− 2u
2
a + u
3
− au + u
2
+ 1
a
12
=
−u
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
7
+ 8u
5
+ 4u
4
− 8u
3
− 4u
2
+ 4u + 14
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
+ 9u
15
+ ··· − 8u
2
+ 1
c
2
, c
4
, c
5
c
6
, c
12
u
16
− u
15
+ ··· + 2u − 1
c
3
, c
9
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
2
c
7
, c
8
, c
10
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
2
c
11
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 5y
15
+ ··· − 16y + 1
c
2
, c
4
, c
5
c
6
, c
12
y
16
− 9y
15
+ ··· − 8y
2
+ 1
c
3
, c
9
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
2
c
7
, c
8
, c
10
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
2
c
11
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.570868 + 0.730671I
a = 0.583515 − 0.832445I
b = −0.234797 + 1.067950I
c = 0.451832 − 0.055667I
d = 1.32053 + 0.63395I
1.04066 + 1.13123I 7.41522 − 0.51079I
u = −0.570868 + 0.730671I
a = −1.063490 + 0.509555I
b = −0.275134 − 0.901574I
c = 0.451832 − 0.055667I
d = 1.32053 + 0.63395I
1.04066 + 1.13123I 7.41522 − 0.51079I
u = −0.570868 − 0.730671I
a = 0.583515 + 0.832445I
b = −0.234797 − 1.067950I
c = 0.451832 + 0.055667I
d = 1.32053 − 0.63395I
1.04066 − 1.13123I 7.41522 + 0.51079I
u = −0.570868 − 0.730671I
a = −1.063490 − 0.509555I
b = −0.275134 + 0.901574I
c = 0.451832 + 0.055667I
d = 1.32053 − 0.63395I
1.04066 − 1.13123I 7.41522 + 0.51079I
u = 0.855237 + 0.665892I
a = 1.003290 + 0.865096I
b = −0.74376 − 2.19413I
c = 0.620212 − 0.418390I
d = −0.547085 + 0.161596I
−2.15941 + 2.57849I 4.27708 − 3.56796I
u = 0.855237 + 0.665892I
a = 1.78504 + 1.17568I
b = −0.28199 − 1.40795I
c = 0.620212 − 0.418390I
d = −0.547085 + 0.161596I
−2.15941 + 2.57849I 4.27708 − 3.56796I
18
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.855237 − 0.665892I
a = 1.003290 − 0.865096I
b = −0.74376 + 2.19413I
c = 0.620212 + 0.418390I
d = −0.547085 − 0.161596I
−2.15941 − 2.57849I 4.27708 + 3.56796I
u = 0.855237 − 0.665892I
a = 1.78504 − 1.17568I
b = −0.28199 + 1.40795I
c = 0.620212 + 0.418390I
d = −0.547085 − 0.161596I
−2.15941 − 2.57849I 4.27708 + 3.56796I
u = 1.09818
a = −0.558131 + 0.380867I
b = 0.612928 + 0.418261I
c = −2.69540
d = 1.87965
6.50273 13.8640
u = 1.09818
a = −0.558131 − 0.380867I
b = 0.612928 − 0.418261I
c = −2.69540
d = 1.87965
6.50273 13.8640
u = −1.031810 + 0.655470I
a = −1.266190 + 0.281077I
b = 1.10166 − 1.54556I
c = −1.56596 − 1.49295I
d = 1.67925 − 0.85124I
2.37968 − 6.44354I 9.42845 + 5.29417I
u = −1.031810 + 0.655470I
a = 1.43867 − 0.58398I
b = −1.12222 + 1.11997I
c = −1.56596 − 1.49295I
d = 1.67925 − 0.85124I
2.37968 − 6.44354I 9.42845 + 5.29417I
19
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.031810 − 0.655470I
a = −1.266190 − 0.281077I
b = 1.10166 + 1.54556I
c = −1.56596 + 1.49295I
d = 1.67925 + 0.85124I
2.37968 + 6.44354I 9.42845 − 5.29417I
u = −1.031810 − 0.655470I
a = 1.43867 + 0.58398I
b = −1.12222 − 1.11997I
c = −1.56596 + 1.49295I
d = 1.67925 + 0.85124I
2.37968 + 6.44354I 9.42845 − 5.29417I
u = −0.603304
a = 0.851522
b = −1.62708
c = 0.683228
d = 0.214962
0.845036 11.8940
u = −0.603304
a = −2.69694
b = 0.513726
c = 0.683228
d = 0.214962
0.845036 11.8940
20
IV. I
u
4
= h−u
6
− u
5
+ · · · + d − 1, 2u
7
a − u
7
+ · · · − 3a + 1, −u
7
− u
6
+ · · · +
b + 1, −u
7
a + u
7
+ · · · + 2a − 2, u
8
+ u
7
+ · · · − 2u − 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
8
=
−2u
7
a + u
7
+ ··· + 3a − 1
u
6
+ u
5
− u
4
+ u
2
a − 2u
3
+ au + u + 1
a
10
=
1
−u
2
a
11
=
−2u
7
a + u
7
+ ··· + 3a − 1
−u
4
a − u
3
a + u
4
+ u
3
+ au − u
2
− 2u − 1
a
1
=
a
u
7
+ u
6
− u
4
+ au − 1
a
3
=
−u
u
a
2
=
−u
7
− u
6
+ u
5
− u
3
a + 2u
4
− u
2
a − u
2
+ a − u
2u
7
+ 2u
6
− u
5
+ u
3
a − 3u
4
+ u
2
a + au + u
2
+ u − 1
a
5
=
−u
7
− u
6
+ u
4
− au − a + 1
2u
7
+ 2u
6
− u
5
+ u
3
a − 3u
4
+ u
2
a + au + u
2
+ u − 1
a
7
=
−2u
7
a + u
7
+ ··· + 3a − 2
u
6
a + u
5
a − u
3
a + u
2
a + au + u
2
+ u + 1
a
6
=
−2u
7
a + u
7
+ ··· + 3a − 2
2u
6
a + u
5
a − u
4
a − u
3
a + u
4
+ 2u
2
a + u
3
+ au + u
2
+ 1
a
12
=
−2u
7
a + u
7
+ ··· + 3a − 1
−u
6
a − u
5
a − u
4
a − au − 2u
2
− a − 2u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
7
+ 8u
5
+ 4u
4
− 8u
3
− 4u
2
+ 4u + 14
21
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
16
+ 9u
15
+ ··· − 8u
2
+ 1
c
2
, c
4
, c
7
c
8
, c
10
u
16
− u
15
+ ··· + 2u − 1
c
3
, c
9
(u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1)
2
c
5
, c
6
, c
12
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
2
c
11
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
2
22
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
− 5y
15
+ ··· − 16y + 1
c
2
, c
4
, c
7
c
8
, c
10
y
16
− 9y
15
+ ··· − 8y
2
+ 1
c
3
, c
9
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
2
c
5
, c
6
, c
12
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
2
c
11
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
2
23
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.570868 + 0.730671I
a = 0.583515 − 0.832445I
b = −0.234797 + 1.067950I
c = 0.755133 + 0.516255I
d = −0.371151 + 0.120354I
1.04066 + 1.13123I 7.41522 − 0.51079I
u = −0.570868 + 0.730671I
a = −1.063490 + 0.509555I
b = −0.275134 − 0.901574I
c = −0.64422 − 2.71770I
d = 1.050620 − 0.754306I
1.04066 + 1.13123I 7.41522 − 0.51079I
u = −0.570868 − 0.730671I
a = 0.583515 + 0.832445I
b = −0.234797 − 1.067950I
c = 0.755133 − 0.516255I
d = −0.371151 − 0.120354I
1.04066 − 1.13123I 7.41522 + 0.51079I
u = −0.570868 − 0.730671I
a = −1.063490 − 0.509555I
b = −0.275134 + 0.901574I
c = −0.64422 + 2.71770I
d = 1.050620 + 0.754306I
1.04066 − 1.13123I 7.41522 + 0.51079I
u = 0.855237 + 0.665892I
a = 1.003290 + 0.865096I
b = −0.74376 − 2.19413I
c = 0.450628 + 0.089664I
d = 1.10695 − 0.96382I
−2.15941 + 2.57849I 4.27708 − 3.56796I
u = 0.855237 + 0.665892I
a = 1.78504 + 1.17568I
b = −0.28199 − 1.40795I
c = −1.48818 + 1.97913I
d = 1.44013 + 0.80222I
−2.15941 + 2.57849I 4.27708 − 3.56796I
24
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.855237 − 0.665892I
a = 1.003290 − 0.865096I
b = −0.74376 + 2.19413I
c = 0.450628 − 0.089664I
d = 1.10695 + 0.96382I
−2.15941 − 2.57849I 4.27708 + 3.56796I
u = 0.855237 − 0.665892I
a = 1.78504 − 1.17568I
b = −0.28199 + 1.40795I
c = −1.48818 − 1.97913I
d = 1.44013 − 0.80222I
−2.15941 − 2.57849I 4.27708 + 3.56796I
u = 1.09818
a = −0.558131 + 0.380867I
b = 0.612928 + 0.418261I
c = 0.518512 − 0.196916I
d = 0.060177 + 0.877586I
6.50273 13.8640
u = 1.09818
a = −0.558131 − 0.380867I
b = 0.612928 − 0.418261I
c = 0.518512 + 0.196916I
d = 0.060177 − 0.877586I
6.50273 13.8640
u = −1.031810 + 0.655470I
a = −1.266190 + 0.281077I
b = 1.10166 − 1.54556I
c = 0.442044 − 0.109789I
d = 0.99859 + 1.19686I
2.37968 − 6.44354I 9.42845 + 5.29417I
u = −1.031810 + 0.655470I
a = 1.43867 − 0.58398I
b = −1.12222 + 1.11997I
c = 0.555142 + 0.391147I
d = −0.677840 − 0.345614I
2.37968 − 6.44354I 9.42845 + 5.29417I
25
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −1.031810 − 0.655470I
a = −1.266190 − 0.281077I
b = 1.10166 + 1.54556I
c = 0.442044 + 0.109789I
d = 0.99859 − 1.19686I
2.37968 + 6.44354I 9.42845 − 5.29417I
u = −1.031810 − 0.655470I
a = 1.43867 + 0.58398I
b = −1.12222 − 1.11997I
c = 0.555142 − 0.391147I
d = −0.677840 + 0.345614I
2.37968 + 6.44354I 9.42845 − 5.29417I
u = −0.603304
a = 0.851522
b = −1.62708
c = 0.593814
d = 0.467894
0.845036 11.8940
u = −0.603304
a = −2.69694
b = 0.513726
c = −6.77192
d = 1.31714
0.845036 11.8940
26
V. I
u
5
= h−u
7
+ u
5
− 2u
3
+ d + u, u
5
+ c + u, −u
7
− u
6
+ · · · + b + 3, u
7
−
2u
5
+ · · · + a − 2, u
8
+ u
7
+ · · · − 2u − 1i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
8
=
−u
5
− u
u
7
− u
5
+ 2u
3
− u
a
10
=
1
−u
2
a
11
=
−u
5
− u
u
5
− u
3
+ u
a
1
=
−u
7
+ 2u
5
+ 2u
4
− 2u
3
− 2u
2
+ 2u + 2
u
7
+ u
6
− 2u
5
− 3u
4
+ 2u
3
+ 4u
2
− 2u − 3
a
3
=
−u
u
a
2
=
u
4
− u
2
+ 1
u
6
− 2u
4
+ 3u
2
− 2
a
5
=
−u
6
+ u
4
− 2u
2
+ 1
u
6
− 2u
4
+ 3u
2
− 2
a
7
=
−u
3
u
3
− u
a
6
=
2u
6
− 2u
4
+ 3u
2
− 1
−2u
6
+ 3u
4
− 4u
2
+ 2
a
12
=
−u
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
7
+ 8u
5
+ 4u
4
− 8u
3
− 4u
2
+ 4u + 14
27
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
8
+ 7u
7
+ 19u
6
+ 22u
5
+ 3u
4
− 14u
3
− 6u
2
+ 4u + 1
c
2
, c
4
, c
5
c
6
, c
7
, c
8
c
10
, c
12
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
c
3
, c
9
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
11
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
28
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
8
− 11y
7
+ 59y
6
− 186y
5
+ 343y
4
− 370y
3
+ 154y
2
− 28y + 1
c
2
, c
4
, c
5
c
6
, c
7
, c
8
c
10
, c
12
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
c
3
, c
9
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
11
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
29
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −0.570868 + 0.730671I
a = −0.66176 + 1.78423I
b = 0.32371 − 3.32741I
c = 0.451832 − 0.055667I
d = 1.32053 + 0.63395I
1.04066 + 1.13123I 7.41522 − 0.51079I
u = −0.570868 − 0.730671I
a = −0.66176 − 1.78423I
b = 0.32371 + 3.32741I
c = 0.451832 + 0.055667I
d = 1.32053 − 0.63395I
1.04066 − 1.13123I 7.41522 + 0.51079I
u = 0.855237 + 0.665892I
a = −1.077860 − 0.708987I
b = 0.771196 + 1.136710I
c = 0.620212 − 0.418390I
d = −0.547085 + 0.161596I
−2.15941 + 2.57849I 4.27708 − 3.56796I
u = 0.855237 − 0.665892I
a = −1.077860 + 0.708987I
b = 0.771196 − 1.136710I
c = 0.620212 + 0.418390I
d = −0.547085 − 0.161596I
−2.15941 − 2.57849I 4.27708 + 3.56796I
u = 1.09818
a = 3.31262
b = −1.60102
c = −2.69540
d = 1.87965
6.50273 13.8640
u = −1.031810 + 0.655470I
a = −2.23610 + 1.61384I
b = 0.70316 − 1.76266I
c = −1.56596 − 1.49295I
d = 1.67925 − 0.85124I
2.37968 − 6.44354I 9.42845 + 5.29417I
30
Solutions to I
u
5
√
−1(vol +
√
−1CS) Cusp shape
u = −1.031810 − 0.655470I
a = −2.23610 − 1.61384I
b = 0.70316 + 1.76266I
c = −1.56596 + 1.49295I
d = 1.67925 + 0.85124I
2.37968 + 6.44354I 9.42845 − 5.29417I
u = −0.603304
a = 0.638815
b = −0.995124
c = 0.683228
d = 0.214962
0.845036 11.8940
31
VI. I
u
6
= hu
5
c − u
4
c + · · · + d − 2c, −3u
5
c + u
5
+ · · · + 6c − 4, −u
4
+ u
2
+
b − u − 1, −u
5
+ u
4
+ · · · + 2a + 4, u
6
− u
5
+ · · · − 2u + 2i
(i) Arc colorings
a
4
=
0
u
a
9
=
1
0
a
8
=
c
−u
5
c + u
4
c + 2u
3
c − 3u
2
c − u
3
− cu + 2c + u
a
10
=
1
−u
2
a
11
=
c
u
5
c − u
4
c − 2u
3
c + 2u
2
c + u
3
+ cu − 2c − u
a
1
=
1
2
u
5
−
1
2
u
4
+ ··· +
1
2
u − 2
u
4
− u
2
+ u + 1
a
3
=
−u
u
a
2
=
1
2
u
5
−
1
2
u
4
+ ··· +
1
2
u
2
−
1
2
u
u
4
− u
3
+ 2u − 1
a
5
=
−
1
2
u
5
−
1
2
u
4
+ ··· −
3
2
u + 1
u
4
− u
3
+ 2u − 1
a
7
=
u
5
c − u
4
c − 2u
3
c + 2u
2
c + u
3
+ cu − c − u
2u
4
c − u
5
− 3u
2
c + cu + 2c + u
a
6
=
u
5
c +
1
2
u
5
+ ··· −
1
2
u − 2
−u
5
c + 2u
4
c − u
5
+ 2u
3
c + u
4
− 3u
2
c − u
2
+ 2c + 2u + 1
a
12
=
u
5
c + u
4
c − u
5
− 2u
3
c − u
2
c + u
3
+ 2cu + c
−u
5
c + 2u
5
+ 3u
3
c − 2u
4
− 2u
2
c − 2u
3
− 3cu + 3u
2
+ 2c − u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2u
5
− 4u
4
+ 8u
3
− 8u + 16
32
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
6
+ 2u
5
+ 3u
4
+ u
3
+ u
2
− u + 1)
2
c
2
, c
4
(u
6
− u
4
+ u
3
+ u
2
− u + 1)
2
c
3
, c
9
(u
6
+ u
5
− u
4
− 3u
3
− u
2
+ 2u + 2)
2
c
5
, c
6
, c
7
c
8
, c
10
, c
12
u
12
− 5u
10
+ 2u
9
+ 9u
8
− 7u
7
− 4u
6
+ 7u
5
− 4u
4
+ 2u
3
+ u
2
− 4u + 4
c
11
(u
6
− 3u
5
+ 5u
4
− 7u
3
+ 9u
2
− 8u + 4)
2
33
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
6
+ 2y
5
+ 7y
4
+ 11y
3
+ 9y
2
+ y + 1)
2
c
2
, c
4
(y
6
− 2y
5
+ 3y
4
− y
3
+ y
2
+ y + 1)
2
c
3
, c
9
(y
6
− 3y
5
+ 5y
4
− 7y
3
+ 9y
2
− 8y + 4)
2
c
5
, c
6
, c
7
c
8
, c
10
, c
12
y
12
− 10y
11
+ ··· − 8y + 16
c
11
(y
6
+ y
5
+ y
4
+ y
3
+ 9y
2
+ 8y + 16)
2
34
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = 0.954425 + 0.469441I
a = −1.127640 − 0.295030I
b = 0.937752 + 0.810947I
c = 0.469359 + 0.113275I
d = 0.790687 − 0.984701I
4.85214 + 1.71504I 13.36090 − 1.32670I
u = 0.954425 + 0.469441I
a = −1.127640 − 0.295030I
b = 0.937752 + 0.810947I
c = −2.14493 + 1.61685I
d = 1.63274 + 0.58144I
4.85214 + 1.71504I 13.36090 − 1.32670I
u = 0.954425 − 0.469441I
a = −1.127640 + 0.295030I
b = 0.937752 − 0.810947I
c = 0.469359 − 0.113275I
d = 0.790687 + 0.984701I
4.85214 − 1.71504I 13.36090 + 1.32670I
u = 0.954425 − 0.469441I
a = −1.127640 + 0.295030I
b = 0.937752 − 0.810947I
c = −2.14493 − 1.61685I
d = 1.63274 − 0.58144I
4.85214 − 1.71504I 13.36090 + 1.32670I
u = −1.130290 + 0.224113I
a = −0.005338 − 0.454789I
b = −0.107958 − 0.512846I
c = 0.532539 + 0.254347I
d = −0.253448 − 0.772641I
6.01369 − 4.89103I 12.12173 + 6.59162I
u = −1.130290 + 0.224113I
a = −0.005338 − 0.454789I
b = −0.107958 − 0.512846I
c = −2.40888 − 0.66651I
d = 1.90853 − 0.29567I
6.01369 − 4.89103I 12.12173 + 6.59162I
35
Solutions to I
u
6
√
−1(vol +
√
−1CS) Cusp shape
u = −1.130290 − 0.224113I
a = −0.005338 + 0.454789I
b = −0.107958 + 0.512846I
c = 0.532539 − 0.254347I
d = −0.253448 + 0.772641I
6.01369 + 4.89103I 12.12173 − 6.59162I
u = −1.130290 − 0.224113I
a = −0.005338 + 0.454789I
b = −0.107958 + 0.512846I
c = −2.40888 + 0.66651I
d = 1.90853 + 0.29567I
6.01369 + 4.89103I 12.12173 − 6.59162I
u = 0.675862 + 0.935235I
a = 0.632981 + 1.174050I
b = 0.67021 − 1.38548I
c = 0.623081 − 0.582789I
d = −0.620351 − 0.230547I
−1.81870 − 5.32947I 4.51738 + 4.54389I
u = 0.675862 + 0.935235I
a = 0.632981 + 1.174050I
b = 0.67021 − 1.38548I
c = 0.428825 + 0.061762I
d = 1.54184 − 0.84534I
−1.81870 − 5.32947I 4.51738 + 4.54389I
u = 0.675862 − 0.935235I
a = 0.632981 − 1.174050I
b = 0.67021 + 1.38548I
c = 0.623081 + 0.582789I
d = −0.620351 + 0.230547I
−1.81870 + 5.32947I 4.51738 − 4.54389I
u = 0.675862 − 0.935235I
a = 0.632981 − 1.174050I
b = 0.67021 + 1.38548I
c = 0.428825 − 0.061762I
d = 1.54184 + 0.84534I
−1.81870 + 5.32947I 4.51738 − 4.54389I
36
VII. I
v
1
= hc, d − 1, b, a + 1, v + 1i
(i) Arc colorings
a
4
=
−1
0
a
9
=
1
0
a
8
=
0
1
a
10
=
1
0
a
11
=
1
−1
a
1
=
−1
0
a
3
=
−1
0
a
2
=
−1
0
a
5
=
−1
0
a
7
=
−1
1
a
6
=
−2
1
a
12
=
1
−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
2
, c
3
c
4
, c
9
, c
11
u
c
5
, c
6
, c
10
u − 1
c
7
, c
8
, c
12
u + 1
38
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
9
, c
11
y
c
5
, c
6
, c
7
c
8
, c
10
, c
12
y −1
39
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = −1.00000
b = 0
c = 0
d = 1.00000
3.28987 12.0000
40
VIII. I
v
2
= ha, d, c − 1, b + 1, v − 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
1
0
a
8
=
1
0
a
10
=
1
0
a
11
=
1
0
a
1
=
0
−1
a
3
=
1
0
a
2
=
1
−1
a
5
=
0
1
a
7
=
1
0
a
6
=
1
1
a
12
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
41
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
12
u − 1
c
3
, c
7
, c
8
c
9
, c
10
, c
11
u
c
4
, c
5
, c
6
u + 1
42
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
6
, c
12
y −1
c
3
, c
7
, c
8
c
9
, c
10
, c
11
y
43
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
2
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
c = 1.00000
d = 0
0 0
44
IX. I
v
3
= ha, d − 1, c + a − 1, b + 1, v − 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
1
0
a
8
=
1
1
a
10
=
1
0
a
11
=
0
−1
a
1
=
0
−1
a
3
=
1
0
a
2
=
1
−1
a
5
=
0
1
a
7
=
0
1
a
6
=
0
1
a
12
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
45
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
10
u − 1
c
3
, c
5
, c
6
c
9
, c
11
, c
12
u
c
4
, c
7
, c
8
u + 1
46
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
7
, c
8
, c
10
y −1
c
3
, c
5
, c
6
c
9
, c
11
, c
12
y
47
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
3
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = −1.00000
c = 1.00000
d = 1.00000
0 0
48
X. I
v
4
= hc, d − 1, av + c + v − 1, bv + 1i
(i) Arc colorings
a
4
=
v
0
a
9
=
1
0
a
8
=
0
1
a
10
=
1
0
a
11
=
1
−1
a
1
=
a
−a − 1
a
3
=
v
0
a
2
=
a + v
−a − 1
a
5
=
−a
a + 1
a
7
=
−1
1
a
6
=
a − 1
−a
a
12
=
1
−1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −a
2
− v
2
− 2a + 7
(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.
49
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
4
√
−1(vol +
√
−1CS) Cusp shape
v = ···
a = ···
b = ···
c = ···
d = ···
1.64493 10.37261 + 0.05860I
50
XI. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u(u − 1)
2
(u
4
+ 3u
3
+ 5u
2
+ 3u + 1)
2
· (u
6
+ 2u
5
+ 3u
4
+ u
3
+ u
2
− u + 1)
2
· (u
8
+ 7u
7
+ 19u
6
+ 22u
5
+ 3u
4
− 14u
3
− 6u
2
+ 4u + 1)
· ((u
16
+ 9u
15
+ ··· − 8u
2
+ 1)
2
)(u
23
+ 10u
22
+ ··· + 88u + 16)
c
2
u(u − 1)
2
(u
4
− u
3
− u
2
+ u + 1)
2
(u
6
− u
4
+ u
3
+ u
2
− u + 1)
2
· (u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
16
− u
15
+ ··· + 2u − 1)
2
· (u
23
− 2u
22
+ ··· + 8u − 4)
c
3
, c
9
u
3
(u
4
+ 2u
3
+ 2u
2
+ u + 1)
2
(u
6
+ u
5
− u
4
− 3u
3
− u
2
+ 2u + 2)
2
· ((u
8
− u
7
+ ··· + 2u − 1)
5
)(u
23
− 2u
22
+ ··· + 4u
2
− 8)
c
4
u(u + 1)
2
(u
4
− u
3
− u
2
+ u + 1)
2
(u
6
− u
4
+ u
3
+ u
2
− u + 1)
2
· (u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)(u
16
− u
15
+ ··· + 2u − 1)
2
· (u
23
− 2u
22
+ ··· + 8u − 4)
c
5
, c
6
, c
12
u(u − 1)(u + 1)(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
3
· (u
8
+ u
7
− 2u
6
− 2u
5
− u
3
+ u
2
+ 2u + 1)
· (u
12
− 5u
10
+ 2u
9
+ 9u
8
− 7u
7
− 4u
6
+ 7u
5
− 4u
4
+ 2u
3
+ u
2
− 4u + 4)
· (u
16
− u
15
+ ··· + 2u − 1)(u
23
+ 2u
22
+ ··· − u − 1)
c
7
, c
8
u(u + 1)
2
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
3
· (u
8
+ u
7
− 2u
6
− 2u
5
− u
3
+ u
2
+ 2u + 1)
· (u
12
− 5u
10
+ 2u
9
+ 9u
8
− 7u
7
− 4u
6
+ 7u
5
− 4u
4
+ 2u
3
+ u
2
− 4u + 4)
· (u
16
− u
15
+ ··· + 2u − 1)(u
23
+ 2u
22
+ ··· − u − 1)
c
10
u(u − 1)
2
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
3
· (u
8
+ u
7
− 2u
6
− 2u
5
− u
3
+ u
2
+ 2u + 1)
· (u
12
− 5u
10
+ 2u
9
+ 9u
8
− 7u
7
− 4u
6
+ 7u
5
− 4u
4
+ 2u
3
+ u
2
− 4u + 4)
· (u
16
− u
15
+ ··· + 2u − 1)(u
23
+ 2u
22
+ ··· − u − 1)
c
11
u
3
(u
4
+ 2u
2
+ 3u + 1)
2
(u
6
− 3u
5
+ 5u
4
− 7u
3
+ 9u
2
− 8u + 4)
2
· (u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
5
· (u
23
− 6u
22
+ ··· + 64u − 64)
51
XII. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y(y − 1)
2
(y
4
+ y
3
+ 9y
2
+ y + 1)
2
· (y
6
+ 2y
5
+ 7y
4
+ 11y
3
+ 9y
2
+ y + 1)
2
· (y
8
− 11y
7
+ 59y
6
− 186y
5
+ 343y
4
− 370y
3
+ 154y
2
− 28y + 1)
· ((y
16
− 5y
15
+ ··· − 16y + 1)
2
)(y
23
+ 6y
22
+ ··· + 1824y −256)
c
2
, c
4
y(y − 1)
2
(y
4
− 3y
3
+ 5y
2
− 3y + 1)
2
· (y
6
− 2y
5
+ 3y
4
− y
3
+ y
2
+ y + 1)
2
· (y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· ((y
16
− 9y
15
+ ··· − 8y
2
+ 1)
2
)(y
23
− 10y
22
+ ··· + 88y −16)
c
3
, c
9
y
3
(y
4
+ 2y
2
+ 3y + 1)
2
(y
6
− 3y
5
+ 5y
4
− 7y
3
+ 9y
2
− 8y + 4)
2
· (y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
5
· (y
23
− 6y
22
+ ··· + 64y −64)
c
5
, c
6
, c
7
c
8
, c
10
, c
12
y(y − 1)
2
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
3
· (y
8
− 5y
7
+ 8y
6
− 10y
4
+ 3y
3
+ 5y
2
− 2y + 1)
· (y
12
− 10y
11
+ ··· − 8y + 16)(y
16
− 9y
15
+ ··· − 8y
2
+ 1)
· (y
23
− 24y
22
+ ··· − 9y −1)
c
11
y
3
(y
4
+ 4y
3
+ 6y
2
− 5y + 1)
2
(y
6
+ y
5
+ y
4
+ y
3
+ 9y
2
+ 8y + 16)
2
· (y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
5
· (y
23
+ 10y
22
+ ··· − 6144y −4096)
52
